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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
(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
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
(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
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
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
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
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)
(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
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
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
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
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
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
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
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
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
(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
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
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
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
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
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
(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
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
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
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
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
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
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
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
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
(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
(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
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
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
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
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
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
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
[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
[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
[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
[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
[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
[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
[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
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
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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
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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
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
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