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PhD Thesis

Thresholds of lasing modes localized on

MUPO and UPO in fully chaotic microcavity

lasers

In-Goo Lee (이 인 구 李 仁 求)

Department of

Emerging Materials Science

DGIST

2020

PhDEM

201631002

이 인 구 In-Goo Lee Thresholds of lasing modes localized

on MUPO and UPO in fully chaotic microcavity lasers

Department of Emerging Materials Science 2020 51p

Advisor Prof Chil-Min Kim Co-Advisor Prof Jung-Wan

Ryu

Abstract

We experimentally observe the lasing of a single-mode group in a rounded D-

shape InGaAsP semiconductor microcavity laser when the cavity is fully chaotic

Although there are numerous unstable periodic orbits and marginally unstable

periodic orbits the lasing mode group is localized on a pentagonal period-5 un-

stable periodic orbit and marginally unstable periodic orbits for each rounded

D-shape microcavity laser with parameters r = 085R φ = 5 and r = 05R

φ = 80 respectively near above the lowest threshold for CW current injection

By means of numerical analyses with ray and wave dynamics we have theoreti-

cally predicted the dominant lasing of the pentagonal mode group by considering

the linear gain condition at the lowest threshold condition We calculate the

threshold parameters depending on the effective pumping region for each peri-

odic orbit and have obtained that the resonance modes localized on the pen-

I

tagonal periodic orbits have the lowest threshold parameter for our experimental

condition which has a 3μm inward gap between the current contact region and

the side-wall of the microcavity Consequently we have confirmed that the path

length and unidirectional emission from the experimental the result well coincide

with our theoretical prediction

Keyword Microdisk laser Microcavity Semiconductor laser Marginally

unstable periodic orbit Scarred mode Threshold

II

Contents

Abstract I

Contents III

List of Figures V

I Introduction 1

II Lasing of scarred mode near above threshold in a rounded D-shape

microcavity laser 3

21 Ray dynamical analysis and unstable periodic orbits 5

22 Wave analysis for the threshold pumping strength 7

23 Fabrication and experimental results 11

24 Discussion 15

241 Quality factor of the scarred mode 15

242 Unidirectional emission from a scarred mode lasing 17

25 Conclusion 17

III Lasing modes localized on marginally unstable periodic orbits in a

rounded D-shape microcavity laser 20

31 Semiconductor RDS microcavity laser with an inward gap 21

32 Unstable periodic orbits in a ray dynamical simulation 23

33 Resonances localized on MUPOs and UPOs 27

34 Linear gain condition with a single-mode approximation 29

341 The threshold condition depending on the clas-

sical path of the periodic orbits 29

342 The threshold condition depending on the spa-

tial wave function localized on the periodic orbits 30

343 The effective pumping region due to the fabri-

cating configuration 33

35 Experimental results and emission characteristics 35

36 Conclusion 39

III

IV Methods 40

41 Fabrication of the two dimensional semiconductor mi-

crocavity laser 40

42 Experimental set-up configuration 42

References 44

Summary (요 약 문) 52

IV

List of Figures

21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R 4

22 The survival probability distribution and the chaotic repeller dis-

tribution of the RDS microcavity (a) is the survival probabil-

ity distribution of rays traveling in counter-clockwise direction in

semi-logarithmic scale There are three emission windows around

ssmax = 04 09 098 marked by the dotted-white circles (b)is

the CRD exhibiting the p4 (purple) the p5 (yellow) and the p6

(cyan) UPOs The UPOs touching the linear section and the UPOs

related to MUPOs are presented by filled circles and empty circles

respectively 6

23 Spatial wave functions of resonances and Husimi functions super-

imposed on the chaotic repeller distribution (a) (b) and (c) are

the resonances localized on the p4 the p5 and the p6 unstable pe-

riodic orbit respectively (d) is Husimi functions of the resonances

localized on the p4 (purple contour) the p5 (yellow contour) and

the p6 unstable periodic orbit (cyan contour) 8

V

24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic

stack layer diagram of an InP-based semiconductor laser along the

vertical cross-section (b) is an SEM image of the fabricated laser

whose radius of main circular arc is R = 30 μm (c) is a zoomed

SEM image of the square region in (b) showing an inward gap of

3 μm 12

25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection cur-

rent which shows a slope change around 16 mA (b) is the peak

intensity of the lasing mode at λ sim 1556 nm depending on injec-

tion current which well exhibits the lasing threshold of about 14

mA The upper and the lower figure in (c) are the optical spectra

at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group

supported by a period-5 UPO 13

26 Sixteen eigenvalues localized on the period-5 UPO around Re(nekR) sim100 The rectangular boxes and triangles are the even and the odd

parity modes respectively Inset A is the eigenvalues of a parity

pair Inset B is the spectral density 15

27 The magnified spectrum of the dominant lasing peak at the thresh-

old of 14 mA The lasing wavelength of λc is 155565 nm and

FWHM of w is 00532 nm 16

ndash VI ndash

28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yel-

low) The FFP of resonance modes is calculated by averaging the

FFPs of the sixteen resonances within Re(nekR) = 97 sim 105

Their main emission directions at θ = 180 are the same The

correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085

for between experiments and resonances 18

31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red

between the smaller circles 21

32 SEM images of the fabricated RDS microcavity laser (a) is an

SEM image of a fabricated RDS microcavity laser (b) is a zoomed

image of the red square region in (a) showing the 3 μm inward gap

between the contact region and the sidewall (c) shows a verti-

cal cross-section of RDS embedding an SCH-MQW active layer

marked by a red band Green bars in (b) and (c) represent the

length of λ 22

33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally

unstable periodic orbits and unstable periodic orbits (b) is the

survival probability distribution in logarithmic scale There are

three emission windows at SSmax = 04 08 10 indicated by

the circles below the critical line 25

ndash VII ndash

34 Semi-logarithmic plots of survival probability time distributions

(discrete symbols) and their linear fittings (straight curves) for

each periodic orbit (a) p13MUPO (b) p14MUPO (c) p14UPO

(d) p15MUPO (e) p15UPO and (f) p16MUPO From the slopes

of the fitting curves the decaying exponent αs with respect to

the effective propagation time of the propagation length with a

constant velocity of rays is obtained 26

35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pen-

tagram) and p16 MUPO (black circle) and p14 (green diamond)

and p15 UPO (purple hexagram) in the region nekR sim 396 The

eigenvalues of representative modes marked with dotted-red boxes

are 1202018minus i38006times10minus5 for p13 MUPO 1190782minus i34248times10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15 MUPO

1197075minusi31429times10minus5 for p16 MUPO 1198308minusi34238times10minus5

for p14 UPO and 1191448minusi31848times10minus5 for p15 UPO (b) is the

spatial wave patterns and their Husimi distributions of represen-

tative modes We overlap the Husimi distribution on the periodic

orbits in CRD to make clear the mode localization 28

ndash VIII ndash

36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain

layer (a) and (b) are threshold parameters of pc and pw respec-

tively which are defined by the classical excited length and the

ratio of the sum of wave function inside the cavity and the sum of

the overlapped wave function Both results show a range of 07 μm

lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO

The insets are the magnified graphs which show the lowest thresh-

old parameters of p15 MUPO 31

37 A schematic diagram for the effective pumping region 34

38 Experimental results of the RDS semiconductor microcavity laser

(a) shows the total output power (red) and major peak intensity

(blue) as functions of injection current A vertical dotted line

shows a threshold of lasing at 44 mA (b) shows the line width w

of the major peak λ = λc (black) fitted by Gaussian curve (red)

at the threshold current 36

39 The spectra of the RDS semiconductor microcavity laser (a) (b)

and (c) are spectra measured at 44 mA 47 mA and 51 mA respec-

tively Two mode groups in the spectra are marked by dashed-red

lines (Group 1) and dashed-green lines (Group 2) which have the

equidistant mode spacing of Δλ1 and Δλ2 37

310 The FFPs of the RDS microcavity laser (a) is the FFP from

the ray dynamical simulation (g) is the averaged FFP from the

resonance modes localized on p15 MUPO and (h) is the FFP

from the experiments All of the far-field patterns indicate three

emission directions around θ = 10 70 and 150 38

ndash IX ndash

41 Fabrication process for semiconductor microcavity laser (a) A

bare InP wafer (b) SiO2 deposition (c) PR coating (d) Mi-

crocavity shaped PR mask patterning (e) SiO2 etching (f) ICP

etching (g) SiO2 passivation layer deposition (h) PR patterning

for contact region (i) SiO2 etching for contact region (j) PR pat-

terning for metal electrode (k) p-contact electrode deposition and

patterning through the lift-off process (l) PR coating for protec-

tion (m) Substrate polishing (n) n-contact electrode deposition

(o) PR removing and heat treatment for completion 41

42 Set-up for measurements of the semiconductor microcavity laser 43

ndash X ndash

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

PhDEM

201631002

이 인 구 In-Goo Lee Thresholds of lasing modes localized

on MUPO and UPO in fully chaotic microcavity lasers

Department of Emerging Materials Science 2020 51p

Advisor Prof Chil-Min Kim Co-Advisor Prof Jung-Wan

Ryu

Abstract

We experimentally observe the lasing of a single-mode group in a rounded D-

shape InGaAsP semiconductor microcavity laser when the cavity is fully chaotic

Although there are numerous unstable periodic orbits and marginally unstable

periodic orbits the lasing mode group is localized on a pentagonal period-5 un-

stable periodic orbit and marginally unstable periodic orbits for each rounded

D-shape microcavity laser with parameters r = 085R φ = 5 and r = 05R

φ = 80 respectively near above the lowest threshold for CW current injection

By means of numerical analyses with ray and wave dynamics we have theoreti-

cally predicted the dominant lasing of the pentagonal mode group by considering

the linear gain condition at the lowest threshold condition We calculate the

threshold parameters depending on the effective pumping region for each peri-

odic orbit and have obtained that the resonance modes localized on the pen-

I

tagonal periodic orbits have the lowest threshold parameter for our experimental

condition which has a 3μm inward gap between the current contact region and

the side-wall of the microcavity Consequently we have confirmed that the path

length and unidirectional emission from the experimental the result well coincide

with our theoretical prediction

Keyword Microdisk laser Microcavity Semiconductor laser Marginally

unstable periodic orbit Scarred mode Threshold

II

Contents

Abstract I

Contents III

List of Figures V

I Introduction 1

II Lasing of scarred mode near above threshold in a rounded D-shape

microcavity laser 3

21 Ray dynamical analysis and unstable periodic orbits 5

22 Wave analysis for the threshold pumping strength 7

23 Fabrication and experimental results 11

24 Discussion 15

241 Quality factor of the scarred mode 15

242 Unidirectional emission from a scarred mode lasing 17

25 Conclusion 17

III Lasing modes localized on marginally unstable periodic orbits in a

rounded D-shape microcavity laser 20

31 Semiconductor RDS microcavity laser with an inward gap 21

32 Unstable periodic orbits in a ray dynamical simulation 23

33 Resonances localized on MUPOs and UPOs 27

34 Linear gain condition with a single-mode approximation 29

341 The threshold condition depending on the clas-

sical path of the periodic orbits 29

342 The threshold condition depending on the spa-

tial wave function localized on the periodic orbits 30

343 The effective pumping region due to the fabri-

cating configuration 33

35 Experimental results and emission characteristics 35

36 Conclusion 39

III

IV Methods 40

41 Fabrication of the two dimensional semiconductor mi-

crocavity laser 40

42 Experimental set-up configuration 42

References 44

Summary (요 약 문) 52

IV

List of Figures

21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R 4

22 The survival probability distribution and the chaotic repeller dis-

tribution of the RDS microcavity (a) is the survival probabil-

ity distribution of rays traveling in counter-clockwise direction in

semi-logarithmic scale There are three emission windows around

ssmax = 04 09 098 marked by the dotted-white circles (b)is

the CRD exhibiting the p4 (purple) the p5 (yellow) and the p6

(cyan) UPOs The UPOs touching the linear section and the UPOs

related to MUPOs are presented by filled circles and empty circles

respectively 6

23 Spatial wave functions of resonances and Husimi functions super-

imposed on the chaotic repeller distribution (a) (b) and (c) are

the resonances localized on the p4 the p5 and the p6 unstable pe-

riodic orbit respectively (d) is Husimi functions of the resonances

localized on the p4 (purple contour) the p5 (yellow contour) and

the p6 unstable periodic orbit (cyan contour) 8

V

24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic

stack layer diagram of an InP-based semiconductor laser along the

vertical cross-section (b) is an SEM image of the fabricated laser

whose radius of main circular arc is R = 30 μm (c) is a zoomed

SEM image of the square region in (b) showing an inward gap of

3 μm 12

25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection cur-

rent which shows a slope change around 16 mA (b) is the peak

intensity of the lasing mode at λ sim 1556 nm depending on injec-

tion current which well exhibits the lasing threshold of about 14

mA The upper and the lower figure in (c) are the optical spectra

at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group

supported by a period-5 UPO 13

26 Sixteen eigenvalues localized on the period-5 UPO around Re(nekR) sim100 The rectangular boxes and triangles are the even and the odd

parity modes respectively Inset A is the eigenvalues of a parity

pair Inset B is the spectral density 15

27 The magnified spectrum of the dominant lasing peak at the thresh-

old of 14 mA The lasing wavelength of λc is 155565 nm and

FWHM of w is 00532 nm 16

ndash VI ndash

28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yel-

low) The FFP of resonance modes is calculated by averaging the

FFPs of the sixteen resonances within Re(nekR) = 97 sim 105

Their main emission directions at θ = 180 are the same The

correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085

for between experiments and resonances 18

31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red

between the smaller circles 21

32 SEM images of the fabricated RDS microcavity laser (a) is an

SEM image of a fabricated RDS microcavity laser (b) is a zoomed

image of the red square region in (a) showing the 3 μm inward gap

between the contact region and the sidewall (c) shows a verti-

cal cross-section of RDS embedding an SCH-MQW active layer

marked by a red band Green bars in (b) and (c) represent the

length of λ 22

33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally

unstable periodic orbits and unstable periodic orbits (b) is the

survival probability distribution in logarithmic scale There are

three emission windows at SSmax = 04 08 10 indicated by

the circles below the critical line 25

ndash VII ndash

34 Semi-logarithmic plots of survival probability time distributions

(discrete symbols) and their linear fittings (straight curves) for

each periodic orbit (a) p13MUPO (b) p14MUPO (c) p14UPO

(d) p15MUPO (e) p15UPO and (f) p16MUPO From the slopes

of the fitting curves the decaying exponent αs with respect to

the effective propagation time of the propagation length with a

constant velocity of rays is obtained 26

35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pen-

tagram) and p16 MUPO (black circle) and p14 (green diamond)

and p15 UPO (purple hexagram) in the region nekR sim 396 The

eigenvalues of representative modes marked with dotted-red boxes

are 1202018minus i38006times10minus5 for p13 MUPO 1190782minus i34248times10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15 MUPO

1197075minusi31429times10minus5 for p16 MUPO 1198308minusi34238times10minus5

for p14 UPO and 1191448minusi31848times10minus5 for p15 UPO (b) is the

spatial wave patterns and their Husimi distributions of represen-

tative modes We overlap the Husimi distribution on the periodic

orbits in CRD to make clear the mode localization 28

ndash VIII ndash

36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain

layer (a) and (b) are threshold parameters of pc and pw respec-

tively which are defined by the classical excited length and the

ratio of the sum of wave function inside the cavity and the sum of

the overlapped wave function Both results show a range of 07 μm

lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO

The insets are the magnified graphs which show the lowest thresh-

old parameters of p15 MUPO 31

37 A schematic diagram for the effective pumping region 34

38 Experimental results of the RDS semiconductor microcavity laser

(a) shows the total output power (red) and major peak intensity

(blue) as functions of injection current A vertical dotted line

shows a threshold of lasing at 44 mA (b) shows the line width w

of the major peak λ = λc (black) fitted by Gaussian curve (red)

at the threshold current 36

39 The spectra of the RDS semiconductor microcavity laser (a) (b)

and (c) are spectra measured at 44 mA 47 mA and 51 mA respec-

tively Two mode groups in the spectra are marked by dashed-red

lines (Group 1) and dashed-green lines (Group 2) which have the

equidistant mode spacing of Δλ1 and Δλ2 37

310 The FFPs of the RDS microcavity laser (a) is the FFP from

the ray dynamical simulation (g) is the averaged FFP from the

resonance modes localized on p15 MUPO and (h) is the FFP

from the experiments All of the far-field patterns indicate three

emission directions around θ = 10 70 and 150 38

ndash IX ndash

41 Fabrication process for semiconductor microcavity laser (a) A

bare InP wafer (b) SiO2 deposition (c) PR coating (d) Mi-

crocavity shaped PR mask patterning (e) SiO2 etching (f) ICP

etching (g) SiO2 passivation layer deposition (h) PR patterning

for contact region (i) SiO2 etching for contact region (j) PR pat-

terning for metal electrode (k) p-contact electrode deposition and

patterning through the lift-off process (l) PR coating for protec-

tion (m) Substrate polishing (n) n-contact electrode deposition

(o) PR removing and heat treatment for completion 41

42 Set-up for measurements of the semiconductor microcavity laser 43

ndash X ndash

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

tagonal periodic orbits have the lowest threshold parameter for our experimental

condition which has a 3μm inward gap between the current contact region and

the side-wall of the microcavity Consequently we have confirmed that the path

length and unidirectional emission from the experimental the result well coincide

with our theoretical prediction

Keyword Microdisk laser Microcavity Semiconductor laser Marginally

unstable periodic orbit Scarred mode Threshold

II

Contents

Abstract I

Contents III

List of Figures V

I Introduction 1

II Lasing of scarred mode near above threshold in a rounded D-shape

microcavity laser 3

21 Ray dynamical analysis and unstable periodic orbits 5

22 Wave analysis for the threshold pumping strength 7

23 Fabrication and experimental results 11

24 Discussion 15

241 Quality factor of the scarred mode 15

242 Unidirectional emission from a scarred mode lasing 17

25 Conclusion 17

III Lasing modes localized on marginally unstable periodic orbits in a

rounded D-shape microcavity laser 20

31 Semiconductor RDS microcavity laser with an inward gap 21

32 Unstable periodic orbits in a ray dynamical simulation 23

33 Resonances localized on MUPOs and UPOs 27

34 Linear gain condition with a single-mode approximation 29

341 The threshold condition depending on the clas-

sical path of the periodic orbits 29

342 The threshold condition depending on the spa-

tial wave function localized on the periodic orbits 30

343 The effective pumping region due to the fabri-

cating configuration 33

35 Experimental results and emission characteristics 35

36 Conclusion 39

III

IV Methods 40

41 Fabrication of the two dimensional semiconductor mi-

crocavity laser 40

42 Experimental set-up configuration 42

References 44

Summary (요 약 문) 52

IV

List of Figures

21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R 4

22 The survival probability distribution and the chaotic repeller dis-

tribution of the RDS microcavity (a) is the survival probabil-

ity distribution of rays traveling in counter-clockwise direction in

semi-logarithmic scale There are three emission windows around

ssmax = 04 09 098 marked by the dotted-white circles (b)is

the CRD exhibiting the p4 (purple) the p5 (yellow) and the p6

(cyan) UPOs The UPOs touching the linear section and the UPOs

related to MUPOs are presented by filled circles and empty circles

respectively 6

23 Spatial wave functions of resonances and Husimi functions super-

imposed on the chaotic repeller distribution (a) (b) and (c) are

the resonances localized on the p4 the p5 and the p6 unstable pe-

riodic orbit respectively (d) is Husimi functions of the resonances

localized on the p4 (purple contour) the p5 (yellow contour) and

the p6 unstable periodic orbit (cyan contour) 8

V

24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic

stack layer diagram of an InP-based semiconductor laser along the

vertical cross-section (b) is an SEM image of the fabricated laser

whose radius of main circular arc is R = 30 μm (c) is a zoomed

SEM image of the square region in (b) showing an inward gap of

3 μm 12

25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection cur-

rent which shows a slope change around 16 mA (b) is the peak

intensity of the lasing mode at λ sim 1556 nm depending on injec-

tion current which well exhibits the lasing threshold of about 14

mA The upper and the lower figure in (c) are the optical spectra

at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group

supported by a period-5 UPO 13

26 Sixteen eigenvalues localized on the period-5 UPO around Re(nekR) sim100 The rectangular boxes and triangles are the even and the odd

parity modes respectively Inset A is the eigenvalues of a parity

pair Inset B is the spectral density 15

27 The magnified spectrum of the dominant lasing peak at the thresh-

old of 14 mA The lasing wavelength of λc is 155565 nm and

FWHM of w is 00532 nm 16

ndash VI ndash

28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yel-

low) The FFP of resonance modes is calculated by averaging the

FFPs of the sixteen resonances within Re(nekR) = 97 sim 105

Their main emission directions at θ = 180 are the same The

correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085

for between experiments and resonances 18

31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red

between the smaller circles 21

32 SEM images of the fabricated RDS microcavity laser (a) is an

SEM image of a fabricated RDS microcavity laser (b) is a zoomed

image of the red square region in (a) showing the 3 μm inward gap

between the contact region and the sidewall (c) shows a verti-

cal cross-section of RDS embedding an SCH-MQW active layer

marked by a red band Green bars in (b) and (c) represent the

length of λ 22

33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally

unstable periodic orbits and unstable periodic orbits (b) is the

survival probability distribution in logarithmic scale There are

three emission windows at SSmax = 04 08 10 indicated by

the circles below the critical line 25

ndash VII ndash

34 Semi-logarithmic plots of survival probability time distributions

(discrete symbols) and their linear fittings (straight curves) for

each periodic orbit (a) p13MUPO (b) p14MUPO (c) p14UPO

(d) p15MUPO (e) p15UPO and (f) p16MUPO From the slopes

of the fitting curves the decaying exponent αs with respect to

the effective propagation time of the propagation length with a

constant velocity of rays is obtained 26

35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pen-

tagram) and p16 MUPO (black circle) and p14 (green diamond)

and p15 UPO (purple hexagram) in the region nekR sim 396 The

eigenvalues of representative modes marked with dotted-red boxes

are 1202018minus i38006times10minus5 for p13 MUPO 1190782minus i34248times10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15 MUPO

1197075minusi31429times10minus5 for p16 MUPO 1198308minusi34238times10minus5

for p14 UPO and 1191448minusi31848times10minus5 for p15 UPO (b) is the

spatial wave patterns and their Husimi distributions of represen-

tative modes We overlap the Husimi distribution on the periodic

orbits in CRD to make clear the mode localization 28

ndash VIII ndash

36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain

layer (a) and (b) are threshold parameters of pc and pw respec-

tively which are defined by the classical excited length and the

ratio of the sum of wave function inside the cavity and the sum of

the overlapped wave function Both results show a range of 07 μm

lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO

The insets are the magnified graphs which show the lowest thresh-

old parameters of p15 MUPO 31

37 A schematic diagram for the effective pumping region 34

38 Experimental results of the RDS semiconductor microcavity laser

(a) shows the total output power (red) and major peak intensity

(blue) as functions of injection current A vertical dotted line

shows a threshold of lasing at 44 mA (b) shows the line width w

of the major peak λ = λc (black) fitted by Gaussian curve (red)

at the threshold current 36

39 The spectra of the RDS semiconductor microcavity laser (a) (b)

and (c) are spectra measured at 44 mA 47 mA and 51 mA respec-

tively Two mode groups in the spectra are marked by dashed-red

lines (Group 1) and dashed-green lines (Group 2) which have the

equidistant mode spacing of Δλ1 and Δλ2 37

310 The FFPs of the RDS microcavity laser (a) is the FFP from

the ray dynamical simulation (g) is the averaged FFP from the

resonance modes localized on p15 MUPO and (h) is the FFP

from the experiments All of the far-field patterns indicate three

emission directions around θ = 10 70 and 150 38

ndash IX ndash

41 Fabrication process for semiconductor microcavity laser (a) A

bare InP wafer (b) SiO2 deposition (c) PR coating (d) Mi-

crocavity shaped PR mask patterning (e) SiO2 etching (f) ICP

etching (g) SiO2 passivation layer deposition (h) PR patterning

for contact region (i) SiO2 etching for contact region (j) PR pat-

terning for metal electrode (k) p-contact electrode deposition and

patterning through the lift-off process (l) PR coating for protec-

tion (m) Substrate polishing (n) n-contact electrode deposition

(o) PR removing and heat treatment for completion 41

42 Set-up for measurements of the semiconductor microcavity laser 43

ndash X ndash

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

Contents

Abstract I

Contents III

List of Figures V

I Introduction 1

II Lasing of scarred mode near above threshold in a rounded D-shape

microcavity laser 3

21 Ray dynamical analysis and unstable periodic orbits 5

22 Wave analysis for the threshold pumping strength 7

23 Fabrication and experimental results 11

24 Discussion 15

241 Quality factor of the scarred mode 15

242 Unidirectional emission from a scarred mode lasing 17

25 Conclusion 17

III Lasing modes localized on marginally unstable periodic orbits in a

rounded D-shape microcavity laser 20

31 Semiconductor RDS microcavity laser with an inward gap 21

32 Unstable periodic orbits in a ray dynamical simulation 23

33 Resonances localized on MUPOs and UPOs 27

34 Linear gain condition with a single-mode approximation 29

341 The threshold condition depending on the clas-

sical path of the periodic orbits 29

342 The threshold condition depending on the spa-

tial wave function localized on the periodic orbits 30

343 The effective pumping region due to the fabri-

cating configuration 33

35 Experimental results and emission characteristics 35

36 Conclusion 39

III

IV Methods 40

41 Fabrication of the two dimensional semiconductor mi-

crocavity laser 40

42 Experimental set-up configuration 42

References 44

Summary (요 약 문) 52

IV

List of Figures

21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R 4

22 The survival probability distribution and the chaotic repeller dis-

tribution of the RDS microcavity (a) is the survival probabil-

ity distribution of rays traveling in counter-clockwise direction in

semi-logarithmic scale There are three emission windows around

ssmax = 04 09 098 marked by the dotted-white circles (b)is

the CRD exhibiting the p4 (purple) the p5 (yellow) and the p6

(cyan) UPOs The UPOs touching the linear section and the UPOs

related to MUPOs are presented by filled circles and empty circles

respectively 6

23 Spatial wave functions of resonances and Husimi functions super-

imposed on the chaotic repeller distribution (a) (b) and (c) are

the resonances localized on the p4 the p5 and the p6 unstable pe-

riodic orbit respectively (d) is Husimi functions of the resonances

localized on the p4 (purple contour) the p5 (yellow contour) and

the p6 unstable periodic orbit (cyan contour) 8

V

24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic

stack layer diagram of an InP-based semiconductor laser along the

vertical cross-section (b) is an SEM image of the fabricated laser

whose radius of main circular arc is R = 30 μm (c) is a zoomed

SEM image of the square region in (b) showing an inward gap of

3 μm 12

25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection cur-

rent which shows a slope change around 16 mA (b) is the peak

intensity of the lasing mode at λ sim 1556 nm depending on injec-

tion current which well exhibits the lasing threshold of about 14

mA The upper and the lower figure in (c) are the optical spectra

at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group

supported by a period-5 UPO 13

26 Sixteen eigenvalues localized on the period-5 UPO around Re(nekR) sim100 The rectangular boxes and triangles are the even and the odd

parity modes respectively Inset A is the eigenvalues of a parity

pair Inset B is the spectral density 15

27 The magnified spectrum of the dominant lasing peak at the thresh-

old of 14 mA The lasing wavelength of λc is 155565 nm and

FWHM of w is 00532 nm 16

ndash VI ndash

28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yel-

low) The FFP of resonance modes is calculated by averaging the

FFPs of the sixteen resonances within Re(nekR) = 97 sim 105

Their main emission directions at θ = 180 are the same The

correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085

for between experiments and resonances 18

31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red

between the smaller circles 21

32 SEM images of the fabricated RDS microcavity laser (a) is an

SEM image of a fabricated RDS microcavity laser (b) is a zoomed

image of the red square region in (a) showing the 3 μm inward gap

between the contact region and the sidewall (c) shows a verti-

cal cross-section of RDS embedding an SCH-MQW active layer

marked by a red band Green bars in (b) and (c) represent the

length of λ 22

33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally

unstable periodic orbits and unstable periodic orbits (b) is the

survival probability distribution in logarithmic scale There are

three emission windows at SSmax = 04 08 10 indicated by

the circles below the critical line 25

ndash VII ndash

34 Semi-logarithmic plots of survival probability time distributions

(discrete symbols) and their linear fittings (straight curves) for

each periodic orbit (a) p13MUPO (b) p14MUPO (c) p14UPO

(d) p15MUPO (e) p15UPO and (f) p16MUPO From the slopes

of the fitting curves the decaying exponent αs with respect to

the effective propagation time of the propagation length with a

constant velocity of rays is obtained 26

35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pen-

tagram) and p16 MUPO (black circle) and p14 (green diamond)

and p15 UPO (purple hexagram) in the region nekR sim 396 The

eigenvalues of representative modes marked with dotted-red boxes

are 1202018minus i38006times10minus5 for p13 MUPO 1190782minus i34248times10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15 MUPO

1197075minusi31429times10minus5 for p16 MUPO 1198308minusi34238times10minus5

for p14 UPO and 1191448minusi31848times10minus5 for p15 UPO (b) is the

spatial wave patterns and their Husimi distributions of represen-

tative modes We overlap the Husimi distribution on the periodic

orbits in CRD to make clear the mode localization 28

ndash VIII ndash

36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain

layer (a) and (b) are threshold parameters of pc and pw respec-

tively which are defined by the classical excited length and the

ratio of the sum of wave function inside the cavity and the sum of

the overlapped wave function Both results show a range of 07 μm

lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO

The insets are the magnified graphs which show the lowest thresh-

old parameters of p15 MUPO 31

37 A schematic diagram for the effective pumping region 34

38 Experimental results of the RDS semiconductor microcavity laser

(a) shows the total output power (red) and major peak intensity

(blue) as functions of injection current A vertical dotted line

shows a threshold of lasing at 44 mA (b) shows the line width w

of the major peak λ = λc (black) fitted by Gaussian curve (red)

at the threshold current 36

39 The spectra of the RDS semiconductor microcavity laser (a) (b)

and (c) are spectra measured at 44 mA 47 mA and 51 mA respec-

tively Two mode groups in the spectra are marked by dashed-red

lines (Group 1) and dashed-green lines (Group 2) which have the

equidistant mode spacing of Δλ1 and Δλ2 37

310 The FFPs of the RDS microcavity laser (a) is the FFP from

the ray dynamical simulation (g) is the averaged FFP from the

resonance modes localized on p15 MUPO and (h) is the FFP

from the experiments All of the far-field patterns indicate three

emission directions around θ = 10 70 and 150 38

ndash IX ndash

41 Fabrication process for semiconductor microcavity laser (a) A

bare InP wafer (b) SiO2 deposition (c) PR coating (d) Mi-

crocavity shaped PR mask patterning (e) SiO2 etching (f) ICP

etching (g) SiO2 passivation layer deposition (h) PR patterning

for contact region (i) SiO2 etching for contact region (j) PR pat-

terning for metal electrode (k) p-contact electrode deposition and

patterning through the lift-off process (l) PR coating for protec-

tion (m) Substrate polishing (n) n-contact electrode deposition

(o) PR removing and heat treatment for completion 41

42 Set-up for measurements of the semiconductor microcavity laser 43

ndash X ndash

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

IV Methods 40

41 Fabrication of the two dimensional semiconductor mi-

crocavity laser 40

42 Experimental set-up configuration 42

References 44

Summary (요 약 문) 52

IV

List of Figures

21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R 4

22 The survival probability distribution and the chaotic repeller dis-

tribution of the RDS microcavity (a) is the survival probabil-

ity distribution of rays traveling in counter-clockwise direction in

semi-logarithmic scale There are three emission windows around

ssmax = 04 09 098 marked by the dotted-white circles (b)is

the CRD exhibiting the p4 (purple) the p5 (yellow) and the p6

(cyan) UPOs The UPOs touching the linear section and the UPOs

related to MUPOs are presented by filled circles and empty circles

respectively 6

23 Spatial wave functions of resonances and Husimi functions super-

imposed on the chaotic repeller distribution (a) (b) and (c) are

the resonances localized on the p4 the p5 and the p6 unstable pe-

riodic orbit respectively (d) is Husimi functions of the resonances

localized on the p4 (purple contour) the p5 (yellow contour) and

the p6 unstable periodic orbit (cyan contour) 8

V

24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic

stack layer diagram of an InP-based semiconductor laser along the

vertical cross-section (b) is an SEM image of the fabricated laser

whose radius of main circular arc is R = 30 μm (c) is a zoomed

SEM image of the square region in (b) showing an inward gap of

3 μm 12

25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection cur-

rent which shows a slope change around 16 mA (b) is the peak

intensity of the lasing mode at λ sim 1556 nm depending on injec-

tion current which well exhibits the lasing threshold of about 14

mA The upper and the lower figure in (c) are the optical spectra

at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group

supported by a period-5 UPO 13

26 Sixteen eigenvalues localized on the period-5 UPO around Re(nekR) sim100 The rectangular boxes and triangles are the even and the odd

parity modes respectively Inset A is the eigenvalues of a parity

pair Inset B is the spectral density 15

27 The magnified spectrum of the dominant lasing peak at the thresh-

old of 14 mA The lasing wavelength of λc is 155565 nm and

FWHM of w is 00532 nm 16

ndash VI ndash

28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yel-

low) The FFP of resonance modes is calculated by averaging the

FFPs of the sixteen resonances within Re(nekR) = 97 sim 105

Their main emission directions at θ = 180 are the same The

correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085

for between experiments and resonances 18

31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red

between the smaller circles 21

32 SEM images of the fabricated RDS microcavity laser (a) is an

SEM image of a fabricated RDS microcavity laser (b) is a zoomed

image of the red square region in (a) showing the 3 μm inward gap

between the contact region and the sidewall (c) shows a verti-

cal cross-section of RDS embedding an SCH-MQW active layer

marked by a red band Green bars in (b) and (c) represent the

length of λ 22

33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally

unstable periodic orbits and unstable periodic orbits (b) is the

survival probability distribution in logarithmic scale There are

three emission windows at SSmax = 04 08 10 indicated by

the circles below the critical line 25

ndash VII ndash

34 Semi-logarithmic plots of survival probability time distributions

(discrete symbols) and their linear fittings (straight curves) for

each periodic orbit (a) p13MUPO (b) p14MUPO (c) p14UPO

(d) p15MUPO (e) p15UPO and (f) p16MUPO From the slopes

of the fitting curves the decaying exponent αs with respect to

the effective propagation time of the propagation length with a

constant velocity of rays is obtained 26

35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pen-

tagram) and p16 MUPO (black circle) and p14 (green diamond)

and p15 UPO (purple hexagram) in the region nekR sim 396 The

eigenvalues of representative modes marked with dotted-red boxes

are 1202018minus i38006times10minus5 for p13 MUPO 1190782minus i34248times10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15 MUPO

1197075minusi31429times10minus5 for p16 MUPO 1198308minusi34238times10minus5

for p14 UPO and 1191448minusi31848times10minus5 for p15 UPO (b) is the

spatial wave patterns and their Husimi distributions of represen-

tative modes We overlap the Husimi distribution on the periodic

orbits in CRD to make clear the mode localization 28

ndash VIII ndash

36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain

layer (a) and (b) are threshold parameters of pc and pw respec-

tively which are defined by the classical excited length and the

ratio of the sum of wave function inside the cavity and the sum of

the overlapped wave function Both results show a range of 07 μm

lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO

The insets are the magnified graphs which show the lowest thresh-

old parameters of p15 MUPO 31

37 A schematic diagram for the effective pumping region 34

38 Experimental results of the RDS semiconductor microcavity laser

(a) shows the total output power (red) and major peak intensity

(blue) as functions of injection current A vertical dotted line

shows a threshold of lasing at 44 mA (b) shows the line width w

of the major peak λ = λc (black) fitted by Gaussian curve (red)

at the threshold current 36

39 The spectra of the RDS semiconductor microcavity laser (a) (b)

and (c) are spectra measured at 44 mA 47 mA and 51 mA respec-

tively Two mode groups in the spectra are marked by dashed-red

lines (Group 1) and dashed-green lines (Group 2) which have the

equidistant mode spacing of Δλ1 and Δλ2 37

310 The FFPs of the RDS microcavity laser (a) is the FFP from

the ray dynamical simulation (g) is the averaged FFP from the

resonance modes localized on p15 MUPO and (h) is the FFP

from the experiments All of the far-field patterns indicate three

emission directions around θ = 10 70 and 150 38

ndash IX ndash

41 Fabrication process for semiconductor microcavity laser (a) A

bare InP wafer (b) SiO2 deposition (c) PR coating (d) Mi-

crocavity shaped PR mask patterning (e) SiO2 etching (f) ICP

etching (g) SiO2 passivation layer deposition (h) PR patterning

for contact region (i) SiO2 etching for contact region (j) PR pat-

terning for metal electrode (k) p-contact electrode deposition and

patterning through the lift-off process (l) PR coating for protec-

tion (m) Substrate polishing (n) n-contact electrode deposition

(o) PR removing and heat treatment for completion 41

42 Set-up for measurements of the semiconductor microcavity laser 43

ndash X ndash

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

List of Figures

21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R 4

22 The survival probability distribution and the chaotic repeller dis-

tribution of the RDS microcavity (a) is the survival probabil-

ity distribution of rays traveling in counter-clockwise direction in

semi-logarithmic scale There are three emission windows around

ssmax = 04 09 098 marked by the dotted-white circles (b)is

the CRD exhibiting the p4 (purple) the p5 (yellow) and the p6

(cyan) UPOs The UPOs touching the linear section and the UPOs

related to MUPOs are presented by filled circles and empty circles

respectively 6

23 Spatial wave functions of resonances and Husimi functions super-

imposed on the chaotic repeller distribution (a) (b) and (c) are

the resonances localized on the p4 the p5 and the p6 unstable pe-

riodic orbit respectively (d) is Husimi functions of the resonances

localized on the p4 (purple contour) the p5 (yellow contour) and

the p6 unstable periodic orbit (cyan contour) 8

V

24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic

stack layer diagram of an InP-based semiconductor laser along the

vertical cross-section (b) is an SEM image of the fabricated laser

whose radius of main circular arc is R = 30 μm (c) is a zoomed

SEM image of the square region in (b) showing an inward gap of

3 μm 12

25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection cur-

rent which shows a slope change around 16 mA (b) is the peak

intensity of the lasing mode at λ sim 1556 nm depending on injec-

tion current which well exhibits the lasing threshold of about 14

mA The upper and the lower figure in (c) are the optical spectra

at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group

supported by a period-5 UPO 13

26 Sixteen eigenvalues localized on the period-5 UPO around Re(nekR) sim100 The rectangular boxes and triangles are the even and the odd

parity modes respectively Inset A is the eigenvalues of a parity

pair Inset B is the spectral density 15

27 The magnified spectrum of the dominant lasing peak at the thresh-

old of 14 mA The lasing wavelength of λc is 155565 nm and

FWHM of w is 00532 nm 16

ndash VI ndash

28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yel-

low) The FFP of resonance modes is calculated by averaging the

FFPs of the sixteen resonances within Re(nekR) = 97 sim 105

Their main emission directions at θ = 180 are the same The

correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085

for between experiments and resonances 18

31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red

between the smaller circles 21

32 SEM images of the fabricated RDS microcavity laser (a) is an

SEM image of a fabricated RDS microcavity laser (b) is a zoomed

image of the red square region in (a) showing the 3 μm inward gap

between the contact region and the sidewall (c) shows a verti-

cal cross-section of RDS embedding an SCH-MQW active layer

marked by a red band Green bars in (b) and (c) represent the

length of λ 22

33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally

unstable periodic orbits and unstable periodic orbits (b) is the

survival probability distribution in logarithmic scale There are

three emission windows at SSmax = 04 08 10 indicated by

the circles below the critical line 25

ndash VII ndash

34 Semi-logarithmic plots of survival probability time distributions

(discrete symbols) and their linear fittings (straight curves) for

each periodic orbit (a) p13MUPO (b) p14MUPO (c) p14UPO

(d) p15MUPO (e) p15UPO and (f) p16MUPO From the slopes

of the fitting curves the decaying exponent αs with respect to

the effective propagation time of the propagation length with a

constant velocity of rays is obtained 26

35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pen-

tagram) and p16 MUPO (black circle) and p14 (green diamond)

and p15 UPO (purple hexagram) in the region nekR sim 396 The

eigenvalues of representative modes marked with dotted-red boxes

are 1202018minus i38006times10minus5 for p13 MUPO 1190782minus i34248times10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15 MUPO

1197075minusi31429times10minus5 for p16 MUPO 1198308minusi34238times10minus5

for p14 UPO and 1191448minusi31848times10minus5 for p15 UPO (b) is the

spatial wave patterns and their Husimi distributions of represen-

tative modes We overlap the Husimi distribution on the periodic

orbits in CRD to make clear the mode localization 28

ndash VIII ndash

36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain

layer (a) and (b) are threshold parameters of pc and pw respec-

tively which are defined by the classical excited length and the

ratio of the sum of wave function inside the cavity and the sum of

the overlapped wave function Both results show a range of 07 μm

lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO

The insets are the magnified graphs which show the lowest thresh-

old parameters of p15 MUPO 31

37 A schematic diagram for the effective pumping region 34

38 Experimental results of the RDS semiconductor microcavity laser

(a) shows the total output power (red) and major peak intensity

(blue) as functions of injection current A vertical dotted line

shows a threshold of lasing at 44 mA (b) shows the line width w

of the major peak λ = λc (black) fitted by Gaussian curve (red)

at the threshold current 36

39 The spectra of the RDS semiconductor microcavity laser (a) (b)

and (c) are spectra measured at 44 mA 47 mA and 51 mA respec-

tively Two mode groups in the spectra are marked by dashed-red

lines (Group 1) and dashed-green lines (Group 2) which have the

equidistant mode spacing of Δλ1 and Δλ2 37

310 The FFPs of the RDS microcavity laser (a) is the FFP from

the ray dynamical simulation (g) is the averaged FFP from the

resonance modes localized on p15 MUPO and (h) is the FFP

from the experiments All of the far-field patterns indicate three

emission directions around θ = 10 70 and 150 38

ndash IX ndash

41 Fabrication process for semiconductor microcavity laser (a) A

bare InP wafer (b) SiO2 deposition (c) PR coating (d) Mi-

crocavity shaped PR mask patterning (e) SiO2 etching (f) ICP

etching (g) SiO2 passivation layer deposition (h) PR patterning

for contact region (i) SiO2 etching for contact region (j) PR pat-

terning for metal electrode (k) p-contact electrode deposition and

patterning through the lift-off process (l) PR coating for protec-

tion (m) Substrate polishing (n) n-contact electrode deposition

(o) PR removing and heat treatment for completion 41

42 Set-up for measurements of the semiconductor microcavity laser 43

ndash X ndash

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic

stack layer diagram of an InP-based semiconductor laser along the

vertical cross-section (b) is an SEM image of the fabricated laser

whose radius of main circular arc is R = 30 μm (c) is a zoomed

SEM image of the square region in (b) showing an inward gap of

3 μm 12

25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection cur-

rent which shows a slope change around 16 mA (b) is the peak

intensity of the lasing mode at λ sim 1556 nm depending on injec-

tion current which well exhibits the lasing threshold of about 14

mA The upper and the lower figure in (c) are the optical spectra

at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group

supported by a period-5 UPO 13

26 Sixteen eigenvalues localized on the period-5 UPO around Re(nekR) sim100 The rectangular boxes and triangles are the even and the odd

parity modes respectively Inset A is the eigenvalues of a parity

pair Inset B is the spectral density 15

27 The magnified spectrum of the dominant lasing peak at the thresh-

old of 14 mA The lasing wavelength of λc is 155565 nm and

FWHM of w is 00532 nm 16

ndash VI ndash

28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yel-

low) The FFP of resonance modes is calculated by averaging the

FFPs of the sixteen resonances within Re(nekR) = 97 sim 105

Their main emission directions at θ = 180 are the same The

correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085

for between experiments and resonances 18

31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red

between the smaller circles 21

32 SEM images of the fabricated RDS microcavity laser (a) is an

SEM image of a fabricated RDS microcavity laser (b) is a zoomed

image of the red square region in (a) showing the 3 μm inward gap

between the contact region and the sidewall (c) shows a verti-

cal cross-section of RDS embedding an SCH-MQW active layer

marked by a red band Green bars in (b) and (c) represent the

length of λ 22

33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally

unstable periodic orbits and unstable periodic orbits (b) is the

survival probability distribution in logarithmic scale There are

three emission windows at SSmax = 04 08 10 indicated by

the circles below the critical line 25

ndash VII ndash

34 Semi-logarithmic plots of survival probability time distributions

(discrete symbols) and their linear fittings (straight curves) for

each periodic orbit (a) p13MUPO (b) p14MUPO (c) p14UPO

(d) p15MUPO (e) p15UPO and (f) p16MUPO From the slopes

of the fitting curves the decaying exponent αs with respect to

the effective propagation time of the propagation length with a

constant velocity of rays is obtained 26

35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pen-

tagram) and p16 MUPO (black circle) and p14 (green diamond)

and p15 UPO (purple hexagram) in the region nekR sim 396 The

eigenvalues of representative modes marked with dotted-red boxes

are 1202018minus i38006times10minus5 for p13 MUPO 1190782minus i34248times10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15 MUPO

1197075minusi31429times10minus5 for p16 MUPO 1198308minusi34238times10minus5

for p14 UPO and 1191448minusi31848times10minus5 for p15 UPO (b) is the

spatial wave patterns and their Husimi distributions of represen-

tative modes We overlap the Husimi distribution on the periodic

orbits in CRD to make clear the mode localization 28

ndash VIII ndash

36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain

layer (a) and (b) are threshold parameters of pc and pw respec-

tively which are defined by the classical excited length and the

ratio of the sum of wave function inside the cavity and the sum of

the overlapped wave function Both results show a range of 07 μm

lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO

The insets are the magnified graphs which show the lowest thresh-

old parameters of p15 MUPO 31

37 A schematic diagram for the effective pumping region 34

38 Experimental results of the RDS semiconductor microcavity laser

(a) shows the total output power (red) and major peak intensity

(blue) as functions of injection current A vertical dotted line

shows a threshold of lasing at 44 mA (b) shows the line width w

of the major peak λ = λc (black) fitted by Gaussian curve (red)

at the threshold current 36

39 The spectra of the RDS semiconductor microcavity laser (a) (b)

and (c) are spectra measured at 44 mA 47 mA and 51 mA respec-

tively Two mode groups in the spectra are marked by dashed-red

lines (Group 1) and dashed-green lines (Group 2) which have the

equidistant mode spacing of Δλ1 and Δλ2 37

310 The FFPs of the RDS microcavity laser (a) is the FFP from

the ray dynamical simulation (g) is the averaged FFP from the

resonance modes localized on p15 MUPO and (h) is the FFP

from the experiments All of the far-field patterns indicate three

emission directions around θ = 10 70 and 150 38

ndash IX ndash

41 Fabrication process for semiconductor microcavity laser (a) A

bare InP wafer (b) SiO2 deposition (c) PR coating (d) Mi-

crocavity shaped PR mask patterning (e) SiO2 etching (f) ICP

etching (g) SiO2 passivation layer deposition (h) PR patterning

for contact region (i) SiO2 etching for contact region (j) PR pat-

terning for metal electrode (k) p-contact electrode deposition and

patterning through the lift-off process (l) PR coating for protec-

tion (m) Substrate polishing (n) n-contact electrode deposition

(o) PR removing and heat treatment for completion 41

42 Set-up for measurements of the semiconductor microcavity laser 43

ndash X ndash

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yel-

low) The FFP of resonance modes is calculated by averaging the

FFPs of the sixteen resonances within Re(nekR) = 97 sim 105

Their main emission directions at θ = 180 are the same The

correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085

for between experiments and resonances 18

31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red

between the smaller circles 21

32 SEM images of the fabricated RDS microcavity laser (a) is an

SEM image of a fabricated RDS microcavity laser (b) is a zoomed

image of the red square region in (a) showing the 3 μm inward gap

between the contact region and the sidewall (c) shows a verti-

cal cross-section of RDS embedding an SCH-MQW active layer

marked by a red band Green bars in (b) and (c) represent the

length of λ 22

33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally

unstable periodic orbits and unstable periodic orbits (b) is the

survival probability distribution in logarithmic scale There are

three emission windows at SSmax = 04 08 10 indicated by

the circles below the critical line 25

ndash VII ndash

34 Semi-logarithmic plots of survival probability time distributions

(discrete symbols) and their linear fittings (straight curves) for

each periodic orbit (a) p13MUPO (b) p14MUPO (c) p14UPO

(d) p15MUPO (e) p15UPO and (f) p16MUPO From the slopes

of the fitting curves the decaying exponent αs with respect to

the effective propagation time of the propagation length with a

constant velocity of rays is obtained 26

35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pen-

tagram) and p16 MUPO (black circle) and p14 (green diamond)

and p15 UPO (purple hexagram) in the region nekR sim 396 The

eigenvalues of representative modes marked with dotted-red boxes

are 1202018minus i38006times10minus5 for p13 MUPO 1190782minus i34248times10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15 MUPO

1197075minusi31429times10minus5 for p16 MUPO 1198308minusi34238times10minus5

for p14 UPO and 1191448minusi31848times10minus5 for p15 UPO (b) is the

spatial wave patterns and their Husimi distributions of represen-

tative modes We overlap the Husimi distribution on the periodic

orbits in CRD to make clear the mode localization 28

ndash VIII ndash

36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain

layer (a) and (b) are threshold parameters of pc and pw respec-

tively which are defined by the classical excited length and the

ratio of the sum of wave function inside the cavity and the sum of

the overlapped wave function Both results show a range of 07 μm

lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO

The insets are the magnified graphs which show the lowest thresh-

old parameters of p15 MUPO 31

37 A schematic diagram for the effective pumping region 34

38 Experimental results of the RDS semiconductor microcavity laser

(a) shows the total output power (red) and major peak intensity

(blue) as functions of injection current A vertical dotted line

shows a threshold of lasing at 44 mA (b) shows the line width w

of the major peak λ = λc (black) fitted by Gaussian curve (red)

at the threshold current 36

39 The spectra of the RDS semiconductor microcavity laser (a) (b)

and (c) are spectra measured at 44 mA 47 mA and 51 mA respec-

tively Two mode groups in the spectra are marked by dashed-red

lines (Group 1) and dashed-green lines (Group 2) which have the

equidistant mode spacing of Δλ1 and Δλ2 37

310 The FFPs of the RDS microcavity laser (a) is the FFP from

the ray dynamical simulation (g) is the averaged FFP from the

resonance modes localized on p15 MUPO and (h) is the FFP

from the experiments All of the far-field patterns indicate three

emission directions around θ = 10 70 and 150 38

ndash IX ndash

41 Fabrication process for semiconductor microcavity laser (a) A

bare InP wafer (b) SiO2 deposition (c) PR coating (d) Mi-

crocavity shaped PR mask patterning (e) SiO2 etching (f) ICP

etching (g) SiO2 passivation layer deposition (h) PR patterning

for contact region (i) SiO2 etching for contact region (j) PR pat-

terning for metal electrode (k) p-contact electrode deposition and

patterning through the lift-off process (l) PR coating for protec-

tion (m) Substrate polishing (n) n-contact electrode deposition

(o) PR removing and heat treatment for completion 41

42 Set-up for measurements of the semiconductor microcavity laser 43

ndash X ndash

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

34 Semi-logarithmic plots of survival probability time distributions

(discrete symbols) and their linear fittings (straight curves) for

each periodic orbit (a) p13MUPO (b) p14MUPO (c) p14UPO

(d) p15MUPO (e) p15UPO and (f) p16MUPO From the slopes

of the fitting curves the decaying exponent αs with respect to

the effective propagation time of the propagation length with a

constant velocity of rays is obtained 26

35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pen-

tagram) and p16 MUPO (black circle) and p14 (green diamond)

and p15 UPO (purple hexagram) in the region nekR sim 396 The

eigenvalues of representative modes marked with dotted-red boxes

are 1202018minus i38006times10minus5 for p13 MUPO 1190782minus i34248times10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15 MUPO

1197075minusi31429times10minus5 for p16 MUPO 1198308minusi34238times10minus5

for p14 UPO and 1191448minusi31848times10minus5 for p15 UPO (b) is the

spatial wave patterns and their Husimi distributions of represen-

tative modes We overlap the Husimi distribution on the periodic

orbits in CRD to make clear the mode localization 28

ndash VIII ndash

36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain

layer (a) and (b) are threshold parameters of pc and pw respec-

tively which are defined by the classical excited length and the

ratio of the sum of wave function inside the cavity and the sum of

the overlapped wave function Both results show a range of 07 μm

lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO

The insets are the magnified graphs which show the lowest thresh-

old parameters of p15 MUPO 31

37 A schematic diagram for the effective pumping region 34

38 Experimental results of the RDS semiconductor microcavity laser

(a) shows the total output power (red) and major peak intensity

(blue) as functions of injection current A vertical dotted line

shows a threshold of lasing at 44 mA (b) shows the line width w

of the major peak λ = λc (black) fitted by Gaussian curve (red)

at the threshold current 36

39 The spectra of the RDS semiconductor microcavity laser (a) (b)

and (c) are spectra measured at 44 mA 47 mA and 51 mA respec-

tively Two mode groups in the spectra are marked by dashed-red

lines (Group 1) and dashed-green lines (Group 2) which have the

equidistant mode spacing of Δλ1 and Δλ2 37

310 The FFPs of the RDS microcavity laser (a) is the FFP from

the ray dynamical simulation (g) is the averaged FFP from the

resonance modes localized on p15 MUPO and (h) is the FFP

from the experiments All of the far-field patterns indicate three

emission directions around θ = 10 70 and 150 38

ndash IX ndash

41 Fabrication process for semiconductor microcavity laser (a) A

bare InP wafer (b) SiO2 deposition (c) PR coating (d) Mi-

crocavity shaped PR mask patterning (e) SiO2 etching (f) ICP

etching (g) SiO2 passivation layer deposition (h) PR patterning

for contact region (i) SiO2 etching for contact region (j) PR pat-

terning for metal electrode (k) p-contact electrode deposition and

patterning through the lift-off process (l) PR coating for protec-

tion (m) Substrate polishing (n) n-contact electrode deposition

(o) PR removing and heat treatment for completion 41

42 Set-up for measurements of the semiconductor microcavity laser 43

ndash X ndash

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain

layer (a) and (b) are threshold parameters of pc and pw respec-

tively which are defined by the classical excited length and the

ratio of the sum of wave function inside the cavity and the sum of

the overlapped wave function Both results show a range of 07 μm

lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO

The insets are the magnified graphs which show the lowest thresh-

old parameters of p15 MUPO 31

37 A schematic diagram for the effective pumping region 34

38 Experimental results of the RDS semiconductor microcavity laser

(a) shows the total output power (red) and major peak intensity

(blue) as functions of injection current A vertical dotted line

shows a threshold of lasing at 44 mA (b) shows the line width w

of the major peak λ = λc (black) fitted by Gaussian curve (red)

at the threshold current 36

39 The spectra of the RDS semiconductor microcavity laser (a) (b)

and (c) are spectra measured at 44 mA 47 mA and 51 mA respec-

tively Two mode groups in the spectra are marked by dashed-red

lines (Group 1) and dashed-green lines (Group 2) which have the

equidistant mode spacing of Δλ1 and Δλ2 37

310 The FFPs of the RDS microcavity laser (a) is the FFP from

the ray dynamical simulation (g) is the averaged FFP from the

resonance modes localized on p15 MUPO and (h) is the FFP

from the experiments All of the far-field patterns indicate three

emission directions around θ = 10 70 and 150 38

ndash IX ndash

41 Fabrication process for semiconductor microcavity laser (a) A

bare InP wafer (b) SiO2 deposition (c) PR coating (d) Mi-

crocavity shaped PR mask patterning (e) SiO2 etching (f) ICP

etching (g) SiO2 passivation layer deposition (h) PR patterning

for contact region (i) SiO2 etching for contact region (j) PR pat-

terning for metal electrode (k) p-contact electrode deposition and

patterning through the lift-off process (l) PR coating for protec-

tion (m) Substrate polishing (n) n-contact electrode deposition

(o) PR removing and heat treatment for completion 41

42 Set-up for measurements of the semiconductor microcavity laser 43

ndash X ndash

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

41 Fabrication process for semiconductor microcavity laser (a) A

bare InP wafer (b) SiO2 deposition (c) PR coating (d) Mi-

crocavity shaped PR mask patterning (e) SiO2 etching (f) ICP

etching (g) SiO2 passivation layer deposition (h) PR patterning

for contact region (i) SiO2 etching for contact region (j) PR pat-

terning for metal electrode (k) p-contact electrode deposition and

patterning through the lift-off process (l) PR coating for protec-

tion (m) Substrate polishing (n) n-contact electrode deposition

(o) PR removing and heat treatment for completion 41

42 Set-up for measurements of the semiconductor microcavity laser 43

ndash X ndash

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

I Introduction

Microcavity lasers have attracted much attention owing to the high-quality

factor of their whispering gallery modes which are supported by total internal

reflection A circular microcavity has been spotlighted because of its extremely

high-quality factor [1] and the potential for highly sensitive label-free detection [2

3 4 5] nano-particle detection [4 5] and gyroscopy [6 7] Currently it has

been reported that deformed microcavities from the circular one have various

outstanding characteristics such as directionality [8 9 10 11 12 13 14 15 16]

and chirality [13 17] with minimized degradation of high-quality factor which

are crucial for wide-range applications

When a deformed semiconductor microcavity laser is excited by current in-

jection it is hard to excite a whole cavity region because the area of the contact

region for pumping is smaller than the cavity surface due to fabricating limi-

tation Because of this reason periodic orbits in a deformed microcavity are

partially overlapped with the contact region Accordingly the modes localized

on them are excited only along the overlapped parts of the periodic orbits near

the threshold (ie we rule out the current diffusion at higher current injection)

This ldquopartial excitationrdquo makes different threshold conditions for lasing of reso-

nance modes localized on the periodic orbits In other words the lasing modes

are conditionally determined by pumping circumstances the lasing threshold of a

certain resonance mode is strongly dependent on its loss and the effective pump-

ing rate [18] The smaller contact region compared to the cavity for injection

current is very natural and general in fabricating two-dimensional semiconduc-

tor lasers However a lasing threshold condition connecting the loss and partial

ndash 1 ndash

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

excitation of periodic orbits in a microcavity laser has not been well analyzed so

far Hence in the present thesis we aim to study their relations explicitly

To study the lowest threshold lasing mode we take a semiconductor micro-

cavity defined by a rounded D-shape (RDS) boundary morphology In this shape

of the microcavity there exist unstable periodic orbits (UPOs) and marginally

unstable periodic orbits (MUPOs) according to a parametric condition In this

thesis we take two types of RDS semiconductor microcavity lasers with and

without MUPOs above a critical line

In chapter II we study the RDS microcavity laser without MUPOs to investi-

gate UPOs in the cavity and their directionality affected by an unstable manifold

structure in a chaotic sea In the single-mode approximation a threshold pump-

ing strength for each resonance mode localized on the UPOs are investigated

according to the gap between the effective pumping region and the sidewall of

the cavity which inevitably occurs in the fabrication process The experimental

results show a unidirectional emission at forward direction and we prove that

it originates from the lowest threshold mode by analyzing the spectrum at the

threshold

In chapter III we apply those analytic methods to the RDS microcavity

laser with both UPOs and MUPOs MUPOs and UPOs above a critical line

are obtained in a ray dynamical simulation and resonances localized on them

are also obtained by employing a boundary element method We develop the

linear gain condition in two cases of classical ray dynamics and resonance modes

to compare the lowest threshold parameters of periodic orbits By considering

the effective pumping region owing to drift current we verify that the both of

period-5 UPO and resonance localized on it have the lowest threshold parameters

in our experimental inward gap condition Also we confirm this result through

experimentation

ndash 2 ndash

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

II Lasing of scarred mode near above

threshold in a rounded D-shape microcavity

laser

Microcavity lasers can be applied to highly sensitive sensors because the

tail of whispering gallery modes (WGMs) can prove small perturbation Here a

high-quality factor is crucial for achieving highly sensitive sensors [1] However

because of the isotropic emission of WGMs collecting the emission laser beam

is another problem to be solved for efficient applications of microcavity lasers

One of the solutions is the use of a deformed microcavity laser [19 20] In recent

studies unidirectional emission has been achieved for efficient collection of laser

emission in such various shapes as a spiral [12 13] a limacon [9 10] a rounded

isosceles triangular [14] a cardioid [15] and ellipse with a notch [21] In these

chaotic microcavities directionality was analyzed based on quantum chaos theory

for unstable manifold structures [22 23]

Unstable manifolds in the chaotic sea are closely related to emission win-

dows in a chaotic microcavity laser When resonance is localized on an unstable

periodic orbit (UPO) named scar the resonance follows the unstable manifolds

and emits out through the window Because of the relation between unstable

manifold structure and strong emission direction of a lasing mode group it is

important to analyze a UPO supporting a scar and the manifold structure in a

fully chaotic microcavity

For the study of manifold structure UPOs and lasing of a resonance mode

group supported by UPOs in this chapter we take a fully chaotic rounded D-

shape (RDS) microcavity laser whose shape is comprised of three circular arcs

ndash 3 ndash

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

r=085RR

= 5o

= 0o

Figure 21 A schematic diagram of the RDS microcavity when the parameters

are fixed at φ = 5 and r = 085R

and one linear section [24 25 26] This shape of microcavity exhibits valuable

chaotic properties such as dynamical tunneling from an islands chain [24] a lasing

mode localized on higher periodic orbit [25] and robustness of lasing modes

localized on marginally unstable periodic orbits (MUPOs) [26] In this shape

depending on the positions and sizes of two smaller arcs such dynamical behaviors

are complicated as UPO MUPO islands chain complex manifold structure For

a certain parametric condition the microcavity is fully chaotic without MUPOs

above the critical line and its emission is unidirectional for valuable applications

The RDS microcavity we adopt is shown in Fig 21 The main circular arc

with a radius of R = 30 μm is tangentially connected to two smaller circular arcs

with a radius of r = 085timesR at an angle φ and a linear section is also tangentially

connected to the two smaller arcs for C1 continuity The direction of emission

can be controlled by tuning r and φ For optimal unidirectional emission the

angle is fixed at φ = 5 for r In this configuration the system is fully chaotic

without marginally unstable periodic orbits above the critical line

ndash 4 ndash

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

21 Ray dynamical analysis and unstable periodic or-

bits

First we obtain a survival probability distribution (SPD) for TE polarized

light as shown in Fig 22(a) For calculation the phase space (s p) is divided

into 103 times 103 cells for s isin [0 smax] and p isin [minus1 1] Here s is the boundary

length from the point θ = 0 smax the total boundary length and p = p0 sin(χ)

the tangential momentum of the ray for the incident angle χ at s with p0 equiv 1

Figure 22(a) following the schemes used in Ref [25] shows the unstable

manifolds of the chaotic saddle [27 28 29 30 31] as the complicated structure

in the chaotic sea In the structure there is no sign of MUPOs and island chains

Nevertheless the figure exhibits three tongues below the critical line (p lt 1ne)

A tongue around ssmax = 04 is the major emission window for the forward

direction (θ = 180) and the others around ssmax = 09 and 098 are the

minor emission windows for the backward direction (θ = 0) Because of the

symmetry of our cavity only counter-clockwise traveling rays are displayed A

tongue around ssmax = 04 is the main emission window for a unidirectional

emission and the others around ssmax = 09 and 098 are the minor emission

windows for the backward direction Similarly clockwise rays exhibit the main

and the minor emission gate around ssmax = 06 01 and 002 respectively

For a clear demonstration of UPOs we obtain a chaotic repeller distribution

(CRD) as shown in Fig 22(b) by following Ref [32] The CRD is a ray dynamical

representation for rays not to reach the leaky region (p lt 1ne) For CRD the

weight factor is set I = 0 when a ray reaches the leaky region and I = 1 when it

does not Then the average weight of each cell wc is given by

wc =1

2m

2msumi=1

Ii (21)

ndash 5 ndash

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

(a)

06

08

10

00

02

04

p=si

n

00 01 02 03 04 05 06 07 08 09 10ssmax

0

1

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(b) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 22 The survival probability distribution and the chaotic repeller distri-

bution of the RDS microcavity (a) is the survival probability distribution of

rays traveling in counter-clockwise direction in semi-logarithmic scale There are

three emission windows around ssmax = 04 09 098 marked by the dotted-

white circles (b)is the CRD exhibiting the p4 (purple) the p5 (yellow) and the

p6 (cyan) UPOs The UPOs touching the linear section and the UPOs related to

MUPOs are presented by filled circles and empty circles respectively

ndash 6 ndash

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

where 2m is the number of initial rays for the forward and the backward

direction from each cell Ii the weight factor for the i-th ray The distribution

of wc is very the CRD As is shown in Fig 22(b) the complicated pattern is a

fractal structure due to the long residence of rays In this CRD we mark the

UPOs touching the linear section as filled circles and the UPOs related to MUPOs

in our previous paper [25] as empty circles While the empty circles are on the

black region where the rays reside long time one of the filled circles are close

to the white region where the rays rapidly decay We note here that the rapid

decay is highly affected by the linear section at ssmax = 0

22 Wave analysis for the threshold pumping strength

To understand the wave dynamics in the RDS microcavity we obtain TE

polarized resonances from Re(nekR) = 97 sim 105 where ne is the effective re-

fractive index of 33 k the vacuum wavenumber and R the radius of the main

circular arc Now the resonances localized on the p4 the p5 and the p6 UPO

are obtained as shown in Fig 23(a) - (c) respectively by employing the bound-

ary element method [33] The eigenvalues nekR of the spatial wavefunctions

shown in Fig 23(a) - (c) are 9458 minus i841 times 10minus3 9854 minus i144 times 10minus4 and

9618minus i749times 10minus5 respectively Their Husimi functions [34] are superimposed

on the CRD as shown by the purple the yellow and the cyan filled contours

in Fig 23(d) The Husimi functions are well-localized on the UPOs related to

MUPOs

In our calculation it is hard to find a resonance touching the linear section

due to its rapidly decaying property Hence the resonances localized on the empty

circles are dominant The waves following the unstable manifolds and emitting

out through the tongues are not presented here because of the low intensity of

the Husimi functions except the UPOs

ndash 7 ndash

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

(a) (b) (c)

06

07

08

09

10

00 01 02 03 04 05 06 07 08 09 10

(d) p5 UPOp4 UPO p6 UPO

p=si

n

ssmax

0

1

Figure 23 Spatial wave functions of resonances and Husimi functions superim-

posed on the chaotic repeller distribution (a) (b) and (c) are the resonances

localized on the p4 the p5 and the p6 unstable periodic orbit respectively (d)

is Husimi functions of the resonances localized on the p4 (purple contour) the p5

(yellow contour) and the p6 unstable periodic orbit (cyan contour)

ndash 8 ndash

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

Figures 23(a)-(c) exhibit broadly distributed spatial patterns of wavefunc-

tions ψ(x y) Hence we need to consider the overlapped region between the

external pumping profile and the wavefunctions by using the optical Bloch equa-

tions In two-level atom model the rate equations are given by Ref [35]

parttN2 = minusγ2N2 + γ12N1 + λ2 + iασ12 minus αlowastσ21

2 (22)

parttN1 = minusγ1N1 + γ21N2 + λ1 minus iασ12 minus αlowastσ21

2 (23)

parttσ12 = minus(γperp + iω0)σ12 + iα(N2 minusN1)

2 (24)

where N is the population of each atomic states σ the coherence between the two

atomic states γ the relaxation rate ω0 the transition frequency λ the pumping

rate and α the Rabi frequency defined as α equiv minuser middot Eexth In the two-level laser

model internal relaxation processes are usually neglected and further assumed

that thw two lasing states relax to the external reservoir at the same rate of γ

Now the pumping rates are

λi =N0

i

γ (25)

where N0 is the population without the external pumping Here we deduce the

rate equations eqs (22) to (24) as the population difference d = N2 minusN1

parttd = minusγ(d0 minus d) + i(ασ12 minus αlowastσ21) (26)

parttσ12 = minus(γperp + iω0)σ12 + iαd2 (27)

where d0 is the population difference in the absence of external pumping that

is the external pumping parameter Assuming that α = αlowast = minuserEh = 2κE

ndash 9 ndash

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

where κ is the coupling strength defined by minuser2h we can rewrite the rate

equations as follows

parttd = minus2iκE(σ minus σlowast)minus γ(dminusWinfin) (28)

parttσ = minusiω0σ minus iκdE minus γperpσ (29)

where σ = σ12 = σlowast21 and Winfin equiv d0 is the external pumping parameter

Now the effective external pumping parameter 〈Winfin〉 is determined by the

effective pumping region By using the schemes in Ref [18 36] we deduce the

threshold pumping strength of resonances for a partial pumping region due to the

inward gap which is inevitable in laser fabrication By assuming that the gain

center is close enough to the resonances [18 36] the threshold pumping strength

Wth can be obtained as follows

Wth = minusn2eγ

2πρκ2h

Im(k)

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

(210)

where ρ is the carrier density κ the coupling strength γ the relaxation rate for

TE polarized light D the cavity region ψ the spatial wave function and Θ the

pumping characteristic function taking ldquo1rdquo inside the effective pumping region

and ldquo0rdquo otherwise Because C equiv n2eγ2πρκ

2h is constant for resonances we can

qualitatively elucidate the lowest threshold mode by calculating WthC

Table 21 is the threshold pumping strengths of the resonances localized on

the p4 the p5 and the p6 UPO shown in Figs 23(a) - (c) Here we consider two

cases of the inward gap of zero and 3 μm for R = 30 μm and λ = 155 μm by

considering InGaAsP semiconductor laser As is shown in table 21 while the

lowest threshold is the p6 UPO resonance for no inward gap that is the p5 one

for 3 μm inward gap

ndash 10 ndash

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

Table 21 The threshold pumping strengths of scarred modes localized on un-

stable periodic orbits

UPOWthC

with gap without gap

p4 11442times 10minus4 89651times 10minus5

p5 21366times 10minus6 14651times 10minus6

p6 22589times 10minus6 77899times 10minus7

23 Fabrication and experimental results

In order to confirm the above theoretical interpretation of the lowest thresh-

old pumping strength we fabricate the InGaAsP multi-quantum well semicon-

ductor RDS microcavity laser as shown in Figs 24(b) and (c) The fabricated

parameters of R = 30 μm r = 255 μm and φ = 5 are fixed for unidirec-

tional emission for the application to an optical source of telecommunication

purposes The inward gap is fixed at 3 μm for the lasing of p5 UPO resonance

For fabrication an InGaAsP separate confinement heterostructure with layers of

7 multi-quantum well (SCH-7-MQW) a p-doped InP cladding and a p-doped

InGaAs ohmic layer are etched into an RDS cross-section by inductively coupled

plasma etching The active region has the MQW structure consisting of seven

07 compressive-strained InGaAsP wells with a bandgap wavelength of 168 μm

and eight 06 tensile-strained barriers with a bandgap wavelength of 13 μm

to reach the threshold gain stably [37 38] The etch depth is 4 μm and the

active layer of SCH-7-MQW is placed at 2 μm from the top of the microcavity A

p-contact metal electrode is deposited through the lift-off process A SiO2 layer

is deposited on the outermost p-InGaAs ohmic layer for passivation and contact

region where the metal electrode contacts to the ohmic layer The fabricated

laser is shown in Fig 24(b) The outer edge of the contact region follows the

perimeter of the microcavity closely along the sidewall with a 3 μm inward gap

ndash 11 ndash

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

3 m

60 m

p-contact

electrode

= 0deg

(b)

(c)

5

50

(a)

n-InP Substrate(500 m)

Metal electrode

n-InP Buffer (03 m)

Metal contact

SCH-MQW Active

p-InP Clad (20 m)

p-InGaAs ohmic (02 m)Et

ch d

epth

(40

m

)

SiO2 Passivation

Figure 24 Schematic diagram of the semiconductor diode structure and SEM

images of fabricated RDS microcavity laser (a) is a schematic stack layer diagram

of an InP-based semiconductor laser along the vertical cross-section (b) is an

SEM image of the fabricated laser whose radius of main circular arc is R = 30

μm (c) is a zoomed SEM image of the square region in (b) showing an inward

gap of 3 μm

as shown in Fig 24(c)

The laser is excited with continuous wave current through the p-contact

metal electrode In a whole experiment we maintain the temperature at 20C

The laser emission is collected with a single-mode lensed fiber at 10 μm away

from the cavity boundary The laser output from the microcavity is distributed

to the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler and the optical power and the

emission spectra are measured in parallel Fig 25(a) shows the optical power

versus the injection current

The emission spectra near and above the threshold are taken as shown in

Fig 25(c) At 15 mA the spectrum clearly exhibits four lasing modes as shown in

the spectrum in Fig 25(c)(i) The lasing modes are periodic with an equidistant

mode spacing of about 372 nm As the current increases one of the lasing modes

ndash 12 ndash

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

(i)

(ii)

times

Figure 25 Emission intensity depending on the injection current and optical

spectra (a) is total output power depending on the injection current which

shows a slope change around 16 mA (b) is the peak intensity of the lasing mode

at λ sim 1556 nm depending on injection current which well exhibits the lasing

threshold of about 14 mA The upper and the lower figure in (c) are the optical

spectra at 15 and 17 mA respectively Lasing modes at 15mA are 154476

154844 155216 and 155588 nm which belong to a mode group supported by

a period-5 UPO

ndash 13 ndash

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

becomes dominant as shown in Fig 25(c)(ii) at 17 mA [39] The orbit length is

obtained to confirm the lasing modes by using the relation between free spectral

range and path length as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi+1 + λi)2]2

ngΔλi (211)

where ng is the group refractive index of about 368 [40] and λi and Δλi are

the wavelengths of lasing modes group and the mode spacing of two neighboring

lasing modes respectively For the four lasing modes the path length Lp is about

17620 μm which coincides with a p5 UPO of 17718 μm within 994 accuracy

As we explain the residing time of rays in above we believe the lasing mode group

is localized on the p5 UPO presented by empty yellow circles in the CRD

According to Eq 210 the threshold pumping strength depends on the spa-

tial wave function and the factor minusIm(k)Re(k) which is inversely proportional

to the quality factor In Tab 21 although the factor minusIm(k)Re(k) of p6 UPO

resonance is smaller than that of p5 one the threshold pumping strength of p6

UPO resonance is larger than that of p5 one for the 3 μm inward gap This result

is different from that of the threshold pumping strengths without the gap Hence

in our experimental condition p5 UPO resonance mode lase [18 36] This inter-

pretation can also apply to the RDS microcavity laser of R = 50 μm r = 05R

φ = 60 and an inward gap of 2 μm where the modes localized on a period-7

UPO lase [25]

To confirm the path length of the lasing modes we also numerically obtain

the mode spacing of resonances localized on p5 UPO In Fig 26 eight rectangular

boxes and eight triangles are the even and the odd resonances belonging to the

mode group localized on p5 UPO respectively Although the eigenvalues of a

parity pair are slightly different as shown in the inset A the real eigenvalues

of each parity group are regularly distributed with the same equidistant spacing

ndash 14 ndash

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

Re(nekR)

A

97 98 99 100 101 102 103 104-20

-15

-10

-05

00

05

10

times10-3

l

d(l)

B

0 5 10 15 20

10

20

Im(nekR)

Figure 26 Sixteen eigenvalues localized on the period-5 UPO around

Re(nekR) sim 100 The rectangular boxes and triangles are the even and the

odd parity modes respectively Inset A is the eigenvalues of a parity pair Inset

B is the spectral density

When we apply Fourier transformation to the resonances the spectral density can

be obtained for complex wavenumbers such that d(l) =sum

n exp(minusiknl) where

kn is the wavenumber of the n-th resonance Inset B in Fig 26 is the spectral

density showing a periodic signal The period corresponds to the path length of

17840 μm [41 42] which well coincides with the experimental and ray dynamical

path length

24 Discussion

241 Quality factor of the scarred mode

At the lasing threshold the current density is 04706 kAcm2 for 30 μm

radius RDS cavity In a microcavity laser the relation between the gain at the

lasing threshold and the quality factor is given by [37 43]

ΓGth = ΓgJth = αi + αs =2neπ

λcQ (212)

where Γ is the confinement factor based on the geometry of the cavity Gth

ndash 15 ndash

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

15556 15557

10

20

30

Inte

nsi

ty (

arb)

times10-8

15555 15558

Wavelength (nm)

FWHM =00532nm

Figure 27 The magnified spectrum of the dominant lasing peak at the threshold

of 14 mA The lasing wavelength of λc is 155565 nm and FWHM of w is 00532

nm

the gain at the lasing threshold g the gain coefficient Jth the threshold current

density αi and αs the internal loss and scattering loss ne the effective refractive

index of the cavity λc the lasing wavelength and Q the quality factor This re-

lation indicates that Q is inversely proportional to the threshold current density

Jth However although the threshold current density of our RDS microcavity

laser has a lower value compared to ones of Jth = 055 sim 200 kAcm2 in the

previous researches [44 40 9 10] it does not guarantee a high quality factor of

p5 UPO resonance because it also depends on many other parameters such as

non-radiative recombination amplification of the active medium output loss

In order to estimate more accurate quality factor we measure the full-width at

half-maximum (FWHM) at the threshold as shown in Fig 27 The quality fac-

tor is given by Q = λcw where w is FWHM The estimated quality factor of

the p5 scarred mode in our RDS microcavity laser is Q 3times 104

ndash 16 ndash

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

242 Unidirectional emission from a scarred mode lasing

Now we compare the directionality of the experiment ray dynamics and

wave dynamics For the far-field pattern (FFP) measurement in the experiment

the fiber is rotated along the cavity boundary at 470 μm away from the sidewall

The yellow curve in Fig 28 is the experimental FFP which shows unidirectional

emission with a divergence angle of about 30 Because of the symmetry of the

cavity shape the FFP in the region from θ = 0 to 180 is displayed The FFP

well coincides with the blue curve of the averaged FFP of the 16 resonances in

Fig 26 Also the experimental FFP well coincides with the red curve of the

ray dynamical FFP Their main emission direction around θ = 180 corresponds

to the emission window around ssmax = 04 in Fig 22(a) Despite the mi-

nor discrepancy of them around θ = 30 which originates from experimental

inaccuracy these results evidently indicate that the resonances localized on the

period-5 UPO emit unidirectionally with a narrow divergence angle To show

the coincidence more clearly we obtain the correlation coefficients among the

FFPs The correlation coefficients are 094 for between rays and resonances 085

for between experiments and resonances and 087 for between experiments and

rays This unidirectional emission guarantees the applicability of our laser to a

telecommunication purpose

25 Conclusion

In conclusion our RDS microcavity laser emits a lasing mode group localized

on a p5 UPO whose configuration is fully chaotic due to the different parametric

conditions of the position and radius of two smaller circular arcs In regard to the

pumping configuration the inward gap is the main factor to determine the lasing

mode group The lasing of p5 resonance mode is theoretically confirmed This

ndash 17 ndash

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

13

13

Figure 28 Far-field patterns of the RDS cavity obtained from ray dynamical

simulation (red) resonance modes (blue) and experiments (yellow) The FFP of

resonance modes is calculated by averaging the FFPs of the sixteen resonances

within Re(nekR) = 97 sim 105 Their main emission directions at θ = 180 are the

same The correlation coefficients of the three FFPs are 094 for between rays

and resonances 087 for between experiments and rays and 085 for between

experiments and resonances

phenomenon can give fabricating benefits because it does not require complicated

patterning for selective pumping to make specific mode lase

ndash 18 ndash

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

ndash 1 ndash9

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

III Lasing modes localized on marginally

unstable periodic orbits in a rounded

D-shape microcavity laser

To study the lowest threshold parameters according to the effective inward

pumping region we take a semiconductor microcavity defined by a rounded D-

shape (RDS) boundary morphology which has both of unstable periodic orbits

(UPOs) and marginally unstable periodic orbits (MUPOs) Here MUPOs exhibit

segmented-continuous curve structures originating from the invariant curves of

non-isolated periodic orbits in phase space of a circular cavity While the modes

localized on MUPOs slowly decay to leaky regions owing to the stickiness of

classical rays in the vicinity of them [29 28] the ones on UPOs decay quickly

This implies that modes localized on MUPOs are advantageous for lasing over

those localized on UPOs known as scarred modes [26] Through numerical and

experimental analyses we investigate lasing threshold conditions of resonance

modes localized on UPOs and MUPOs In numerical investigations we make a

prediction for the lowest threshold lasing modes which is deduced in terms of

the effective pumping region (ie partial excitation) in both ray dynamics and

resonance mode analysis To obtain the threshold condition in ray dynamics

we calculate the gain due to the effective pumping region and the total loss

consisting of internal and scattering losses Also in the resonance mode analysis

the threshold pumping strength is obtained by using the spatial wave function and

complex wavenumber of resonance modes This numerical prediction is clearly

confirmed in the experiments

ndash 20 ndash

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

31 Semiconductor RDS microcavity laser with an in-

ward gap

An RDS microcavity comprises three circular arcs and one linear section

as shown in Fig 21(a) The main circular arc with a radius of R = 30 μm is

tangentially connected to two smaller circular arcs with a radius of r = 05R at

an angle φ from the y-axis The linear section is tangentially connected to the

two smaller arcs We set the angle φ = 80 where the cavity has various MUPOs

and UPOs Our system is fabricated based on seven-layered multi-quantum-well

(MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH)

This core structure is interposed between the n-doped InP substrate at the bottom

layer with a buffer zone and by a p-doped InP cladding layer at the top layer

Finally a p-doped InGaAs ohmic layer is grown at the uppermost surface The

whole structure is illustrated in Fig 24(a)

The active layer consists seven pairs of a 08 compressively strained In-

GaAsP well (λg = 168 μm 72 A thick) and a 060 tensile strained InGaAsP

Rr

S

Figure 31 A schematic diagram of the rounded D-shaped microcavity with

φ = 80 and r = 05R which has a linear part highlighted in red between the

smaller circles

ndash 21 ndash

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

(a)

50

(b)

5

(c)

5

Active layer

Figure 32 SEM images of the fabricated RDS microcavity laser (a) is an SEM

image of a fabricated RDS microcavity laser (b) is a zoomed image of the red

square region in (a) showing the 3 μm inward gap between the contact region and

the sidewall (c) shows a vertical cross-section of RDS embedding an SCH-MQW

active layer marked by a red band Green bars in (b) and (c) represent the length

of λ

ndash 22 ndash

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

barrier (λg = 131 μm 105 A thick) and undoped InGaAsP separate confine-

ment layers (see Ref [45 46] for details) The laser was etched 4 μm deep into an

RDS cross-section by inductively coupled plasma etching The SiO2 passivation

layer is deposited on the outermost ohmic layer by using the PECVD to define

the current contact region where the electrode and ohmic layer adjoin The inner

boundary of this contact region was defined along the sidewall of the microcav-

ity with a 3 μm inward gap A p-contact TiAu metal electrode was deposited

through the lift-off process [47] to set the current contact layer The scanning

electron microscope (SEM) image of the full structure of our laser is shown in

Fig 32(a) Fig 32(b) shows the 3 μm gap between the sidewall and electrode

Although the electrode was slightly misaligned this misalignment does not affect

the experimental results because the electrode is large enough to cover the entire

contact region

32 Unstable periodic orbits in a ray dynamical sim-

ulation

An RDS cavity is a fully chaotic cavity which has only resonance modes

localized on UPOs such as MUPOs and UPOs To obtain the MUPOs and UPOs

we numerically simulate a ray dynamics in a phase space of the RDS microcavity

The phase space (q p) equiv (SSmax p0 sinχ) of our cavity is constructed by

the canonical conjugate variables S and p S is the boundary length from the

point θ = 0 Smax the total boundary length and p the tangential momentum

of the ray for the incident angle χ at S with the initial momentum of p0 equiv1 Because there is no recognizable stable periodic orbits the overall structure

of phase space follows a fully chaotic property To reveal MUPOs and UPOs

we numerically obtain the chaotic repeller distribution (CRD) [32] and survival

probability distribution (SPD) [48 28 29] as shown in Fig 33 through ray

ndash 23 ndash

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

Table 31 Path lengths of periodic orbits for our RDS microcavity

Periodic orbit 〈ΔkR〉 Lresp (μm) Lray

p (μm)

MUPO

p13 03665 15585 15588

p14 03352 17041 16971

p15 03246 17597 17634

p16 03195 17878 18000

UPOp14 03362 16990 16950

p15 03293 17346 17582

dynamical simulations For calculations of CRD we follow the schemes used in

Ref [32] MUPOs and UPOs above the critical line are observed and highlighted

by red-straight curves and green squares respectively MUPOs and UPOs in

Fig 33(a) are labeled by the number of rotations (r) and reflections (t) as follows

prt = p13 p14 p15 p16 MUPOs and p14 p15 UPOs

The emission properties of the RDS cavity are investigated by means of

the survival probability distributions for TE polarized rays applying the Fresnel

equations [48 28 29] SPD in a logarithmic scale in Fig 33(b) elucidates the

emission windows below the critical line through the unstable manifolds of chaotic

saddles [27 28 29 30 31] The figure shows three emission windows marked by

dotted circles around SSmax = 04 08 and 10 for the counter-clockwise (CCW)

traveling rays (p gt 0) Due to the mirror-reflection symmetry of RDS three

emission windows for clockwise (CW) traveling rays are located at SSmax = 00

02 and 06

From the CRD and SPD we can also compute the classical path length of

the MUPOs and UPOs which are shown in Tab 31 To estimate the scattering

loss of each periodic orbit it is useful to implement the survival probability time

distribution (SPTD) [49] of ray dynamical simulation Because the scattering

loss is usually expressed in decibels per centimeter (dBcm) our SPTD at time

t = lc where c is the speed of light is obtained as an averaged survival intensity

ndash 24 ndash

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

p=si

np=

sin

04

06

08

10

02

06

08

10

00

04

p14

p15

p16

p13

1ne

Figure 33 Ray dynamical simulation of the RDS cavity (a) is the chaotic re-

peller distribution which shows chaotic saddles containing marginally unstable

periodic orbits and unstable periodic orbits (b) is the survival probability distri-

bution in logarithmic scale There are three emission windows at SSmax = 04

08 10 indicated by the circles below the critical line

ndash 25 ndash

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

s 2759 dBcm(a)

s 1899 dBcm(b)

s 1612 dBcm(c)

s 1253 dBcm(d)

s 1631 dBcm(e)

s 1223 dBcm(f)

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-6

0

-3

-60 1 2 3 4 5 6

l (cm)

L I

(dB)

Figure 34 Semi-logarithmic plots of survival probability time distributions (dis-

crete symbols) and their linear fittings (straight curves) for each periodic orbit

(a) p13MUPO (b) p14MUPO (c) p14UPO (d) p15MUPO (e) p15UPO and

(f) p16MUPO From the slopes of the fitting curves the decaying exponent αs

with respect to the effective propagation time of the propagation length with a

constant velocity of rays is obtained

normalized by an initial intensity of I0 in decibel unit as follows

〈LI〉l equiv 10 log 〈INI0〉l = 10 log

⎛⎜⎝

1

N

Nsumn

rn

⎞⎟⎠

l

asymp minusαsl (31)

where I is the survival intensity given by the reflection probability r at the prop-

agating time l N is the number of rays taken into account for our computation

and αs is the decaying ratio giving rise to the scattering loss

αs = minusd 〈LI〉dl

(32)

Figure 34 shows SPTDs computed by N = 100 initial rays of I0 = 1 starting

from the vicinity of periodic orbits In the figure the linear fitting curves of

ndash 26 ndash

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

SPTDs deduce the scattering loss αs It is important to note that even a small

roughness of cavity boundaries can affect the lasing of modes [50] However as

is shown in Fig 41(b) and (c) the surface roughness of our microcavity is much

tiny compared to the wavelength emitted from our microcavity so that we can

rule out the scattering loss factor provoked by the surface roughness in our mod-

eling

33 Resonances localized on MUPOs and UPOs

Resonance modes in our RDS cavity are obtained by using the boundary

element method [33] in the region Re(nekR) sim 396 which matches the exper-

imental one Here ne k and R denote the effective refractive index of 33

the vacuum wavenumber and the radius of the main circle in Fig 31 respec-

tively Wavenumbers of resonance mode groups localized on MUPOs and UPOs

are shown in Fig 35(a) The regular interval Re(ΔkR) of each mode group

identified in Fig 35 corresponds to a periodic orbit in the sense of the relation

Lresp R = 2πneΔkR where Lres

p is the periodic orbit length The average in-

tervals 〈ΔkR〉 of the resonance mode groups and corresponding path lengths are

shown in Table 31 for R = 30 μm This result agrees well with the classical path

lengths Lrayp obtained by ray dynamical calculations ie geometries of periodic

orbits By exemplifying representative highest-Q modes which are marked by

dotted-red squares in Fig 35(a) it is confirmed that spatial wave patterns and

their Husimi distributions are well-matched with the periodic orbits inferred as

are shown in Fig 35(b) [33 34]

ndash 27 ndash

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

(a)

(b)119 1195 120 1205 121

Re( )kR

Im(

)

kR

-50

-45

-40

-35

-30

Figure 35 Passive resonance modes for the RDS cavity (a) is the eigenvalue

distribution of p13 (blue triangle) p14 (cyan square) p15 (red pentagram) and

p16 MUPO (black circle) and p14 (green diamond) and p15 UPO (purple hex-

agram) in the region nekR sim 396 The eigenvalues of representative modes

marked with dotted-red boxes are 1202018 minus i38006 times 10minus5 for p13 MUPO

1190782 minus i34248 times 10minus5 for p14 MUPO 1197007 minus i32675 times 10minus5 for p15

MUPO 1197075minus i31429times 10minus5 for p16 MUPO 1198308minus i34238times 10minus5 for

p14 UPO and 1191448 minus i31848 times 10minus5 for p15 UPO (b) is the spatial wave

patterns and their Husimi distributions of representative modes We overlap

the Husimi distribution on the periodic orbits in CRD to make clear the mode

localization

ndash 28 ndash

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

34 Linear gain condition with a single-mode approx-

imation

Now we discuss the robustness of the lasing of resonance modes supported

by MUPOs and UPOs in terms of the lowest threshold conditions The general

lasing condition for a semiconductor laser is expressed by the following inequality

G gt αi + αs = αi +1

Lplog

1

R (33)

where G is the gain αi the internal loss αs the scattering loss Lp the orbit

path length of a mode and R the survival ratio per cycle This inequality simply

implies that when the gain of certain modes in the active medium is greater

than the total loss of the modes the mode can lase above the threshold ie

G = αi + αs From now on we will use the single-mode approximation and

disregard modal interactions to simplify our problems in analyzing a lasing mode

at the lowest threshold condition of the RDS microcavity laser [18 36] Note

that strictly speaking the full version of the MaxwellndashBloch equations should

be solved numerically taking nonlinear modal interactions [51 36] into account

to investigate the general lasing modes

341 The threshold condition depending on the classical path of

the periodic orbits

Because an inward gap exists between the contact region and the sidewall

of our microcavity laser the effective pumping region is limited by the contact

region and only partial trajectories of periodic orbits are overlapped with this

effective pumping region To reflect this partial pumping condition in our deriva-

tions we define an effective excited length Le then the gain in the threshold

ndash 29 ndash

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

condition can be modified as follows

Gprimeth =

Le

LptimesGth = Γg

Le

Lp(Jth minus J0) = αi + αs (34)

where Le is the excited length Lp the path length Gth the gain at the thresh-

old Γ the confinement factor g the gain coefficient Jth the threshold current

density and J0 the transparency current density The transparency current and

the internal loss can be assumed to be constant for the same sample because

they depend solely on material compositions of InGaAsP SCH-MQW gain layer

In other words the scattering loss is the only functional factor to determine the

threshold current density for each periodic orbit Under this assumption the

threshold current density can be expressed as follows [52 43]

Jth = J0 +LpLe

Γg(αi + αs) = J0 + C1pc (35)

where C1 equiv 1Γg is a constant and pc equiv (αi+αs)LpLe is the threshold parame-

ter Note that the parameter pc depends both on the trajectories of the periodic

orbits and the effective pumping region

342 The threshold condition depending on the spatial wave

function localized on the periodic orbits

Unlike the classical path of the periodic orbits wave functions ψ(x y) of

resonance modes are broadly distributed though they are localized mainly along

the classical periodic orbits as shown in Fig 35(b) Hence based on the schemes

used in Ref [18] we can derive the threshold condition of lasing modes by further

assuming that the real part of a wavenumber (frequency) of a mode is almost the

same as the center of the gain spectrum (or the gain spectrum is broad enough to

cover the resonance modes that we obtained) [18 36] Subsequently the threshold

ndash 30 ndash

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

lg ( m)

(a) (b)

4 3 2 1 0

p c

6 5 4 3 2 1 0

p w

10-4

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

10-5

(13) MUPO

(14) MUPO(14) UPO

(15) MUPO(15) UPO

(16) MUPO

40

30

20

10

20 1015 05

8

10

12

101520

26

27

28

25

20

15

10

05

35

25

15

30

Figure 36 Threshold parameters depending on the length of the inward gap

between the sidewall and the effective pumping region at the gain layer (a) and

(b) are threshold parameters of pc and pw respectively which are defined by

the classical excited length and the ratio of the sum of wave function inside the

cavity and the sum of the overlapped wave function Both results show a range

of 07 μm lt lg lt 18 μm for the lowest threshold parameter of p15 MUPO The

insets are the magnified graphs which show the lowest threshold parameters of

p15 MUPO

ndash 31 ndash

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

pumping strength Wth is expressed by

Wth =n2eγ

2πρκ2h

minus Im(k) + β

Re(k)

intD dxdy|ψ(x y)|2

intD dxdy|ψ(x y)|2Θ(x y)

= C2pw (36)

where ρ is the carrier density κ the coupling strength γ the relaxation rate

for TE polarized light k the resonant wavenumber for a passive cavity β the

background absorption D the cavity region ψ the spatial wave function and Θ

the pumping characteristic function which takes rdquo1rdquo inside the effective pumping

region and rdquo0rdquo otherwise Here β can be expressed by the internal loss αi as

β = ηαi2ne where η is the scale factor of 3 times 10minus5 in this case The above

equation can be simplified by reducing the constant as C2 equiv n2eγ2πρκ

2h and by

defining the threshold parameter as

pw equiv [(minusIm(k) + ηαi2ne)

intD dxdy|ψ(x y)|2] [Re(k) intD dxdy|ψ(x y)|2Θ(x y)

]

Obviously the main factor to determine the threshold pumping strength Wth

is the threshold parameter pw which depends on the fraction of wave functions

overlapped with the effective pumping region This effective pumping region

is controlled by adjusting the profiles of the function Θ(x y) in Eq (36) for

theoretical prediction Figure 36 shows the parameters pc in Eq (35) and pw in

Eq (36) as a function of the ldquoeffective inward gaprdquo equiv lg distance between the

effective pumping region and the sidewall In obtaining pc and pw in Fig 36

we exemplify the case of αi = 68 cmminus1 which is reported in Ref[53] for the

eight-quantum-well structures

The mode which has the lowest threshold parameter can be the first-

appearing lasing mode for each lg In the range of 07 μm lt lg lt 18 μm

both parameters of pc and pw for p15 MUPO exhibit the lowest values compared

with the ones of other periodic orbits

ndash 32 ndash

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

343 The effective pumping region due to the fabricating con-

figuration

According to the analyses in the previous section the mode group with

the lowest threshold parameter might be the firstly appearing lasing mode for a

certain value of lg because the threshold parameters pc and pw are proportional to

the threshold pumping Figure 36 shows the parameters pc and pw corresponding

to the length of the inward gap lg between the pumping region and the cavity

boundary Now we consider the true effective pumping region and in turn

the effective inward gap lg taking into account the realistic parameters which

are used in our experiments The effective pumping region inside the cavity is

well defined as shown in Fig 37 by considering Ohmrsquos law and the continuity

equation in Eq 37 for drift current

J = qρμE middot middot middotOhmrsquos law

I(x) = Aeff(x)J(x) = I0 middot middot middot Continuity equation

(37)

Here J is the current density q the charge of an electron μ the mobility of

the semiconductor E the electric field x the depth variable from the top I(x) the

current at x Aeff(x) the area of the effective pumping region at x J(x) the current

density at x and I0 the injection current [54 55 56] In the procedures the

effects of small amount of direct diffusion current is neglected Further assuming

that the electric field inside a semiconductor drift region decreases linearly as

the depth increases from the top electrode to the bottom ground and that the

effective pumping region has a discrete profile instead of a Gaussian profile Now

the effective pumping region Aeff is expressed by

Aeff =I0

qρμ

1

(Et minus Eb)(1minus xd) + Eb=

Ac

1 + (AcAe minus 1)(1minus xd) (38)

ndash 33 ndash

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

Figure 37 A schematic diagram for the effective pumping region

where Et the electric field at the top electrode Eb the electric field at the bottom

of the cavity d the etch depth of the microcavity Ac the area of the cavity and Ae

the area of the contact region Because the active layer ie the multi-quantum

wells of fabricated cavities [see Fig 24(a) and Fig 32(c)] was placed at a depth

of 22 μm from the top the effective inward gap lg between the effective pumping

region and the sidewall is obtained as 177 μm although the physical inward gap

between the contact region and the sidewall is 3 μm for our RDS microcavity laser

with R = 30 μm The effective pumping region coincides well with the result of

the threshold condition This result implies that indeed the lasing modes localized

on p15 MUPO firstly appear at the lowest threshold injection current compared

with the others in our experimental condition

ndash 34 ndash

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

35 Experimental results and emission characteristics

We experimentally investigate the first lasing mode to verify our theoreti-

cal prediction Our RDS laser is uniformly pumped through the electrode with

continuous-wave injection current at the temperature of 20C In order to take

the maximized laser output we align a single-mode lensed fiber at θ = 150

which is the predetermined direction fixed by preliminary experiments The laser

output is collected 10 μm away from the sidewall for the measurement of out-

put power and emission spectra Then the collected laser output is split with a

5050 optical coupler and equally distributed to an optical power meter (Newport

818-IR) and an optical spectrum analyzer (Agilent 86142B) A far-field pattern

(FFP) is measured 470 μm away from the sidewall

Figure 38(a) shows the total output power and major peak intensity versus

injection current From the total output power (red) in Fig 38(a) we can identify

the lasing threshold near 44 mA To see the threshold of the first lasing mode

the peak intensity of the first main lasing mode is measured as a function of the

injection current (blue) and it is revealed that the threshold of this mode is almost

the same as the one of the total output power as 44 mA Because the sum of the

internal and scattering losses is inversely proportional to the quality factor (= Q)

of modes [37 43] we estimate it to obtain the total loss at the lasing threshold

By measuring a center wavelength λc sim 1551 nm of the major peak and its

full width at half maximum w = 0077 nm [see Fig 38(b)] it is obtained that

Q = λcw asymp 20times 104 Now the internal loss αi can be extracted by subtracting

the scattering losses of p15MUPO αs = 1253 cmminus1 [see Fig 34(d)] from the

total loss deduced from the experimental quality factor [37 43] as follows

αi + αs =2πne

λQasymp 736 cmminus1 rarr αi sim 61 cmminus1 (39)

ndash 35 ndash

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

This value of the internal loss is consistent with the previous results of semicon-

ductor lasers [53 57 58] as well as the one used in Fig 36

Figure 39(a) shows the spectrum at the threshold current where the promi-

nent lasing peak is not noticeable Above the threshold a main lasing mode

group (Group 1) is observed at 47 mA as shown in Fig 39(b) which lases at

154740 155114 155593 nm Also we can find another mode group (Group

2) at 154790 155157 155547 nm The path length of each mode group can

be calculated using the relation between the path length and the free spectral

Current injection (mA)

(a)

5

010 30 50 7020 40 60 80

10

15

20

Outp

ut

(arb

)

05

00

10

15

20

Pea

k i

nte

nsi

ty (

arb)

25

30

(b)

Wavelength (nm)15510 15511 15512

02

03

04

05

Inte

nsi

ty (

arb)

06

wc

Figure 38 Experimental results of the RDS semiconductor microcavity laser (a)

shows the total output power (red) and major peak intensity (blue) as functions

of injection current A vertical dotted line shows a threshold of lasing at 44 mA

(b) shows the line width w of the major peak λ = λc (black) fitted by Gaussian

curve (red) at the threshold current

ndash 36 ndash

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

(a)

(b)

(c)

10

1

1545 1550 1555 1560

10

1

10

1

1 2

Group 1

Group 2

Figure 39 The spectra of the RDS semiconductor microcavity laser (a) (b) and

(c) are spectra measured at 44 mA 47 mA and 51 mA respectively Two mode

groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green

lines (Group 2) which have the equidistant mode spacing of Δλ1 and Δλ2

range as follows

Lp =1

N minus 1

Nminus1sumi=1

[(λi + λi+1)2]2

ngΔλ (310)

where N is the number of peaks in the mode group ng the group refractive index

of 368 [40] λi the wavelength of the i-th peak in the mode group and Δλ the

free spectral range The resulting path lengths are 17366 μm for the first lasing

mode group and 17299 μm for the other mode group these values agree well

with the path length of p15 MUPO and p15 UPO respectively with an accuracy

above 98

As a final remark FFPs of the RDS microcavity laser obtained experimen-

tally and numerically are compared Figures 310(a)-(c) show FFPs of the ray

dynamical simulation the resonance modes localized on p15 MUPO and the

experiments respectively All FFPs clearly reveal three emission directions to-

ndash 37 ndash

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

(a)

(b)

(c)

0 30 60 90 120 150 18001

04

071001

04

071001

04

0710

Figure 310 The FFPs of the RDS microcavity laser (a) is the FFP from the

ray dynamical simulation (g) is the averaged FFP from the resonance modes

localized on p15 MUPO and (h) is the FFP from the experiments All of the

far-field patterns indicate three emission directions around θ = 10 70 and

150

ward sim 150 sim 70 and sim 10 with the strongest emission at θ = 150 These

emission directions originate from the emission windows of SSmax = 06 (CW)

10 (CCW) and 02 (CW) respectively Although the emission directions of the

FFPs from the ray and wave simulation are slightly different the nice overall

agreement among the three FFPs indicates that the first lasing mode in our RDS

microcavity laser is the mode localized on p15 MUPO

ndash 38 ndash

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

36 Conclusion

We have studied a rounded D-shape semiconductor microcavity laser excited

by the continuous-wave injection current Based on the fundamental linear gain

condition with a single-mode lasing approximation we predict that the modes

localized on the pentagonal-shaped marginally unstable periodic orbit has the

lowest threshold for lasing in our rounded D-shape microcavity laser which has

a 3 μm inward gap between the contact region and the sidewall This prediction

of the lowest threshold condition for lasing has been transparently verified in

experiments The path length deduced from ray dynamics resonance modes

and experiments agreed well with the length of the pentagonal-shaped unstable

and marginally unstable periodic orbits Nice agreement of emission directions

toward θ sim 150 sim 70 and sim 10 is obtained for all results of ray dynamics

resonance modes and experiments The results of the present work clearly reveal

the fact that the lasing thresholds of the resonance modes are decisively affected

by the effective pumping region and cavity morphology In the same context

the lasing of scars reported in our previous study [25] can be explained without

contradictions

ndash 39 ndash

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

IV Methods

41 Fabrication of the two dimensional semiconductor

microcavity laser

The fabrication process of the semiconductor microcavity laser is consists

of 15 steps as shown in Fig 41 First we deposit the SiO2 layer by using a

plasma-enhanced chemical vapor deposition (PECVD) Using photolithography

A photoresist (PR) patterning mask is constructed on the SiO2 layer by using

the photolithography and then the SiO2 layer is etched to make the cavity region

by using a deep reactive ion etching (DRIE) The two-dimensional microcavities

are obtained through an inductively coupled plasma etching (ICPE) using SiO2

layer as a mask for ICPE Second to make a contact region and p-electrode for

current pumping a thin SiO2 or Al2O3 passivation layer is deposited by using

the PECVD (sim 10 nm) This passivation layer is used to avoid surface current

and to prevent the deposition of a metal electrode on the side-wall To make

a contact region we open the center of the passivation layer with an inward

gap by using photolithography and DRIE PR is coated on the passivation layer

and the metal electrode region is defined by photolithography A Ti-Au metal

electrode is deposited on the PR layer and contact region by using the electron

beam evaporator Through the lift-off process the metal electrode is formed on

the top of the microcavity Finally fill the PR to fix all the microcavities on the

wafer and grind the bottom surface to make the substrate has a thickness of 500

μm Then an n-electrode is formed on the bottom surface using an electron beam

evaporator After removing PR heat treatment is performed with a furnace to

bond the electrodes and the microcavities

ndash 40 ndash

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

InP

SiO

2

Photo

resi

st

Ti-

Au

Pas

sivat

ion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

4 i

nch

waf

er

4

m

500

m500

m

3

m

Figure

41Fabricationprocess

forsemiconductormicrocavitylaser

(a)A

bare

InP

wafer(b)SiO

2dep

osition(c)PR

coating

(d)Microcavityshaped

PR

mask

patterning(e)SiO

2etching(f)IC

Petching(g)SiO

2passivationlayerdep

osition(h)PR

patterningforcontact

region(i)SiO

2etchingforcontact

region(j)PR

patterningformetalelectrode

(k)p-contact

electrode

dep

ositionandpatterningthroughthelift-off

process

(l)PR

coatingforprotection

(m)Substrate

polishing

(n)n-contact

electrodedep

osition(o)PR

removingan

dheattreatm

entforcompletion

ndash 41 ndash

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

42 Experimental set-up configuration

The schematic diagram for our microcavity laser measurement set-up is

shown in Fig 42 To inject current into the semiconductor microcavity a sharp

probe tip is used for p-contact of the top electrode The sample stage is made

of brass for n-contact at the bottom electrode The current source (LDC-3724C)

for CW current is connected to the probe tip and the stage for current injection

Also the temperature controller (TEC) in LDC-3724C is connected to a temper-

ature sensor and a Peltier element in the sample stage to measure temperature

and maintain constant temperature of 20C To collect the output laser beam

we set a lensed single-mode fiber which has a divergence angle of 70 at the front

of the microcavity laser We fix the distance between the lensed fiber and the

side-wall of the microcavity as 10 μm for L-I curve and spectra measurement and

470 μm for far-field measurement The collected laser output is distributed to

the optical power detector (Newport 818-IR) and the optical spectrum analyzer

(Agilent 86142B) by using a 5050 optical coupler for simultaneous measurement

of the spectrum and the output power For the FFP measurement a rotational

stage controlled by a stepping motor is mounted under the sample stage All set-

ups are installed in a closed space to minimize experimental error factors and all

devices in the experiment are connected to a computer so all the experimental

process is automatically controlled without manual operation

ndash 42 ndash

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

Figure

42Set-upformeasurements

ofthesemiconductormicrocavitylaser

ndash 43 ndash

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

References

[1] D K Armani T J Kippenberg S M Spillane and K J Vahala

ldquoUltra-high-Q toroid microcavity on a chiprdquo Nature vol 421 no 6926

p 925 2003

[2] A M Armani R P Kulkarni S E Fraser R C Flagan and K J Vahala

ldquoLabel-free single-molecule detection with optical microcavitiesrdquo Science

vol 317 no 5839 pp 783ndash787 2007

[3] F Vollmer and S Arnold ldquoWhispering-gallery-mode biosensing label-free

detection down to single moleculesrdquo Nature methods vol 5 no 7 p 591

2008

[4] T Lu H Lee T Chen S Herchak J-H Kim S E Fraser R C Flagan

and K Vahala ldquoHigh sensitivity nanoparticle detection using optical micro-

cavitiesrdquo Proceedings of the National Academy of Sciences vol 108 no 15

pp 5976ndash5979 2011

[5] L He S K Ozdemir J Zhu W Kim and L Yang ldquoDetecting single viruses

and nanoparticles using whispering gallery microlasersrdquo Nature Nanotech-

nology vol 6 no 7 p 428 2011

[6] S Sunada and T Harayama ldquoDesign of resonant microcavities application

to optical gyroscopesrdquo Opt Express vol 15 pp 16245ndash16254 Nov 2007

[7] J Li M-G Suh and K Vahala ldquoMicroresonator brillouin gyroscoperdquo Op-

tica vol 4 pp 346ndash348 Mar 2017

ndash 44 ndash

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

[8] J Wiersig and M Hentschel ldquoUnidirectional light emission from high-Q

modes in optical microcavitiesrdquo Phys Rev A vol 73 p 031802 Mar 2006

[9] C-H Yi M-W Kim and C-M Kim ldquoLasing characteristics of a limacon-

shaped microcavity laserrdquo Applied Physics Letters vol 95 no 14 p 141107

2009

[10] C Yan Q J Wang L Diehl M Hentschel J Wiersig N Yu C Pflugl

F Capasso M A Belkin T Edamura M Yamanishi and H Kan ldquoDi-

rectional emission and universal far-field behavior from semiconductor lasers

with limacon-shaped microcavityrdquo Applied Physics Letters vol 94 no 25

p 251101 2009

[11] J-W Ryu and M Hentschel ldquoDesigning coupled microcavity lasers for high-

Q modes with unidirectional light emissionrdquo Opt Lett vol 36 pp 1116ndash

1118 Apr 2011

[12] G D Chern H E Tureci A D Stone R K Chang M Kneissl and N M

Johnson ldquoUnidirectional lasing from InGaN multiple-quantum-well spiral-

shaped micropillarsrdquo Applied Physics Letters vol 83 no 9 pp 1710ndash1712

2003

[13] C-M Kim J Cho J Lee S Rim S H Lee K R Oh and J H Kim

ldquoContinuous wave operation of a spiral-shaped microcavity laserrdquo Applied

Physics Letters vol 92 no 13 p 131110 2008

[14] M S Kurdoglyan S-Y Lee S Rim and C-M Kim ldquoUnidirectional lasing

from a microcavity with a rounded isosceles triangle shaperdquo Opt Lett

vol 29 pp 2758ndash2760 Dec 2004

ndash 45 ndash

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

[15] I-G Lee S-M Go J-H Ryu C-H Yi S-B Kim K R Oh and C-M

Kim ldquoUnidirectional emission from a cardioid-shaped microcavity laserrdquo

Optics Express vol 24 no 3 pp 2253ndash2258 2016

[16] J-W Lee C-H Yi I-G Lee J-H Kim H-H Yu K-R Oh and C-M

Kim ldquoExtremely high Q and unidirectional laser emission due to combi-

nation of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a

dielectric microdiskrdquo Optics Letter vol 43 pp 6097ndash6100 Dec 2018

[17] J Ryu J-W Lee C-H Yi J-H Kim I-G Lee H-S Kim S-B Kim

K R Oh and C-M Kim ldquoChirality of a resonance in the absence of

backscatteringsrdquo Optics Express vol 25 pp 3381ndash3386 Feb 2017

[18] Y Kawashima S Shinohara S Sunada and T Harayama ldquoSelf-adjustment

of a nonlinear lasing mode to a pumped area in a two-dimensional micro-

cavityrdquo Photon Res vol 5 pp B47ndashB53 Dec 2017

[19] J U Nockel and A D Stone ldquoRay and wave chaos in asymmetric resonant

optical cavitiesrdquo Nature vol 385 no 6611 p 45 1997

[20] C Gmachl F Capasso E E Narimanov J U Nockel A D Stone J Faist

D L Sivco and A Y Cho ldquoHigh-power directional emission from micro-

lasers with chaotic resonatorsrdquo Science vol 280 no 5369 pp 1556ndash1564

1998

[21] Q J Wang C Yan N Yu J Unterhinninghofen J Wiersig C Pflugl

L Diehl T Edamura M Yamanishi H Kan and F Capasso ldquoWhispering-

gallery mode resonators for highly unidirectional laser actionrdquo Proceedings

of the National Academy of Sciences vol 107 no 52 pp 22407ndash22412 2010

[22] T Harayama and S Shinohara ldquoTwo-dimensional microcavity lasersrdquo Laser

amp Photonics Reviews vol 5 no 2 pp 247ndash271 2011

ndash 46 ndash

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

[23] H Cao and J Wiersig ldquoDielectric microcavities Model systems for wave

chaos and non-Hermitian physicsrdquo Reviews of Modern Physics vol 87 no 1

p 61 2015

[24] M-W Kim K-W Park C-H Yi and C-M Kim ldquoDirectional and low-

divergence emission in a rounded half-moon shaped microcavityrdquo Applied

Physics Letters vol 98 no 24 p 241110 2011

[25] J-W Lee C-H Yi M-W Kim J Ryu K-R Oh and C-M Kim ldquoUni-

directional emission of high-Q scarred modes in a rounded D-shape micro-

cavityrdquo Optics Express vol 26 no 26 pp 34864ndash34871 2018

[26] C-H Yi J-W Lee J Ryu J-H Kim H-H Yu S Gwak K-R Oh

J Wiersig and C-M Kim ldquoRobust lasing of modes localized on marginally

unstable periodic orbitsrdquo Phys Rev A vol 101 p 053809 May 2020

[27] J Wiersig and M Hentschel ldquoCombining directional light output and ul-

tralow loss in deformed microdisksrdquo Physical Review Letters vol 100 no 3

p 033901 2008

[28] E G Altmann ldquoEmission from dielectric cavities in terms of invariant sets

of the chaotic ray dynamicsrdquo Physical Review A vol 79 no 1 p 013830

2009

[29] E G Altmann J S E Portela and T Tel ldquoLeaking chaotic systemsrdquo

Rev Mod Phys vol 85 pp 869ndash918 May 2013

[30] H G L Schwefel N B Rex H E Tureci R K Chang A D Stone

T Ben-Messaoud and J Zyss ldquoDramatic shape sensitivity of directional

emission patterns from similarly deformed cylindrical polymer lasersrdquo J

Opt Soc Am B vol 21 pp 923ndash934 May 2004

ndash 47 ndash

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

[31] J Kullig and J Wiersig ldquoFrobeniusndashperron eigenstates in deformed mi-

crodisk cavities non-hermitian physics and asymmetric backscattering in

ray dynamicsrdquo New Journal of Physics vol 18 no 1 p 015005 2016

[32] J Wiersig and J Main ldquoFractal weyl law for chaotic microcavities Fres-

nelrsquos laws imply multifractal scatteringrdquo Physical Review E vol 77 no 3

p 036205 2008

[33] J Wiersig ldquoBoundary element method for resonances in dielectric micro-

cavitiesrdquo Journal of Optics A Pure and Applied Optics vol 5 no 1 p 53

2002

[34] M Hentschel H Schomerus and R Schubert ldquoHusimi functions at di-

electric interfaces Inside-outside duality for optical systems and beyondrdquo

Europhysics Letters vol 62 pp 636ndash642 jun 2003

[35] G J d Valcarcel E Roldan and F Prati ldquoSemiclassical theory of amplifi-

cation and lasingrdquo Revista mexicana de fısica E vol 52 no 2 pp 198ndash214

2006

[36] T Harayama S Sunada and K S Ikeda ldquoTheory of two-dimensional mi-

crocavity lasersrdquo Physical Review A vol 72 no 1 p 013803 2005

[37] A Yariv Quantum electronics Wiley 1989

[38] C Weisbuch C Vinter and B Vinter Quantum Semiconductor Structures

Fundamentals and Applications Elsevier Science 1991

[39] S Sunada S Shinohara T Fukushima and T Harayama ldquoSignature of

wave chaos in spectral characteristics of microcavity lasersrdquo Physical Review

Letters vol 116 no 20 p 203903 2016

ndash 48 ndash

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

[40] C-H Yi S H Lee M-W Kim J Cho J Lee S-Y Lee J Wiersig and

C-M Kim ldquoLight emission of a scarlike mode with assistance of quasiperi-

odicityrdquo Physical Review A vol 84 no 4 p 041803 2011

[41] M Lebental N Djellali C Arnaud J-S Lauret J Zyss R Dubertrand

C Schmit and E Bogomolny ldquoInferring periodic orbits from spectra of

simply shaped microlasersrdquo Physical Review A vol 76 no 2 p 023830

2007

[42] E Bogomolny N Djellali R Dubertrand I Gozhyk M Lebental

C Schmit C Ulysse and J Zyss ldquoTrace formula for dielectric cavities ii

regular pseudointegrable and chaotic examplesrdquo Physical Review E vol 83

no 3 p 036208 2011

[43] J Faist C Gmachl M Striccoli C Sirtori F Capasso D L Sivco and

A Y Cho ldquoQuantum cascade disk lasersrdquo Applied Physics Letters vol 69

no 17 pp 2456ndash2458 1996

[44] L Zhang and E Hu ldquoLasing from InGaAs quantum dots in an injection

microdiskrdquo Applied Physics Letters vol 82 no 3 pp 319ndash321 2003

[45] S H Oh C-W Lee J-M Lee K S Kim H Ko S Park and M-H Park

ldquoThe design and the fabrication of monolithically integrated GaInAsP MQW

laser with butt-coupled waveguiderdquo IEEE Photonics Technology Letters

vol 15 no 10 pp 1339ndash1341 2003

[46] S H Oh J-U Shin Y-J Park S Park S-B Kim H-K Sung Y-S Beak

and K-R Oh ldquoFabrication of WDM-PON OLT source using external cavity

laserrdquo in COIN-NGNCON 2006 - The Joint International Conference on

Optical Internet and Next Generation Network pp 217ndash219 July 2006

ndash 49 ndash

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

[47] M Hatzakis B J Canavello and J M Shaw ldquoSingle-step optical lift-off

processrdquo IBM Journal of Research and Development vol 24 pp 452ndash460

July 1980

[48] S-Y Lee S Rim J-W Ryu T-Y Kwon M Choi and C-M Kim

ldquoQuasiscarred resonances in a spiral-shaped microcavityrdquo Phys Rev Lett

vol 93 p 164102 Oct 2004

[49] J-W Ryu S-Y Lee C-M Kim and Y-J Park ldquoSurvival probability

time distribution in dielectric cavitiesrdquo Phys Rev E vol 73 p 036207

Mar 2006

[50] A I Rahachou and I V Zozoulenko ldquoEffects of boundary roughness on a

Q factor of whispering-gallery-mode lasing microdisk cavitiesrdquo Journal of

applied physics vol 94 no 12 pp 7929ndash7931 2003

[51] A Cerjan Y Chong L Ge and A D Stone ldquoSteady-state ab initio laser

theory for n-level lasersrdquo Optics express vol 20 no 1 pp 474ndash488 2012

[52] T R Chen L E Eng Y H Zhuang and A Yariv ldquoExperimental determi-

nation of transparency current density and estimation of the threshold cur-

rent of semiconductor quantum well lasersrdquo Applied Physics Letters vol 56

no 11 pp 1002ndash1004 1990

[53] T Namegaya N Matsumoto N Yamanaka N Iwai H Nakayama and

A Kasukawa ldquoEffects of well number in 13-μm GaInAsPInP GRIN-SCH

strained-layer quantum-well lasersrdquo IEEE Journal of Quantum Electronics

vol 30 no 2 pp 578ndash584 1994

[54] P Aloısi Power MOSFET Transistors ch 1 pp 1ndash56 John Wiley amp Sons

Ltd 2010

ndash 50 ndash

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

[55] V S Pereira and S Blawid ldquoDrift-diffusion simulation of leakage currents in

unintentionally doped organic semiconductors with non-uniform interfacesrdquo

Journal of Computational Electronics vol 18 pp 120ndash129 Mar 2019

[56] Y Sun X Kang Y Zheng J Lu X Tian K Wei H Wu W Wang

X Liu and G Zhang ldquoReview of the recent progress on GaN-based vertical

power Schottky barrier diodes (SBDs)rdquo Electronics vol 8 no 5 p 575

2019

[57] D Garbuzov L Xu S R Forrest R Menna R Martinelli and J C Con-

nolly ldquo15 μm wavelength SCH-MQW InGaAsPInP broadened-waveguide

laser diodes with low internal loss and high output powerrdquo Electronics Let-

ters vol 32 pp 1717ndash1719(2) August 1996

[58] L Xu D Garbuzov S Forrest R Menna R Martinelli and J Connolly

ldquoVery low internal loss 15 μm wavelength SCH-MQW InGaAsPInP laser

diodes with broadened-waveguidesrdquo in Conference Proceedings LEOSrsquo96 9th

Annual Meeting IEEE Lasers and Electro-Optics Society vol 1 pp 352ndash353

vol1 1996

ndash 51 ndash

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

요 약 문

완전 혼돈 마이크로 공진기 레이저에서의 불안정 주기 궤도와 한계적 불안정

주기 궤도에 맺힌 레이저 모드의 임계값

2차원 마이크로 디스크 공진기 레이저는 일반적으로 안정적 주기 궤도(stable

periodic orbit)를 비롯한 주기궤도(periodic orbit) 및 준주기궤도(quasi-periodic

orbit)에 국한된 공명 상태가 레이저 모드로써 발진된다 그러나 완전 혼돈 마이크

로 디스크 공진기는 안정적 주기 궤도를 가지지 않으며 위상 공간 내 혼돈 바다에

존재하는 불안정 주기 궤도(unstable periodic orbit)에 국한되어 레이저 발진을 이

루게 된다 이러한 불안정 주기 궤도에 국한된 상태를 스카 모드(scarred modes)

라고 부르며 많은 연구를 통해 혼돈 공진기의 스카 모드 레이저 발진이 보고되어

왔다

2차원 원형 공진기의 변형을 조절함에 따라 한계적 불안정 주기 궤도(marginally

unstable periodic orbit)가 나타나는데 이는 원형 공진기의 불변 곡선(invariant

curve)으로 부터 기인한 것이다 따라서 회랑 모드(whispering gallery mode)와 같

이 전반사를 반복하기에 기대되는 품위값(quality factor) 또한 높으며 비록 고정점

(fixed point)가 존재하지 않아 완전한 주기 궤도를 이루지 못하지만 궤도의 끈적임

(stickiness)이 강하여 파동 모드가 국한되기에 좋은 조건을 갖는다

이러한 불안정 주기 궤도와 한계적 불안정 주기 궤도를 동시에 가질 수 있는 둥근

D자형 공진기(rounded D-shape cavity)를 통해 불안정 주기 궤도와 한계적 불안정

주기 궤도를 수치해석적 방법과 실험을 통하여 조사하였다 둥근 D자형 공진기의

변형 인자를 조절하여 불안정 주기 궤도만이 존재하는 상태로 만들어 분석했을 때

오각형 불안정 주기 궤도에 국한되어 레이저 발진이 일어나 단방향성 방출이 나타

ndash 52 ndash

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

나는 것을 확인할 수 있었다 육각형 이상의 불안정 주기 궤도가 더 높은 품위값을

가지고 있음에도 불구하고 우리가 공정한 샘플은 전극과 공진기 경계 사이에 3 마

이크로미터의틈이존재하기에육각형이상의불안정주기궤도에충분한양의유효

전류 주입이 이뤄지지 않고 때문에 해당 모드의 이득(gain)이 줄어들어 육각형 대신

오각형 궤도에 맺힌 레이저 모드가 발진된다 이러한 현상은 선택적 펌핑을 통한

레이저 모드 선택에 필요한 공정 수고를 줄이고 기대하는 레이저 특성을 얻어내는

데에 이용할 수 있으며 본 논문에서도 마이크로 공진기 레이저에서의 단방향성

방출을 달성하여 여러 응용으로의 가능성을 확인할 수 있었다

또한 불안정 주기 궤도와 그에 짝이 맞는 한계적 불안정 주기 궤도를 동시에 갖는

다른 변형 인자를 사용한 둥근 D자형 공진기를 통해 두 종류의 궤도 간 임계 조건

(threshold condition)을 비교하였으며 전극과 경계면 사이의 틈은 불안정 주기 궤

도만 존재하는 공진기와 마찬가지로 3 마이크로미터로 공정하였다 고전적 유효

경로 길이와 공진기 내 파동 함수 분포를 통하여 각 주기 궤도의 고전역학적 파동

역학적 임계 변수(threshold parameter)를 비교하였으며 이를 통해 유효 전류 주입

영역과 공진기 경계면 사이의 거리에 따라 어떠한 주기 궤도가 가장 낮은 임계 변

수를 가지는지 확인할 수 있었다 약 17 마이크로미터일 때 오각형 한계적 불안정

주기 궤도의 임계 변수가 가장 낮게 나왔으며 이는 단일 모드 근사 조건에서 17

마이크로미터의 유효 펌핑 간격이 존재할 경우 오각형 한계적 불안정 주기 궤도에

국한된 레이저 발진이 일어날 것을 암시한다 17 마이크로미터는 우리의 공정조건

인 3 마이크로미터보다 작은 값이지만 낮은 전류에서 옴의 법칙과 연속 방정식을

따르는 전류 확산 과정을 고려할 때 이는 이득 매질(gain medium)의 위치에서의

유효 전류 주입 영역과 일치하는 결과이다

본논문에서는유효전류주입영역에따른선형이득조건식을통해각주기궤도

간의 임계 조건을 직접 비교하였으며 이를 통해 둥근 D자형 공진기에서의 오각형

궤도 발진을 추정하고 실험을 통해 검증하였다 특히 한계적 불안정 주기 궤도와

ndash 53 ndash

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

불안정 주기 궤도 간의 경쟁에서 별도의 선택적 전류 주입 없이 한계적 불안정 주기

궤도의 레이저 발진을 수치해석적 방법과 실험적 검증을 통하여 확인할 수 있었다

핵심어 마이크로 디스크 레이저 한계적 불안정 주기 궤도 스카 모드 마이크로

공진기 레이저 임계값

ndash 54 ndash

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt

• I Introduction
• II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser
• • 21 Ray dynamical analysis and unstable periodic orbits
  • 22 Wave analysis for the threshold pumping strength
  • 23 Fabrication and experimental results
  • 24 Discussion
  • • 241 Quality factor of the scarred mode
    • 242 Unidirectional emission from a scarred mode lasing
    • • 25 Conclusion
      • • III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser
        • • 31 Semiconductor RDS microcavity laser with an inward gap
          • 32 Unstable periodic orbits in a ray dynamical simulation
          • 33 Resonances localized on MUPOs and UPOs
          • 34 Linear gain condition with a single-mode approximation
          • • 341 The threshold condition depending on the classical path of the periodic orbits
            • 342 The threshold condition depending on the spatial wave function localized on the periodic orbits
            • 343 The effective pumping region due to the fabricating configuration
            • • 35 Experimental results and emission characteristics
              • 36 Conclusion
              • • IV Methods
                • • 41 Fabrication of the two dimensional semiconductor microcavity laser
                  • 42 Experimental set-up configuration
                  • • References
                    • Summary (요약문)

                    • ltstartpagegt17I Introduction 1II Lasing of scarred mode near above threshold in a rounded D-shape microcavity laser 3 21 Ray dynamical analysis and unstable periodic orbits 5 22 Wave analysis for the threshold pumping strength 7 23 Fabrication and experimental results 11 24 Discussion 15 241 Quality factor of the scarred mode 15 242 Unidirectional emission from a scarred mode lasing 17 25 Conclusion 17III Lasing modes localized on marginally unstable periodic orbits in a rounded D-shape microcavity laser 20 31 Semiconductor RDS microcavity laser with an inward gap 21 32 Unstable periodic orbits in a ray dynamical simulation 23 33 Resonances localized on MUPOs and UPOs 27 34 Linear gain condition with a single-mode approximation 29 341 The threshold condition depending on the classical path of the periodic orbits 29 342 The threshold condition depending on the spatial wave function localized on the periodic orbits 30 343 The effective pumping region due to the fabricating configuration 33 35 Experimental results and emission characteristics 35 36 Conclusion 39IV Methods 40 41 Fabrication of the two dimensional semiconductor microcavity laser 40 42 Experimental set-up configuration 42References 44Summary (요약문) 52ltbodygt