Dissertations in Forestry and Natural Sciences...Dissertations in Forestry and Natural Sciences No...

uef.fi

PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLAND

Dissertations in Forestry and Natural Sciences

ISBN 978-952-61-2350-9ISSN 1798-5668


DIS

SE

RT

AT

ION

S | D

EN

IS K

AR

PO

V | R

ES

ON

AN

CE

PH

EN

OM

EN

A IN

NO

NL

INE

AR

AN

D A

CT

IVE

NA

NO

PH

OT

ON

ICS

| No

251

DENIS KARPOV

RESONANCE PHENOMENA IN NONLINEAR AND ACTIVE NANOPHOTONICS


This work is dedicated to theoretical and

experimental investigation of the resonance optical phenomena occurring in nonlinear and active photonics nanostructures. By

using a wide range of theoretical and experimental techniques we studied glass-metal nanocomposites, whispering gallery mode semiconductor quantum dots lasers and exciton-polaritons lasing in the bias-

controlled heterostructures.

DENIS KARPOV

DENIS KARPOV

Resonance phenomena in

nonlinear and active

nanophotonics

Publications of the University of Eastern Finland


No 251

Academic Dissertation

To be presented by permission of the Faculty of Science and Forestry for public

examination in the Auditorium F100 in Futura Building at the University of

Eastern Finland, Joensuu, on December 8, 2016, at 12 o’clock noon.

Department of Physics and Mathematics

Grano Oy

Jyväskylä, 2016

Editors: Research Dir. Pertti Pasanen,

Pekka Toivanen, Jukka Tuomela, Matti Vornanen

Distribution:

University of Eastern Finland Library / Sales of

publications P.O.Box 107, FI-80101 Joensuu, Finland

tel. +358-50-3058396

www.uef.fi/kirjasto

Print

ISBN: 978-952-61-2350-9 (Print)

ISSNL: 1798-5668

ISSN: 1798-5668

Online

ISBN: 978-952-61-2351-6 (PDF)

ISSNL: 1798-5668

ISSN: 1798-5676

http://www.uef.fi/kirjasto

Author’s address: University of Eastern Finland

Department of Environmental Sciences

P.O.Box 1627

70211 KUOPIO

FINLAND

email: [email protected]

Supervisors: Professor Yuri Svirko, Ph.D.

University of Eastern Finland

Department of Physics and Mathematics

P.O.Box 111

80100 JOENSUU

FINLAND


Professor Andrei Lipovskii, Ph.D.

St. Petersburg Academic University

Department of Physics and Technology of

Nanoheterostructures,

194021 ST. PETERSBURG

RUSSIA


Reviewers: Professor V. A. Makarov, Ph.D

Moscow State University

International laser center

Leninskiye Gory

119991 MOSCOW

RUSSIA


Professor Erik Vartiainen, Ph.D

Lappeenranta University of Technology

School of Engineering Science

P.O.Box 20

FI-53851 LAPPEENRANTA

FINLAND


Opponent: Professor Stefano Pelli, PhD

Institute of Applied Physics nello Carrara

Department of Optoelectronics and Photonics

Via Madonna del Piano 10

50019 SESTO FIORENTINO

ITALY


mailto:[email protected]





ABSTRACT

This work is dedicated to theoretical and experimental

investigation of the resonance optical phenomena occurring in

nonlinear and active photonics nanostructures. By using a wide

range of theoretical and experimental techniques we studied the

interaction of intense laser pulses with glass-metal

nanocomposites and individual metal nanoparticles deposited on

a dielectric surface. In particular, by performing the light-induced

transmission measurements we reveal the modification of the

metal nanoparticle shape under irradiation with intense

femtosecond laser pulses. The numerical simulation allowed us

to reveal the contribution of the sharp edges to the polarization

and intensity of the second harmonic wave generated by

individual metal semispheres deposited onto dielectric

substrates. Stimulated emission of InAs quantum dots embedded

in the semiconductor ring/disc microcavities with Q-factor as high as 20000 was studied by microphotoluminescence microphotoluminescence measurements, while silicon carbide 2-dimensional photonic

crystals were fabricated and used for development of new nitride

growth technology. We also performed theoretical investigation

of the exciton polaritons dynamics in a semiconductor

microcavity with a saturable absorber. In particular, the role of

the dissipative nonlinearity due to emergence of bistability of the

polariton condensate was studied. We also develop protocols of

soliton formation and destruction in such structures. A

microscopic theory of the lasing in the bias-controlled

heterostructure was developed. In particular, we simulated the

dynamics of the exciton-polariton ensemble and revealed

threshold dependence of the number of quasiparticles on the

applied bias.

Universal Decimal Classification: 534.242, 535.14, 544.532.122, 538.9,

539.122, 620.3

Library of Congress Subject Headings: Photonics; Nanophotonics; Optical

resonance; Nanotechnology; Nanostructured materials; Nanocomposites

(Materials); Nanoparticles; Metals; Glass; Semiconductors; Quantum dots;

Optoelectronics; Plasmons (Physics); Polaritons; Solitons; Solid state physics;

Second harmonic generation; Nonlinear optics; Bose-Einstein condensation;

Femtosecond lasers; Numerical analysis

Preface

I wish to thank my supervisors Prof. Yuri Svirko and Prof. Andrei

Lipovskii for support during my PhD studies. I also wish to thank

the Head of the Department of Physics and Mathematics

Professor Timo Jääskeläinen for opportunity to work in such a

pleasant atmosphere.

Special thanks to Dr. Ivan Savenko, my friend and long time

collaborator for his guiding in the field of exciton-polaritons. I

want to thank Dr. Janne Laukkanen who have taught me

everything in the field of fabrication of micro- and nanostructures

and good cleanroom practice, Dr. Victor Prokofiev and Dr. Olga

Svirko for guiding me through cleanroom facilities. I wish also to

thank Dr. Natalia Kryzhanovskay for guiding me throughout

micro lasers research. Many thanks to my friends including Dr.

Viatcheslav Vanyukov, Mrs Feruza Tuyakova and Mr Semen

Chervinskii for their friendship and help both inside and outside

the University.

I very grateful to Mrs Hannele Karppinen, Mrs Katri

Mustonen and Dr. Noora Heikkilä for their assistance and

backing during my PhD studies.

I wish to express my gratitude to my mother Olga and my wife

Elena for their strong and permanent support through all my

studies.

Joensuu December 8, 2016 Denis Karpov

LIST OF ABBREVIATIONS

ALD atomic layer deposition

BEC Bose-Einstein condensate

DBR distributed Bragg reflectors

DDE drift diffusion equations

DS dissipative soliton

EBL electron beam lithography

EHR electron hole recombination

EMA effective medium approximation

EP exciton polariton

ES excited state

FEM finite elements method

FWHM full width at half-maximum

GMN glass-metal composite

GPE Gross-Pitaevskii equation

GS ground state

ICP inductively coupled plasma

MBE molecular beam epitaxy

MG Maxwell Garnet effective medium approximation

NP nano particle

Q-factor resonance quality factor

QD quantum dot

QW quantum well

RIE reactive ion etching

SA saturable absorber

SERS surface enhanced Raman scattering

SESAM semiconductor saturable absorber mirror

SHG second harmonic generation

SP surface plasmon

SPR surface plasmon resonance

TIR total internal reflection of wave

WGM whispering gallery mode

LIST OF ORIGINAL PUBLICATIONS

This thesis is based on data presented in the following articles,

referred to by the Roman numerals I–VI.

I D.V. Karpov and I. G. Savenko, “Operation of a

semiconductor microcavity under electric excitation”, Applied

Physics Letters 109(6), 061110 (2016)

II D. V. Karpov, I. G. Savenko, H. Flayac, and N. N. Rosanov,

“Dissipative soliton protocols in semiconductor microcavities

at finite temperatures”, Physical Review B 92, 075305 (2015)

III D.V. Karpov, S. A. Scherbak, Y.P. Svirko and A.A. Lipovskii,

“Second harmonic generation from hemispherical metal

nanoparticle covered by dielectric layer”, Journal of Nonlinear

Optical Physics & Materials 25, 1650001 (2016)

IV S. Chervinskii, R. Drevinskas, D. V. Karpov, M. Beresna, A.

A. Lipovskii, Yu. P. Svirko & P. G. Kazansky, “Revealing the

nanoparticles aspect ratio in the glass-metal nanocomposites

irradiated with femtosecond laser”, Scientific Reports 5, 13746

(2015)

V M.V.Maximov, N.V. Kryzhanovskay, A.M.Nadtochiy, E.I.

Moiseev, I.I. Shostak, A.A. Bogdanov, Z.F.Sadrieva,

A.E.Zhukov, A.A. Lipovskii, D.V. Karpov, J. Laukkanen, J.

Tommila, “Ultrasmall microdisk and microring lasers based

on InAs/InGaAs/GaAs quantum dots”, Nanoscale Research

Letters 9:657 (2014)

VI A.E.Zhukov, M.V.Maximov, N.V. Kryzhanovskay,

A.M.Nadtochiy, E.I. Moiseev, I.I. Shostak, A.A. Bogdanov,

Z.F.Sadrieva, A.A. Lipovskii, D.V. Karpov, J. Laukkanen, J.

Tommila, “Lasing in microdisks of an ultra-small diameter”,

Semiconductors 48(12), 1666-1670 (2016)

Another papers which are not included in this thesis

V.N.Bessolov, D.V.Karpov, E. V. Konenkova , A.А. Lipovskii,

A.V. Osipov, A. V. Redkov, I.P. Soshnikov, S.A. Kukushkin,

“Pendeo-epitaxy of stress-free AlN layer on a profiled SiC/Si

substrate”, Thin Solid Films 606, 74–79 (2016)

I. Reduto, S. Chervinskii, A. Kamenskii, D. Karpov and A. A.

Lipovskii, “Self-Organized Growth of Small Arrays of Metal

Nanoislands on the Surface of Poled Ion-Exchange Glasses”,

Technical Physics Letters 42(1), (2016)

The publications I-V have been included at the end of this thesis

with their copyright holders’ permission.

AUTHOR’S CONTRIBUTION

Author formulated the problem for paper I. In papers I, II, III

and IV, the author conducted theoretical analysis and performed

numerical simulation. In papers II and III, the author performed

parallel computing using supercomputer facilities. The papers I,

II and III were written by the author. In papers V, VI as well as

two papers not included in the Thesis, author has designed and

fabricated the studied micro- and nanostructures and

participated in the writing of the parts of the papers related to the

fabrication.

Contents

Preface .............................................................................................. 7

Contents ......................................................................................... 11

1 Introduction ................................................................................ 13

2 Surface plasmon resonance in glass-metal composite ....... 21

2.1 Linear plasmonics ................................................................. 21

2.1.1 Maxwell’s equations ........................................................... 22

2.1.2 Localized surface Plasmon: Eigenmode expansion ............... 22

2.1.3 Metallic nanoparticles: free electron gas model .................... 24

2.1.4 Quasi-static approximation ................................................. 25

Example 1: Sphere ....................................................................... 26

Example 2: Spheroid .................................................................... 27

Example 3: Bisphere .................................................................... 29

Example 4: Hemisphere ............................................................... 29

2.2 Effective medium approximation ........................................ 30

2.2.1 Composite with spherical inclusions .................................... 31

2.2.2 Composite with prolate spheroidal inclusions ...................... 34

2.3 Conclusion of chapter 2 ........................................................ 36

3 Nonlinear optics of glass-metal composite .......................... 38

3.1 Nonlinear response of glass metal composites ................... 38

3.2 Second order nonlinearity .................................................... 39

3.3 Hydrodynamic theory of electron gas motion.................... 40

3.4 Hyperpolarizability of metal particle .................................. 42

3.5 Third order nonlinearity....................................................... 43


4 Disc/ring microcavities with quantum dots ........................ 47

4.1 Whispering gallery mode resonance ................................... 47

4.2 Disc/ring microcavities with active medium based on

quantum dots ................................................................................. 49

4.3 Semiconductor nanostructure fabrication ........................... 50

4.3.1 Molecular beam epitaxy ...................................................... 51

4.3.2 Electron beam lithography .................................................. 52

4.3.3 Atomic layer deposition ...................................................... 52

4.3.4 Reactive ion etching ............................................................ 53

4.4 Microphotoluminescense measurements ............................ 54

4.5 Threshold characteristics of QD ring microcavities ........... 55

4.6 Temperature dependence ..................................................... 56


5 Nonlinear phenomena in exciton-polariton condensate .... 59

5.1 Electric and optical properties of semiconductor

nanostructures ............................................................................... 60

5.2 Exciton-photon strong coupling .......................................... 61

5.3 Exciton-polariton condensation ........................................... 63

5.4 Semiconductor microcavity at the electrical excitation ...... 64

5.7 Saturable absorption ............................................................. 69

5.8 Dissipative solitons in microcavity ...................................... 70

5.9 Dissipative soliton protocol ................................................. 72

5.10 Conclusion of chapter 5 ......................................................... 73

6 Summary ..................................................................................... 75

7 References ................................................................................... 79

Dissertations in Forestry and Natural Sciences No 251

13

1 Introduction

Linear and nonlinear optical properties of media comprised of

metallic and semiconductor nanostructures are of great interest

for photonics. This is mainly because the properties of such

composite materials are strongly influenced by the resonances

associated with their mesoscopic nature. For example, inclusion

of metal nanoparticles (NP) into dielectric matrix allows one to

excite surface plasmons [1-4], which can be coupled to resonances

of molecules or ions in the vicinity of nanoparticles. This effect,

which gave birth to the surface enhanced Raman scattering [5-8],

also results in the drastic change of the linear and nonlinear

absorption [9-14] and can lead to the enhancement of the second

harmonic generation in nanocomposites [15-18]. It is worth

noting that position and strength of the plasmon resonance in a

nanocomposite strongly depend on the shape of nanoparticles.

This makes it possible e.g. to determine the aspect ratio and

concentration of spheroidal inclusions by measuring the

differential optical density spectra of the glass-metal

nanocomposite, see Fig. 1.

The enhanced local electric field in the vicinity of metal

inclusion results in the electron ejection from the inclusion to the

glass and respectively, the accumulation of the electric charge.

Thus irradiation with intense laser pulses, which results in

increasing the local temperature, can give rise to elongation of the

nanoparticle by the Coulomb force along the polarization of the

incident beam [paper IV, 19-20]. The elongation makes the

position of surface plasmon resonance dependent on the light

polarization allowing one to deduce the aspect ratio from the

transmittance spectrum measured for two orthogonal

polarizations. Knowledge of the aspect ratio allows one to define

the characteristics of the nanocomposite, as well as to visualize

Denis Karpov: Resonance phenomena in nonlinear and active nanophotonics

14 Dissertations in Forestry and Natural Sciences No 251

the metal nanoparticle elongation under external influence (e.g.

stress and bending at the elevated temperature).

Figure 1. (a) Absorbance spectrum of the GMN with spherical inclusion fitted using

Maxwell Garnet model. (b) Simulation of differential optical density as a function of the

aspect ratio c/a of spheroidal inclusion and wavelength for the GMN modified with an

intense laser pulse. Insets show GMN before and after modification. Picture from paper

IV.

The plasmon resonance can also have a strong influence on the

nonlinear optical response of the glass-metal nanocomposites.

For example, metal hemisphere on the glass surface have a non-

zero dipole hyperpolarizability because the inversion symmetry

is lifted due to its shape and proximity of the interface. This leads

to the second harmonic generation upon irradiation of the

hemisphere with intense light pulse. The local field enhancement

in the vicinity of the sharp edges of the hemisphere [21-23] is of

special importance in this respect. Since the coating with a

dielectric shell shifts the plasmon resonance, the dependence of

the second harmonic intensity on the shell thickness may reveal

the effects of surface plasmon contribution to the second-order

nonlinear response. Identification of the hemisphere regions

which provide maximum contribution of the SHG intensity is

important for potential applications.

Lasing in ring/disc microcavities with different active media is

a subject of considerable interest during last three decades [24-

26]. In semiconductor disk and ring microcavities embedded with

InAs/InGaAs quantum dots (QD) [paper V, 27-29], the resonant

coupling between whispering gallery modes (WGM) and emitted

Introduction

Dissertations in Forestry and Natural Sciences No 251 15

photons makes it possible to design advanced laser sources due

to the high WGM Q-factor. Such structures can be considered as

an alternative to lasers based on quantum wells [30] because of

their good temperature stability due to 3D carrier confinement in

QD. In such microcavities, threshold pump power is as low as 5

𝜇W at room temperature and can be adjusted by changing the

geometry of the microcavity.

Figure 2. (a) Scanning electron microscope image of the ring microcavity. (b) Micro

photoluminescence spectrum of the ring microcavity with diameter of 2μm and inner

diameter of 0.8μm for different pumping powers. Lasing takes place at TE12,1 mode with.

Picture is borrowed from paper V.

Lasers based on ring microcavities (Fig. 2.) are promising for

interchip data transfer, modulators [31], switchers [32] and filters

[33]. Due to the high Q-factor of WGM [34] they can also be used

as a frequency standard in integrated optics. Due to in-plane

localization of the WGM modes in ring microcavity [35,36],

electrically-pumped ring resonators coupled to the planar

waveguide can be can be employed used as light sources and

modulators in photonic circuits. Development methods of

fabrication of the ultra small resonators is an important task in

terms of their integration into optical circuits and minimizing

energy consumption. The major role is played by the quality of

the wall of the ring/disc. This is because whispering modes

propagate along the wall surface, and the roughness of the

surface leads to large optical losses thus increasing the lasing

threshold. This problem can be resolved by using the electron-

beam lithography, which allows one to achieve smooth walls.



Correspondingly, the further advancing in fabrication of the

ultra-smooth wall cavities by electron-beam and/or UV

lithography is of a strong importance.

Synthesis of new materials (nitrides, organic semiconductors),

in which the interband dipole moment is large in comparison

with the conventional GaAs, makes possible a condensation of

exciton polaritons in microcavities at room temperature [37-50].

This makes the study of the optical properties of such structures

extremely important for applications. In particular, the lasing

threshold for polaritons is much lower than that for photons thus

allowing one to reduce the energy consumption of data

transmission devices.

Cavity polaritons ensemble is highly nonequilibrium system

due to short exciton lifetime (10-100ps). Therefore description of

the spatial-temporal evolution of such a system requires kinetic

approach. Introduction of saturable absorber into microcavity

leads to additional dissipation that makes this nonequilibrium

system nonlinear and leads to formation of solitary waves

(solitons). Such nonequlibrium and nonlinear system is stabilized

by pumping, which compensates the dissipation losses. This is

very different from conventional conservative nonlinear systems,

in which soliton formation is possible when dispersion

compensates nonlinearity, and we usually have a family of

solitons. In contrast, in dissipative nonlinear systems the only

soliton can exist if gain compensates losses (see Fig. 3) [51-53].

Semiconductor microcavities under incoherent pump

(electrical or optical) can have different fields of application, such

as optical routers [54,55], transistors [56], sources of terahertz

radiation [57,58], elements of optical circuits [59], high-speed

optical switches of polarization [60]. In this context, the study of

microcavities under optical pumping is an important task. The

introduction of the saturable absorber into a microcavity leads to

the dependence of the polariton dissipation rate on the polariton

density and can significantly affect the operation of the polariton

devices. This can be described via including nonlinear dissipative

term in the Gross-Pitaevskii equation and studying of optical

solitons at a finite temperature. It is worth mentioning that the

Introduction


recently reported polariton condensation in WGM microcavity

strongly links the parts of this Thesis making it more consistent.

Figure 3. Scheme of nonlinear problem solution for (a) Hamiltonian and (b) dissipative

systems

Electrically pumped microcavities with polariton condensate

are highly interesting for applications. Description of

nonequilibrium Bose-Einstein condensation (BEC) in

semiconductor heterostructures under electrical excitation (e.g.

electrically pumped polariton laser) requires solution of the

Boltzmann equation for the exciton reservoir supplemented with

drift-diffusion equations for charge carriers and the Gross-

Pitaevskii equation for polaritons. We consider wide-band-gap

semiconductor InAlGaN alloy, which is a promising material for

room-temperature BEC and, thus, lasing [61,62]. The large

oscillator strength and exciton binding energy and giant Rabi

splitting (more than 30meV) lead to robust polariton BEC at 300K

(see Fig. 4). The oscillator strength of the InGaN QW excitons is

found to be one order of magnitude higher than that of GaAs QW

excitons.



Figure 4. Exciton-polariton density in the vicinity of k = 0 as a function of the forward

U for the InGaN quantum-well diode. Inset shows color map of the particle distribution

in momentum space at (a) U = 2.2 V (under threshold, left) and (b) U = 2.3 V (above

threshold, right). Picture was taken from paper I.

The second chapter of the Thesis describes the linear

properties of the plasmonic nanostructures and macroscopic

properties of media based on metallic inclusions. Plasmon

resonances for metallic sphere, bi-sphere, ellipsoid and

hemisphere are described using epsilon-method.

The third chapter presents theoretical description of second-

harmonic generation in plasmonic nanostructures based on the

hydrodynamic theory of the optical nonlinearity of the

conduction electrons in a metal. The latter, being combined with

the electrostatic approach, made it possible to obtain quasi-

analytical expressions for the hyperpolarizability tensor of metal

hemisphere. On the basis of the developed approach we predict

increase of second harmonic generation at the frequency of the

plasmon resonance in hemispherical metal nanoparticles coated

with a dielectric layer.

In the fourth chapter of the Thesis we describe the fabrication

technology of the ring/disc semiconductor microcavity of

diameter as small as 2𝜇m. Here we present the fabrication

technique, which provides a small size combined with low

roughness of the ring/disc walls. This combination allowed us to

obtain cavity with a high quality factor and to achieve lasing at

room temperature. Methods of characterization on the basis of

micro photoluminescence are presented together with the

Introduction


experimental results. Developed method of modifying the

surface of the silicon carbide by means of electron beam

lithography, in order to optimize subsequent growth of

aluminum nitride and gallium nitride is also described.

In the fifth part of the Thesis, a model of a semiconductor

heterostructure with quantum well (QW) and a saturable

absorber (NP) is presented. The effect of the absorption saturation

in a microcavity leads to the formation of dissipative solitons in

the polariton ensemble. We also consider the effect of acoustic

phonon-polariton interaction. We demonstrate protocol (laser

pulse consequence and regime of incoherent pumping) of

formation and destruction of a dissipative soliton at finite

temperatures. We present microscopic theory of polariton laser

(for Indium nitride heterostructure), in which we imply the

microscopic description of the exciton reservoir.


21

2 Surface plasmon resonance

in glass-metal composite

When a metal nanoparticle is embedded into a dielectric matrix,

the electric field strength in its vicinity can be strongly enhanced

by collective oscillations conduction electrons in the nanoparticle

(surface plasmons). Calculation of the local filed amplitude at the

surface plasmon resonance (SPR) is important for various

applications such as surface enhanced Raman scattering (SERS)

and second harmonic generation (SHG). In this chapter,

eigenmode expansion for Maxwell equations is introduced. This

technic allows us to compute SPR features for nanoparticles of

different shapes. Effective medium approximation based on

Maxwell Garnett approach describes the SPR dependence on the

nanoparticles concentration. This approach allows one also to

describe optical properties of the GMN composed of spheroids.

Such a nanostructure can be produced by irradiating the GMN

with intense laser pulse that leads to transformation of spherical

nanoparticles to spheroids. Position of the SPR in the spheroid is

depends on light polarization.

2.1 LINEAR PLASMONICS

Surface plasmons (SPs) are collective oscillations of conducting

electrons on the metallic surfaces coupled to an external

electromagnetic field [1-4,63]. Alternatively SPs can be

understood as the electromagnetic eigenmodes of the metal-

dielectric interfaces. Since in GMN, SPs are localized at the

nanoparticle and do not propagate, they often referred to as

localized surface plasmons. In this Chapter, we present plasmon



eigenmodes analysis for subwavelength NP. This theory in

combination with effective medium approximation was used to

describe the linear and nonlinear optical properties of the glass-

metal nanocomposites.

2.1.1 Maxwell’s equations

Maxwell’s equations [64] describe relations between the electric

field E , electric displacement vector D , magnetic induction B ,

magnetic field vector H , charge density and current density j :

D (2.1a)

0 B (2.1b)

t

BE (2.1c)

t

DH j (2.1d)

Maxwell’s equations should be supplemented with constitutive

relations, which for isotropic linear media can be presented in the

following form:

0 D E (2.2a)

0 B = H (2.2b)

j E (2.2c)

where , and and are permittivity, permeability and

conductivity of the medium, 0 and 0 are vacuum permittivity

and permeability.

2.1.2 Localized surface Plasmon: Eigenmode expansion

Consider nanocomposite consisting of metallic inclusions in a

dielectric matrix. In presence of the light wave, oscillations of free

electrons at the metal-dielectric interface can be resonantly

coupled with incident photons and form surface plasmon. At the

surface plasmon resonance, the momentum of the coupled

electron-photon excitation can be much bigger than photon

momentum, i.e. it becomes localized. Resonance frequencies of

such localized plasmons depend on the shape of the nanoparticle,

Surface plasmon resonance in glass-metal composite


distance between nanoparticles forming ensemble and materials

of the inclusion and the host. Most general concept of localized

plasmons can be introduced using eigenvalue problem

formulation for dielectric constant also referred to as the epsilon

method [3, 65].

In the framework of the epsilon method, the solution of the

Maxwell equations reduces to finding the eigenvalues and

eigenfunctions of the boundary problem for a specified geometry.

Epsilon method permits calculation of the SPR frequencies and

electromagnetic field distribution for both individual NPs and

their ensembles. The method allows us to describe the properties

of localized plasmons and enhancement of the electric field at the

plasmon resonance, with a focus on the dependence of the SPR

on the nanoparticle shape. In the framework of the epsilon

method, the SPR emerges as the frequency corresponding to the

eigenvalues n of the permittivity. In the quasi-static

approximation, when the characteristic size of particles much

smaller than the incident light wavelength, epsilon method

allows one to reduce the solution of Maxwell's equations down to

the finding eigenfunctions of the Laplace boundary problem.

In spectral representation the eigenfunctions ne and nh of the

boundary problem satisfy the following equations:

0

0

( ) ( ) 0

( ) ( ) 0

n n n

n n

i

i

h r e r

e r h r , (2.3)

where is external light frequency, n is the eigenvalue. The

eigenfunctions are orthogonal,

,n m nm n m nm

V V

dV dV e e h h (2.4)

and the electric field in the medium can be present as: 0

n n

n

A E E e (2.5)

where 0E is external field and



0

2

( ( ) 1)

( ( )) ( )

n

Vn

n n

V

dV

AdV

e E

e.

The most important feature of the solutions is the presence of

resonant factor in the denominator , ( ) 0n res . That is at

the resonance frequency, 0

2

( ( ) 1)

( ( ))

n

res Vres n

n res n

V

dV

dV

e E

E E ee

(2.6)

It is worth noting that 𝜀𝑛 and 𝑒𝑛do not depend on material and

are determined by pure geometrical reasons. In particular, this

approach is valid for quite close particles allowing one to

understand mechanisms of plasmon hybridization.

2.1.3 Metallic nanoparticles: free electron gas model

Conducting electrons in metals can be considered to move freely.

With this assumption, most of the electronic and optical

properties of metals can be described in terms of the Drude-

Lorentz-Sommerfeld model [66, 67].

This model allows one to describe motion of conduction

electrons along x-axes in terms of the damped harmonic oscillator:

( ) ( ) ( )x

ex t x t E t

m (2.7)

with e as the elementary charge, m as the electron mass, and γ as

the damping constant. The x-component of the medium

polarization defined as 0 ( 1)x xP enx E , here n is electron

concentration. By solving Eq. (2.7) one can arrive at the following

equation for medium permittivity:

2

1 p

i

, (2.8)

where2

0

p

ne

m

is the plasma frequency.

Drude model does not consider bound electrons contribution

to the permittivity that may be important for noble metals. To



account this contribution one need to modify Eq. (2.8) as the

following:

2

p

i

, (2.9)

where is the high frequency permittivity. Plasmon resonance

frequencies can be obtained from Eq. (2.9) from the condition

res n where n is the eigenvalue of Eqs. (2.3).

2.1.4 Quasi-static approximation

Eigenvalue problem for Maxwell equation presented above

considers all types of resonance including whispering gallery

modes, radiative modes, Mie resonances, etc. However, if the size

of the nanoparticles is much smaller than the light wavelength

the Maxwell equations can be solved in the framework of the

quasi-static approximation, which is valid if the size of the

nanoparticle l is

- much smaller than the wavelength of the exciting

field 𝜆 and

- much bigger than the mean free path of the electron le,

Debye radius rD and the wavelength of an electron 𝜆F

at the Fermi surface.

These requirements can be presented by the following inequality:

e D Fl l r

If the particle size is comparable to or less than the electron

mean free path, the important role is played by the scattering of

the conduction electrons by the NP surface. This scattering will

lead to decrease in the relaxation time of the electron. Specifically,

in this case the scattering of the conduction electrons by the

surface increases the electron relaxation rate by

FvA

R (2.10)

where νF, R and A are Fermi velocity, typical size of nanoparticle,

while constant A is between 0 and 0.7 depending on the NP shape

[68,69]. Fermi velocity for the silver and gold can be estimated as:

1.4Fv nm/fs . When the particle size is comparable to the Debye

radius, the spatial dispersion begins to play an important role.



When the particle size is comparable to the electron’s wavelength,

an important role is played by spatial quantization [11,12].

By neglecting effects of the spatial dispersion i.e. assuming

21kl l

one can reduce Eqs.(2.3) down to

0,

0.

n n

n

e

e (2.11)

That is in this quasi-static approximation, for particle embedded

in host with permittivity host Maxwell equations are reduced to

the Laplace equations for potential

0,n (2.12)

where n n e . The continuity of the normal component of

the electric displacement across the NP/host interface is given the

following boundary conditions for the potential:

.in out

n n S host n S a a (2.13)

Where a is unit vector along the NP surface normal, while

superscripts “in” and “out” label potential inside and outside the

nanoparticle, respectively.

Example 1: Sphere

Let us consider a sphere with radius 0R embedded in a host

medium with permittivity host . Eigen functions of Laplace

operator in spherical coordinates are:

0

0

1

00

( , ),

( , ),

n

nm

nmn

nm

rY r R

R

RY r R

r

where nmY is spherical harmonics, ( , , )r are spherical

coordinates. Electrostatic problem is linear, hence the electric

potential is written as:

1 0

n

nm

n m



Using the boundary conditions (2.13), we can find the

eigenvalues of the permittivity as

1n

host

n

n

(2.14)

For example, for Drude dispersion relation (2.10), neglecting

damping parameter we obtain well-known formula for plasmon

resonances of a sphere:

1

p

host

n

n

(2.15)

Example 2: Spheroid

In order to solve the Laplace boundary problem for a prolate

spheroid it is convenient to use spheroidal coordinates ( , , )

[70]:

2 2 2 2(1 )( 1) cos , (1 )( 1) sin ,x a y a z a

Eigenfunctions of Laplace operator in new coordinates are:

(1) (1)

(2) (2)

( ) ( )( cos sin ),

( ) ( )( cos sin ),

n n

m m nm nm

nm n n

m m nm nm

P Q m m outside

P P m m inside

where ( )n

mP and ( )n

mQ associated Legendre polynomials of

first and second type respectively. The potential inside spheroid

can be presented in the following form:

1 0

n

nm

n m

Using boundary conditions for continuity of the tangential

component of E and the normal component of D we eliminate

, coefficients and obtain relation for plasmon eigenvalues:

0 0

0 0

( )( ( ))

( ( )) ( )

m n n

n m m

n n

host m m

P Q

P Q



where 0 can be expressed using spheroid axes (a<c) as

02 2

c

c a

In dipolar case, when only mode with n=1 is nonzero, the linear

polarizability can be expressed from:

0 0 0 0( ( ) ) ( ( ) )host host hostV

dV V d E E E

where E can be expressed from (2.5) as

10

1 ( )

m

hostm m m

E E

and thus polarizability can be expressed as: 2

1

1

( )4 1

3 ( )

m

mm m

hos

host

t

ca

(2.16)

For oblate spheroid, using the same approach we have:

0 0

0 0

( )( ( ))

( ( )) ( )

m n n

n m m

n n

host m m

P i Q i

P i Q i

and further steps to obtain polarizability are the same as

presented above, for more details see [3].

Figure 5. SPR wavelengths for oblate and prolate silver spheroids in the glass matrix as functions of the aspect ratios. Red and black solid lines show SPR wavelength for the light polarized along a- and c-axis, respectively. The following parameters were used for the numerical simulations: ε∞ = 4, λp = 135 nm, γ/ωp = 0.1, εhost = 7.4. Figure borrowed from our paper IV.



Example 3: Bisphere

The eigenvalue problem can be solved using coordinate

transformation to the bispherical coordinates ( , , ) [71]:

sin cos sin sin, ,

cos cos cos

shx a y a z a

ch ch ch

Continuity of the tangential component of E and the normal

component of D allows us to arrive at matrix relation for

eigenvalues that could be solved for zero interparticle gap

analytically [63,65,72-73].

Example 4: Hemisphere

In case of hemisphere, we have no orthogonality for basic

functions and cannot obtain eigenvalues analytically. However,

by applying boundary conditions one can arrive at algebraic

equations for coefficients in the expansion of the potential in

series of Legendre polynomials and first associated Legendre

polynomials [70]. For example, when the external electric field is

directed normal and parallel to the interface, respectively:

0

1

,n

n nnn

rr aE B P cos

a

(2.17a)

1

|| 0||

1

, ,n

n nnn

rr aE C P cos cos

a

(2.17b)

Electric field can be found using equality ||, E r ,

Coefficients ,n nB C in Eqs. (2.17) can be found using the

boundary conditions for the potential and electric field [21-23].

Calculation of dipole moment and polarizability is quite

straightforward. For more details, see Paper III.



Figure 6. Linear absorption spectrum of the silver hemisphere on dielectric

substrate. The normal incident light wave is polarized in plane of substrate surface.

Silver permittivity data were taken from [79].

The epsilon method is much better in terms of the numerical

burden than the finite elements method (FEM) for hemisphere.

This is because the singularity associated with the sharp edge of

the hemisphere greatly complicates the numerical solution. For

example, it may take several hours to get results with COMSOL

but it takes a few minutes using the epsilon method.

2.2 EFFECTIVE MEDIUM APPROXIMATION

Optical properties of metallic nanoparticles embedded in a

transparent host are the subject of considerable experimental and

theoretical interest. Much of this attention is due to the possibility

to control dielectric function and optical properties of these

composite media through the concentration and geometry of the

metal inclusions. When the concentration of the inclusions is

quite low, the Maxwell Garnett (MG) effective media

approximation (EMA) [74-77] is conventionally used for

calculation the dielectric function of such media. This approach is

based on presentation of the polarizability of the composite



media as the sum of polarizabilities of non-interacting

nanoparticles. Generally speaking, the higher metal volume

fraction, the higher probability. This is because a small distance

between nanoparticles implies strong contributions of the dipole,

quadrupole, and higher multipole interactions between them.

The dipole interaction between separated nanoparticles, which

has been analyzed in many papers and books [3,65], allows one

to obtain first correction of Maxwell Garnett formula as a virial

expansion of series of concentration.

Nanoparticle dimers are of considerable importance in this

context because of the first step of considering interaction

between nanoparticles is two-body approximation. One may

expect that nanodimers make a major contribution to

macroscopic dielectric constant of the nanocomposite. The

plasmonic properties of nanoparticle dimers are briefly discussed

in this chapter.

The effective dielectric constant of a GMN with constituent

inclusions depends upon the average local field acting in the

interior of an inclusion. This average field is not in general equal

to the macroscopic field E , entering into the macroscopic field

equations. Below we present analysis of the averaged electric

field in the ensemble of metal nanoparticles.

2.2.1 Composite with spherical inclusions

Maxwell Garnett model [75] describe the macroscopic properties

of composite materials by averaging the multiple values of the

constituents’ dipole moments. This model considers shift of

surface plasmon resonance as function of inclusion’s

concentration and shape. Maxwell Garnett formula is valid for

dilute composites, in which concentration metal volume fraction

is low so that the inter-particle interaction does not change

plasmon resonance of individual particle and macroscopic field

is uniform. In the framework of the Maxwell Garnett model the

nanocompsite can be characterized by the effective values of

conductivity and permittivity. In the paragraph, we assume that

inclusions have spherical shape, are uniformly distributed in the

bulk, and have the same size.



Figure 7. Sketch of the glass-metal nanocomposite. Metal spheres with

permittivity 𝝐 and polarizabilities α are embedded into glass matrix with

permittivity 𝝐𝒉𝒐𝒔𝒕. is the effective permittivity of the GMN, E0 is external electric

field.

Following Kirkwood [77], we consider an ensemble of N equal

spherical particles embedded in a dielectric matrix. The

magnitude of the induced dipole moment of i-th particle is

determined by the external field E0 and the interaction with the

other particles of the ensemble:

0 ,i j

j i

G

i jp E r r p , (2.18)

where is the particle polarizability, ,G i jr r is the dyadic

Green function describing dipole interaction between i and j

particles. In quasi static approximation the Green function reads

3

0

3,

4

IG

ij ij

i j

ij

r rr r

r (2.19)

where ij i ir r r , denotes tensor multiplication, I is identity

matrix.

Solution of the Eq. (2.18) can be presented in terms of the

expansion in series of the polarizability as the following:

2 3

0, , ,ij ij i

k j

p G G G E

i j i j j kr r r r r r



(2.20)

Polarization of the nanocomposite can be obtained by averaging

dipole moment of the individual particle over their space

distribution. In an isotropic composite one can arrive at the

following equation for the polarization:

2 3

12 1 2 1 2 2

3 3

12 1 2 1 2 2 1 2

3 3 3

123 1 2 3 1 2 2 3 2 3 0

, ,

( , ) , ,

, , , ,

n n n G d

n G G d

n G G d d

P p r r r r r

r r r r r r r

r r r r r r r r r E

(2.21)

where n , 12 1 2,n r r and 123 1 2 3, ,n r r r are concentration, two-

and three-particles distribution functions, respectively.

Solving (2.21) for the external electric field and applying

statistical averaging we arrive at the following equation:

3

12 1 2 1 2 2 1 22

0

3 312 1 2 23 2 3123 1 2 3 1 2 2 3 2 32

1( , ) , ,

3

( , ) ( , ), ), , ,

n G G dn n

n nn G G d d

n n

E r r r r r r r

r r r rr r r r r r r r r

P

P

(2.22)

Rewriting Eq. (2.22) in terms of the permittivity of the composite

one can arrive at the virial expansion in terms of powers of

nanoparticles concentration. In particular, by taking in to account

two lowest order terms in the virial expansion one can arrive at:

2

0 0

ε1

ε 2 3

host

host

nBn

(2.23)

where B is an analog of the second virial coefficient in the

statistical theory of the equation of non-ideal gas. It depends only

on the interaction of two or more particles and in the framework

of the Kirkwood approximation [77] can be presented in the

following form:

12

2 4

2 n r drB

n r



(2.24)

If concentration of the nanoparticles is low, one can neglect B and

arrive at the conventional MG equation for the permittivity:

(2 ) 2ε

(2 )

host host

host

host host

f

f

(2.25)

2.2.2 Composite with prolate spheroidal inclusions

The polarizability tensor of an isolated spheroid with the radii of

a and c (rotation axis) can be presented in the following form:

||,

||,

host

hostN

(2.26)

where 24 / 3V ca is the volume of the spheroid. If z axis is

directed along the rotation axis of spheroid, polarization tensor

ij is diagonal, xx yy and ||zz . Here ||,N are

depolarization factors of the spheroid [3],

||,||

0 00,

2

0

11( 1) ln 1

2 11 host

N

,

where 0

2 2

c

c a

and ||1 / 2N N . Applying Maxwell Garnett approach

(MGA) [78] for effective composite permittivity tensor the same

way as it was presented above for spherical inclusion, but using

the polarizability tensor of the ellipsoids (2.16) we have effective

permittivity:

,||

,||

||, ||,

(1 )1

host

f

fN

(2.27)

and f is the volume fraction of the spheroidal inclusions. Thus the

dielectric constants of the Maxwell-Garnett GMN consisting of

metal spheroids depends on both concentration and anisotropy



of the metal inclusions. Then we carry on extremum analysis of

previous formula and obtain:

2 3Δ 1 3Δ

1 3Δ 1

SPR

p

N f N

N f

(2.28a)

4 3Δ 2 2 3Δ

2 3Δ 1

SPR

p

N f N

N f

(2.28b)

where 2 /p pc is the plasma wavelength, Δ 1/ 3N N .

It is worth to noting that we obtain SPR position as a function of

volume fraction, that is differ from formula (2.16) where is no

dependence of volume fraction. This is the main motivation of

using effective medium theory for real composite in which metal

concentration can vary.

For , p range of frequency, we show that SPR has

Lorentzian shape,

1

2 2

, , ,Im ΔSPR SPR

, (2.29)

where the linewidth is determined by the electron scattering

rate, , , Δ .

2

SPR SPR

p



Figure 8. SPR wavelengths for light polarized along a-axis (red) and c-axis

(blue) (⊥ and ∥ respectively) as functions of (a) the aspect ratio at the silver volume fraction f = 0.01(solid lines) and f = 0.1(dashed lines).(b) SPR

wavelengths as function of volume fraction at the aspect ratio c/a = 2. Johnson-

Christy data for silver were used for simulation [79].

2.3 CONCLUSION OF CHAPTER 2

SPR dominates the optical properties of GMNs. Since the SPR

spectral position critically depends on the size, shape and

concentration of metal inclusions, one can tailor the linear and

nonlinear optical response of the GMN by modifying the shape

of the nanoparticles. Vice versa, based on the SPR position one

can deduce the shape parameters of ellipsoidal nanoparticles

forming the composite, and their concentration. In particular, the

modification of the inclusion shape by ultrafast lasers opens the

way for optical engineering of GMN.


38

3 Nonlinear optics of glass-

metal composite

The composite media with embedded metal nanoparticles has

strong nonlinear response at the plasmon resonance due to

enhancement of the local field in the vicinity of the metal

nanoparticles. Although in the electric dipole approximation the

second-order nonlinear processes including SHG are forbidden

in GMN, the electric quadrupole and magnetic dipole

mechanisms of the optical nonlinearity still contribute to the SHG

in nanocomposites composed of spherical nanoparticles. It is

worth noting that the broken inversion symmetry at the metal-

dielectric interface may also result in the dipole SHG in GMN. For

NPs without inversion symmetry, the SHG in dipole

approximation is also allowed.

In this chapter, we consider SHG by metallic NP. Specifically,

by using hydrodynamic theory for SHG at metal surfaces [80-85]

we express the hyperpolarizability of a NP through local electric

field.

3.1 NONLINEAR RESPONSE OF GLASS METAL COMPOSITES

Optical properties of media with embedded nanoparticles with

strong nonlinearity are extensively studied since 80s. Composite

materials, in which size of inclusions is much smaller wavelength,

show visible nonlinear response due to local field enhancement

associated with plasmon resonance. Such media demonstrate

various optical phenomena originating from the third-order

nonlinearity [11-14] including optical Kerr effect, stimulated

Brillouin and Raman scattering, and harmonics generation [15-

18]. Small size of inclusions and large distance between them

Nonlinear optics in glass-metal composite


allows one to use electrostatic approximation [see chapter 2] to

describe nonlinear interaction of light and GMN.

Optical properties of metallic nanoparticles (NP) critically

depend on their size, shape and host media. In particular, shape

of the NP can considerably modify the nonlinear optical response

of the composite media. This property of GMN opens a way to

create metamaterial with desired optical properties. Although in

the electric dipole approximation, SHG is forbidden in bulk

medium with inversion symmetry, NP may show a strong SHG

response because the inversion symmetry is broken at the meta-

dielectric interface. Apart from the surface dipole nonlinearity

[85], bulk electric quadrupole and magnetic dipole contribute to

the second harmonic generation [83,85] in centrosymmetric

media.

In metals, the second harmonic is generated in the skin layer

of the metal, in which electromagnetic field is not zero. When size

of metal particle is of order of the skin layer, SHG can be

enhanced by local electric field due to surface plasmon resonance.

Metal island films, which are widely used in SERS

measurements [8], can be see and two dimensional glass-metal

nanocomposite. Metal nanoislands on the glass surface are often

have hemispherical shape, i.e. they do not possess inversion

symmetry. Linear optical properties of the hemisphere can be

developed by expanding electrostatic potential in series of

Legendre polynomials and first associated Legendre polynomials

(see Eqs. (2.17)). This approach can be employed to describe SHG

from small metallic hemisphere, whose second-order

nonlinearity originates from the oscillations of the conduction

electrons at metal surfaces [80].

3.2 SECOND ORDER NONLINEARITY

Electric displacement vector for nonlinear medium can be written

in form considering previous moments of time [86,87]:

0( ) ( ) ( )

t

t t d

D E J (3.1)



t

J P Q M (3.2)

in which P, Q and M are electric dipole (ED), electric quadrupole

(EQ) and magnet dipole (MD) polarizations, respectively. By

taking into account the second order nonlinearity of the medium

Fourier components of P, Q and M vectors can be presented in

the following form:

1 2 0 1 2 1 2( , ) ; , ( , ) ( , )ED

i ijk j kP ( )E E r r r (3.3)

1 2 0 1 2 1 2( , ) ; , ( , ) ( , )EQ

i ijkl j k lQ ( )E E r r r (3.4)

1 2 0 1 2 1 2( , ) ; , ( , ) ( , )MD

i ijk j kM ( )E E r r r (3.5)

where , ,ED EQ MD

ijk ijkl ijk are second order susceptibilities, subscripts

i, j, k, and l label coordinates of the Cartesian laboratory frame.

The tensor of the nonlinear dipole susceptibility 1 2; ,ED

ijk ( )

is zero for centrosymmetric materials, in which electric

quadrupole and magnetic dipole contributions dominate to

second order nonlinear response.

Constitutive equations (3.1) and (3.2) lead to the nonlinear

wave equation

2 2 2

02 2 2 2

NL

c t t t t

E E = P Q M , (3.6)

which describe in particular the SHG in centrosymmetric and

non-centrosymmetric media.

3.3 HYDRODYNAMIC THEORY OF ELECTRON GAS MOTION

To obtain the current density (3.2) for the ensemble of conduction

electrons interacting with an intense light wave one can employ

the theory for SHG at metal surfaces [80-84]. This theory is based

on the following classical equation of motion for the conduction

electron [86]:



1 e

pt nm m

uu u u E u B , (3.7)

where u is electron velocity, n is the electron density, ,e m and

are mass, charge, and collision rate for the conduction electron in

metal, respectively, p is pressure, 𝑬 and 𝑩 are electric and

magnetic field in metal. It can be shown [82,84] that at optical

frequencies the first term in the right-hand side of (3.7) is

negligible in comparison with other terms. In such a case the

nonlinear optical response of the electron gas in metal is

governed by the magnetic part of the Lorentz force.

When an intense light wave at frequency ,

( ) { }, , . .t r r exp i t c c E E

( ) { }, , . .t r r exp i t c c B B

propagates in the metal, the perturbative solution of Eq.(3.7)

yields:

, , 2 , 2exp i t exp i t u r t u r u r ,

where

, ,

e

m i

E ru r

11

2 , , , , , 22 2

ei

m

u r u r B r u r u r

By using the Maxwell equation , ,r i r E B , we

arrive at the following equation for the current density at the

frequency of the second harmonic:

3

222

4 2

2

ine,

m i i

i, , , ,

J r

E r E r E r E r

(3.8)

Rewriting vector product in the right-had-side of this equation

we can arrive at the following equation for the current density at

the frequency of the second harmonic:



3

228 2

2 2

2ne

, ,( ,rm i i

i i, , , ,

)

E r E r

E r E r E r E r

P

(3.9)

Comparing expression above with (3.3-5) one can determine

tensors of the nonlinear susceptibilities of the isotropic ensemble

of conduction electrons in the framework of the hydrodynamic

approach.

3.4 HYPERPOLARIZABILITY OF METAL PARTICLE

Let us consider the metal nanoparticles irradiated by intense light

beam. The nonlinearity of the conduction electron ensemble

results in the generation of the electric current inside the

nanoparticle oscillating at the second harmonic frequency. This

current results in the light wave at the frequency 2 , which can

be described in terms of the vector potential. In the Coulomb

gauge, the vector potential in the point R of the laboratory frame

2 ,A R can be presented in the following form [64]:

302 , 2 ,4

ike

dr

r R

A R J rr R

, (3.10)

where is fundamental wavelength and 𝑘 = 2𝜋/𝜆, integration

takes place over volume of metal particle. In the dipole

approximation for the second harmonic current, we keep only the

first term in (3.9). Further expanding of the exponent under the

integral and keeping only the first nonzero term give the

expression for dipole moment on second harmonic frequency:

3

33

22

12 2 , dr

2

1, ,

2 4 2

i

nedr

m i i

p J r

E r E r



(3.11)

It can be shown that the dipole moment is zero for

centrosymmetric particles and can be presented in the following

form:

0 02 ijk j kp E E i , (3.12)

where the subscripts label coordinates of the Cartesian laboratory

frame, ijk is the nanoparticle hyperpolarizability tensor, and

0 E is the amplitude of the incident light wave at the

fundamental frequency. For example, hemisphere belongs to the

point symmetry group vC , the following components of the

hyperpolarizability are nonzero: , , zzz xxz yyz zxx zyy

where Z-axis is along the substrate normal. Thus for s-polarized

incident wave, only z-component of the dipole moment oi the

second harmonic frequency is nonzero, i.e. 2p is directed

along the substrate normal.

In order to calculate the amplitude of the dipole moment at the

frequency 2 and, hence, the hyperpolarizability of the

hemisphere, we need to find the amplitude of the fundamental

wave at the frequency inside NP, see Eq. (3.13). This can be

done in the framework of the electrostatic approximation, i.e. by

assuming that nanoparticle size is much smaller than the

wavelength, see chapter 2 and paper III for more details.

3.5 THIRD ORDER NONLINEARITY

In the centrosymmetric medium, the third order nonlinearity

gives rise to the modification of the absorption coefficient and

refractive index in the presence of an intense light wave. This

phenomenon can be described in terms of effective dielectric

tensor of the medium [87]: (1) (3)( ) 1 ( ) ( ; , , ) ( , ) ( , )ij ij ijkl k lE E r r

(3.13)



When a linearly polarized intense light wave propagates an

isotropic medium, the intensity dependent refractive index n and

absorption coefficient can be presented in the following form:

0 2

0 2

n n n I

I

, (3.14)

where I is the light intensity, 0n and 2n ( 0 and 2 ) are linear

and nonlinear refraction indices (absorption coefficients),

respectively. They can be presented in terms of the components

of the nonlinear susceptibility tensor, which depends on the

mechanism of the optical nonlinearity. If the saturation

mechanism dominates, the nonlinear absorption is

conventionally described in terms of the saturation intensity as

the following [87]:

0

01

sn

I I

, (3.15)

where n and s are non-saturable and saturable parts of the

linear absorption coefficient, respectively, and 0I is saturation

intensity. This mechanism dominates in the vicinity of excitonic

resonance in semiconductors and is described in details in

Chapter 5.


Surface plasmon resonance, which is associated with collective

oscillations of the conduction electrons in metal nanoparticles,

can strongly manifest itself in the nonlinear optical response of

the GMN. Such nonlinear optical effects as second harmonic

generation and optical Kerr effect can be strongly enhanced when

the frequencies of participating waves approaching SPR. This is

caused by the resonant enhancement of the local field in the

vicinity of the nanoparticle and can lead also to e.g. SERS and a

giant two photon absorption. The second-order nonlinear

response is crucially dependent on the shape of the nanoparticles,



and especially on its asymmetry with respect to inversion

operation. The lack of inversion center in an individual

nanoparticle enables the electric dipole of the second-order,

which can be much stronger than electric quadrupole of the first

order. In paper III these effects were studied by considering the

second harmonic generation by silver hemispheres covered by a

dielectric shells and placed on a metal-dielectric interface.


47

4 Disc/ring microcavities

with quantum dots

This chapter presents results on the investigation of the

whispering gallery mode (WGM) resonance in the semiconductor

disc and ring microcavities. This includes theoretical

background, major fabrication technology steps and

experimental results on the WGM light emission from the GaAs

microcavities containing InAs quantum dots. Specifically, atomic

layer deposition, electron beam lithography and plasma dry

etching techniques were used in context of fabrication

microdisk/ring lasers based on quantum dots. The developed

technology has allowed us to fabricate and achieve lasing in

ring/disc resonators of ultra small diameter.

4.1 WHISPERING GALLERY MODE RESONANCE

The term “whispering gallery modes” has been coined by lord

Rayleigh in 1910 when he discovered that in a round cavity

beneath St Paul's Cathedral dome shown in Fig. 9(a), the sound

could propagated over a long distance experiencing multiple

total internal reflection [88]. Similar phenomenon can be

observed for electromagnetic waves [89-92], which can propagate

along e.g. cylindrical or spherical surface with very low

dissipation rate. This implies that such surfaces can be employed

as high quality factor electromagnetic cavities.



Figure 9. (a) Photo of St Paul's Cathedral dome1, (b) Dielectric disc surrounded by

another material with a smaller refractive index.

Light ray experiences total internal refraction (TIR) at the

boundary between two media when refractive index of first

material higher than refractive index of the second one.

Conditions of TIR follows from Snell’s law, if angle of incidence,

inc satisfy condition c inc where 1 2arcsinc n n then TIR

takes place and the light wave does not leave medium 2. Fig. 9(b)

shows whispering gallery wave propagation in the disc cavity

with refractive index n2 embedded into dielectric media with

refractive index n1. For waves satisfying conditions of total

internal reflection we have standing wave condition which

means that perimeter of the wall should be of order of several

light wavelengths in medium:

2

2 R Nn

, (4.1)

where N is an integer. Other words, the integer number of waves

lie on perimeter of circle. Equation (4.1) allows one to estimate the

resonance frequencies, however more detailed calculations can

be found in [28].

For a cylindrical surface, optical modes are characterized by

azimuthal direction number m, radial direction number q and

normal direction number p:

2,

2m q

nR T

where Tm,q is the q-th root of the mth order Bessel function. Here

we consider 2D problem, z-direction ignored. It is justified for

1 http://www.explore-stpauls.net/oct03/textMM/DomeConstructionN.htm

http://www.explore-stpauls.net/oct03/textMM/DomeConstructionN.htm

Disc/ring microcavities with quantum dots


disk microstructures, which are quite thin and have a single

(fundamental) vertical TE-like mode. In this notation TM(TE)

modes correspond to the field with zero projection of the

magnetic (electric) field on the cylinder axis.

4.2 DISC/RING MICROCAVITIES WITH ACTIVE MEDIUM

BASED ON QUANTUM DOTS

Elements of future photonic integrated circuits have been actively

investigated during recent years. In this field, ring or disk

microcavity can be the included into photonic circuits as low

threshold microlasers [35,36], modulators [31] and add/drop

filters [33]. The main feature of ring/disc microcavity is a high

quality factor of the spreading around the surface whispering

gallery modes. The high-Q WGMs open the possibility to achieve

low-threshold lasing in the structures with size much smaller

compared to the Fabry – Perot cavities. Also, an ultra small

microcavity provides a large distances between neighboring

resonance wavelengths, which facilitates obtaining single

frequency oscillation. From the viewpoint of integrated optics,

ultra small microcavity reduces an active region volume enabling

high density of components in an optical circuit.

Figure 10. (a) Scanning electron microscopy image of QD microring with

remaining of dielectric mask on the top. In the schematic layout (b), brown shows

GaAs, green layers are AlGaAs barriers, InAs/InGaAs quantum dot region is

presented by red/grey. Inset shows sketch of the active region.



However, fabrication procedures and especially etching may

increase the number of defects. This is increasing non-radiative

carriers recombination rate that results in a higher lasing

threshold. This difficulty can be avoided by using self-organized

quantum dots (QDs) as an active medium. In QDs, even at 300 K

the electron and hole confinement in QDs is sufficiently strong

[27] to provide deep localization of charge carriers. In such case

the non-radiative recombination at sidewalls is the main source

of the losses.

4.3 SEMICONDUCTOR NANOSTRUCTURE FABRICATION

We manufactured 2 μm in diameter quantum dots

microdisc/microring lasers working at room temperature under

optical pumping. In the fabrication experiments we used a chip

grown with molecular beam epitaxy (MBE), which contained five

layers of InAs/In0.15Ga0.95As quantum dots and AlGaAs barriers.

The set of the 2 μm ring/disk resonators of inner diameter from 0

(microdisks) to 1.4 μm was patterned with electron beam

lithography and reactive ion beam etching through the atomic

layer deposition made 100 nm thick SiO2 mask. Scanning electron

microscopy and photoluminescence measurements were applied

to characterize the laser structures emitting in 1.3 μm range. Laser

generation threshold as the function of the ring thickness was

studied.

The fabrication of the ring/disc microcavity QD lasers

includes the following technology steps (see Fig. 11):

(a) Molecular beam epitaxy

(b) Deposition of SiO2 by ALD (at~300C, thickness ~150-

200 nm) and spinning of negative e-beam Resist (150 nm)

(c) E-beam lithography

(d) Resist development

(e) Etching of SiO2 mask (RIE etching)

(f) Etching of GaAs structure mask (ICP etching)



Figure 11. Technology steps necessary for fabrication a semiconductor ring

microcavity (a-f)

4.3.1 Molecular beam epitaxy

The structure was grown by molecular beam epitaxy on semi-

insulating GaAs(100) substrate. An active region contains five

layers of InAs/In0.15Ga0.85As QDs separated by 35-nm-thick GaAs

spacers. The active region was placed in the middle of a 210-nm-

thick GaAs waveguiding layer confined from both sides by 20nm

thick Al0.3Ga0.7As barriers. A 450-nm-thick Al0.98Ga0.02As cladding

layer was grown beneath the waveguiding layer, Fig 12(a).

Figure 12. (a) Schematic representation InAs/InGaAs planar structure. (b)

Photoluminescence spectrum of InAs/InGaAs QDs at T=300K (c) Sketch of the

semiconductor heterojunction bandsrtucture.

Fig. 12(b) shows photoluminescence spectrum which indicate

ground state (GS), first excited (ES1) and second excited (ES2)



optical transitions of the QDs, Fig. 12(c). Lasing can takes place

on GS and ES1 transitions, but here we describe only GS lasing.

0 0GS e h bgE E E E (4.2)

where 0 0,e hE E and bgE are electron, hole fundamental states in

conductive and valence band, and band gad energy, respectively

[93].

4.3.2 Electron beam lithography

Electron beam lithography (EBL) is a technology that allows one

to create structures with a spatial resolution as low as several

nanometers [94,95]. EBL has become a conventional technique for

obtaining masks for subsequent use in photolithography to

produce monolithic chips. This also includes projection

photolithography masks for mass production of ultra-large chips.

We used negative resist nLOF as the second mask. For electron

lithography we use Gaussian beam vector scanning system. The

electron source is a high current density thermal field emission

gun. The pattern generator of our system can work with up to 50

MHz frequency making the exposures of big areas really fast.

Quality of produced resist mask is very high that was verified by

SEM. The following e-beam parameters were used: dose= 100

uC/cm2, current= 4 nA, frequency= 49 MHz. nLOF development

is done with pure AR 300-47 90 seconds developer. After

development the sample is rinsed in water for 30 seconds. After

and before e-beaming sample heated till 110C.

4.3.3 Atomic layer deposition

Atomic layer deposition (ALD) is a thin film deposition

technique, which is based on a sequence of chemical reactions

between the vapor and solid body and has the ability to self-

restraint [96-99]. Most of the reactions use two chemical

compounds that are commonly referred to as precursors.

Precursors alternately react with the surface giving rise to the thin

film growth.

ALD is a self-regulating process (number of precipitated

material in each reaction cycle is constant) in which sequential

chemical reactions occur resulting in a uniform thin film



deposited on a substrate. ALD process is similar to that of

chemical vapor deposition and possesses properties of self-

limiting surface reactions. ALD process of growing thin films

makes it possible to control deposition at the atomic level.

Keeping the precursors in the deposition process alone one can

achieve process control at ~ 0.1 Å (10 picometers) per cycle.

Separation is carried out by pulses of precursor cleaning gas

(usually nitrogen or argon) after each pulse of the precursor to

remove residual precursor from the reactor and prevent the

"parasitic" chemical reactions on the substrate.

We make on the surface of GaAs hetero-structure a stack of

two masks, silicon dioxide (SiO2) mask and resist mask. High

quality SiO2 mask was produced by atomic layer deposition

(ALD) setup. It’s supposed to be used in GaAs etching process as

main mask. The secondary mask (resist mask) is employed to

form the main mask.

Figure 13. SiO2 with NLOF negative resist on the top, mask formed using ALD

setup

We produced the array of disc/ring microcavities of different size

and with different inner diameter.

4.3.4 Reactive ion etching

Reactive ion plasma dry etching (RIE) is based on the etching of

materials under the influence of a gas-discharge plasma [100,101].

The advantage of this method compared with wet etching is high

anisotropy and the absence of need in removing the reaction

products from the surface.



We proceed dry etching of SiO2 with Plasmalab 80 from

Oxford Plasma Technology that is a reactive ion etching system

featuring a loadlock and an Inductive coupled plasma (ICP) unit.

Dry etching of GaAs structure was done with Plasmalab 100 from

Oxford Plasma Technology that is a reactive ion etching system

featuring a loadlock and an ICP unit, which allows us to increase

the ion density in the process. For SiO2 etching we employ

standard parameters CHF3=12 sccm, Ar=38 sccm, Pressure=45

mtorr, Power= 220 W. For GaAs etching we used SiCl4/Ar based

chemistry and ICP-RIE. More detailed etch parameters were,

power: 60W(RIE), 200W(ICP), pressure: 3mTorr, plate

temperature: -50C, the etching was performed at low

temperatures.

The fabrication process started with SiO2 coating using ALD.

SiO2 layer is used as a main mask during semiconductor etching

process. Then SiO2 layer covering with electron resist and EBL

patterning were performed. The pattern was transferred from

resist to SiO2 layer by dry etching (Fig. 14a). Then GaAs ICP

etching was produced, resulting mask can be seen in Fig. 14b.

Figure 14. (a) SEM image of SiO2 mask before GaAs etching and (b) after GaAs etching

in SiCl4/Ar plasma.

4.4 MICROPHOTOLUMINESCENSE MEASUREMENTS

Photoexcitation of the QDs in the ring and disc microcavities was

carried out through an Olympus 100 microscope objective using

the second harmonic of a YAG:Nd laser at the wavelength of

532nm and beam power of 10 - 200mW). The photons emitted by



QDs passed through a Horiba FHR1000 monochromator, were

collected by the objective and detected by InGaAs CCD array

(Horiba Symphony).

Micro photoluminescence (‘light-in-light-out’ characteristic)

spectrum was obtained under optical pumping at room and 100 0C temperature.

Figure 15. Microphotoluminescence spectrum of the QD disc laser pump power of 1 𝜇W

(blue), 6 𝜇W (red) and 25 𝜇W (blue); (b) intensity and FWHM of the of TE12,1 mode as

a function of pump power. Pictures taken from Paper V.

Photoluminescence spectra of experimental microcavities are

presented on Fig. 15a. The spectra show spontaneous emission

from QDs with sharp WGM resonances. Lasing is visualized by

measuring dependence of the mode intensity on the excitation

power. The resonance height and full width at half-maximum

(FWHM) as functions of optical pumping power are presented in

Fig. 15(b); this shows laser generation threshold at the pump

power of about 5μW. Specifically, the lasing threshold, the

intensity of a single mode strongly increases indicating transition

from spontaneous emission to lasing. Cavity geometry, in general,

determines WGM wavelength position.

4.5 THRESHOLD CHARACTERISTICS OF QD RING

MICROCAVITIES

Measured photoluminescence spectra shows critical influence

the inner diameter of the ring microcavity on the threshold

characteristics. In particular, we studied lasing at room



temperature in QD-embedded ring microcavities with outer

diameters of 1.5 and 2 μm and different inner diameters. The

results of the measurements shown in Fig. 16 allow us to conclude

that the dependence of the lasing threshold on the inner ring

diameter in non-monotonous. Specifically, at the inner ring

diameter smaller than 0.8 μm, the lower diameter of the ring

provides the lower the lasing threshold. That is the threshold goes

down when the active region volume decreases. However further

thinning of the ring leads to increasing of the lasing threshold.

This can be explained by increasing the non-radiative

recombination centers at the ring inner walls, see Fig. 16.

Figure 16. Threshold pump power as a function of inner diameter of the ring cavity.

Pictures from Paper V.

4.6 TEMPERATURE DEPENDENCE

When the size of the nanostructure is comparable with the

electron de Broglie wavelength, quantization of the energy levels

occurs [102,103]. As it was mentioned in the Chapter 1, the QD

emission shows better – in comparison with QW emission -

temperature stability due to full charge confinement. In contrast

to QW, radiative recombination in QD does not depend on

temperature. Probability of electron-hole recombination (EHR) is

much higher when they are localized in a small region (see Fig.

17).



Figure 17. Carrier motion geometry for (a) bulk semiconductor (b) quantum well (c)

quantum dots array. (d-f) electron density of states for these structures.

Lasing threshold and photoluminescence spectra of ring

microcavities at temperatures of 20 0C and 100 0C are

demonstrated in Fig. 18. At room temperature, the TE12,1 mode

position is close to the GS transition, see Fig. 18(b), and the

generation threshold is low. Under increasing the temperature

the position of the GS shifts toward longer wavelengths, that is

detuning of WGM resonance and GS transition resonance takes

place, hence lasing threshold increases. In the range of 40-80 0C

there is no generation at TE12,1 mode. But after 80 0C the TE12,1

mode perfectly matches the ES1 transition and we again have low

threshold. Threshold power for TE12,1 mode increases with the

increase of detuning, but TE11,1 moves closer to GS transition, and

it is perfectly tuned at 80 0C, that corresponds to lower threshold,

see Fig. 18(a).

Figure 18. (a) Threshold pump power as a function of temperature for TE12,1 (blue

squares) and TE11,1 (red circles) modes. (b) Photoluminescence spectrum at temperature

20 0C (black) and 100 0C (red). Figures are take from paper V.




Optical properties of semiconductor nanostructures (e.g. QW and

QD) are determined by their size, shape and properties of the host

medium [93,102-103]. Using semiconductor nanostructures as the

active medium for lasers is possible as soon as smooth adjustment

of the radiative transition resonance in the particle and the optical

cavity mode is provided. In disc-shaped cavities, the WGM

resonance with extremely high Q-factor can be the tuned to the

GS transition in InAs quantum dots.

In ultra small WGM cavities, the inter-mode distance is

relatively large and allows one to achieve single-mode lasing.

WGM modes positions are determined by the size and shape of

the microcavity while the lasing threshold essentially depends on

the internal diameter of the ring, see paper V.


59

5 Nonlinear phenomena in

exciton-polariton

condensate

In this Chapter, we discuss resonant nonlinear phenomena in

semiconductor heterostructures when strong exciton-photon

coupling can lead to the formation of cavity exciton polaritons

(EPs). Condensation of these Bose quasi-particles is one of the

most interesting phenomena in the modern solid state physics.

Polariton-polariton scattering leads to nonlinearity originating

from the Coulomb interaction between excitons’ constituencies.

This cubic nonlinearity plays an important role in condensate

formation under optical or electrical pumping. In this chapter a

theory of electrically pumped microcavity with Kerr-like

nonlinearity (exciton-polariton laser) is presented. In order to

describe the condensation in real semiconductor heterostructure,

drift diffusion equations for electrons and holes are employed

together with the Gross-Pitaevskii equation for EPs and

Bolzmann equation for exciton reservoir.

The third-order nonlinearity corresponds to repulsive

(defocusing) exciton-exciton interaction and leads to the

formation of solitary waves (conservative solitons) in the

condensate. Another type of solitons can arise due to the effect of

saturable absorption in the system. We will demonstrate that in a

microcavity with a saturable absorber inside the Bragg mirrors

(in the so-called SESAM configuration), EP can forms spatial

dissipative soliton (DS) in visible spectrum (as opposed to regular

conservative solitons). DS manifests itself as spatially stable picks

of the EP density preserving their shape and amplitude in the

nanosecond time scale. It will also be shown that the DS



originates from the increase of the effective photon lifetime in the

regions of high polariton density that leads to bistability and

formation of spatial domains with different polariton densities.

The regimes, which allow for creation and destruction of DSs,

are referred to as dissipative soliton protocols, which critically

depend on the excitation conditions. In this Chapter, the DS

protocol at finite temperatures is presented. DS in the presence of

strong polariton-polariton interaction is also discussed.

5.1 ELECTRIC AND OPTICAL PROPERTIES OF

SEMICONDUCTOR NANOSTRUCTURES

Fundamental optical characteristics of semiconductors are

dictated by the crystal symmetry and the band structure [104-

106]. For the sake of effective light-matter interaction, the

maximum of the valence band and the minimum of the

conduction band should correspond to the same momentum, see

Fig. 19(a). Such materials can effectively absorb and emit photons

due to the selection rules, and they are called direct band

semiconductors. In the vicinity of the band edge, direct band

semiconductors usually show excitonic resonances, which

represent bound states of an electron and a hole in the crystal

lattice. Excitons at low enough concentrations can be considered

as Bose quasiparticles due to integer total spin of the electron-hole

pair forming them. Beside this, conventionally excitonic

resonances can be observed at low temperature, when the energy

of the electron-hole interaction exceeds the thermal energy. If the

distance between the bound electron and hole is much larger than

the crystal lattice constant, the exciton is called the Wannier-Mott

exciton [107-109], see Fig. 19(b). When, instead, the distance

between the electron and hole is smaller than the crystal lattice

constant, the exciton is referred to as the Frenkel exciton [110], see

Fig. 19(c).

Nonlinear phenomena in exciton-polariton condensate


Figure 19. (a) Energy band gap diagram for semiconductor zinc blende crystal

structure. Eg, ESO and EX are bandgap, spin-orbit coupling energy and exciton energy,

respectively. (b) the Wannier-Mott exciton in a crystal lattice; (c) the Frenkel exciton in

a crystal lattice.

5.2 EXCITON-PHOTON STRONG COUPLING

When confined electrons are placed into an optical microcavity,

one may achieve strong coupling between excitons and photons

if the energy of the cavity mode is close to the exciton energy, see

Fig. 20(a). The planar microcavity is usually formed by a pair of

Distributed Bragg Reflectors (DBRs), shown in Fig. 20(b). Yellow

region here corresponds to heterostructure barriers for electron

and holes from the active region. To produce such structures, one

can use MBE technology, described in Chapter 4.

When a quantum well (QW) is embedded in a high-quality

factor microcavity, strong photon-exciton coupling leads to the

formation of EPs. These bosonic quasiparticles [111-113] have

half-photonic character, which allow particle wavelets propagate

with high velocity, and half-excitonic character allowing for

interactions, as has been mentioned before.

Exciton and cavity photon dispersions are: [114-115]: 2 2 ( )

(0)2 2

XX X

X

kkE E i

m

(5.1)

2 2 2 2 2

2

( )(0) (0)

2 2

cc c c

c c

kc c k kE E E i

n n m

, (5.2)



where k is the electron wave vector, X cm m and X c are the

exciton (photon) masses and damping rates, respectively, cn is

the effective refractive index of the heterostructure of the

microcavity.

The Hamiltonian of the ensemble of excitons coupled with

cavity photons is:

† † † †( ) ( ) ( )2

RX k k c k k k k k k

k k k

H E k a a E k c c i a c a c

, (5.3)

where † †, , ,k k k ka a c c are creation and annihilation operators for

excitons and photons, respectively, /R d Ε is the Rabi

frequency that characterizes the strength of the coupling; here d

and E are the dipole matrix element of the transition and electric

field. Eigenmodes of the system, E(k), represent the upper (UP)

and lower (LP) polariton branches and they can be found via the

diagonalization of the Hamiltonian above:

2 2

2 2

( )2 2

0

( )2 2

R

X

R

c

kE k

m

kE k

m

,

2

2 2 2 2 2 2 2 22 2

,

1 1( )

2 2 2 2 2 2UP LP R

X c X c

k k k kE k

m m m m

(5.4)

where ”+” and ”–” correspond to UP and LP. Rigorous quantum

theory of exciton-photon coupling can be found, for example, in

[116,117].



Figure 20. (a) Cavity photon dispersion Ec (blue dashed line), exciton dispersion EX

(green dashed line), UP and LP are disperions of the upper- and lower-branch polaritons

(red lines). (b) In the quantum well microcavity, photons with frequency ωc are localized

between two DBRs, excitons are localized in the QW. Yellow layers correspond to

potential barriers that confine exciton. γc is the radiative loss rate and is light field

amplitude.

5.3 EXCITON-POLARITON CONDENSATION

When a macroscopic number of EPs is accumulated in the single-

particle ground state of the system, such state can be referred to

as EP quasi-condensate. It is worth noting that the EP ensemble

can never reach a perfect thermal equilibrium due to the finite

lifetime of the particles (10-100 ps). Therefore, here a kinetic

approach is required for the description instead of a

thermodynamic one. The temporal evolution of the system can be

described with the Gross-Pitaevskii equation for the

microscopically averaged EP wave function 𝜓 also referred to as

the macroscopic wave function or the order parameter [118]:

2( ( ) ( ) )

2LP R c ci E k i Rn V g P

t

, (5.5)

where Rn is the exciton density, c is the cavity photon decay

rate, which determines the polariton lifetime [119,120] , ,R V g

and Pc are system-reservoir interaction strength, electric potential,



two particle interaction coefficient and coherent pumping power,

respectively. It is worth noting that g is determined by the

exciton-exciton interaction, which leads to the Kerr-like optical

nonlinearity in condensate [121].

EP condensate is an open quantum system that interacts

with the exciton reservoir, continuously exchanging particles

(excitations). This process is described by the second term in the

right-hand-side of Eq. (5.5). The temporal evolution of the exciton

density is determined by the excitation method. In the Thesis we

will consider QW microcavity with electrical and optical

excitations.

5.4 SEMICONDUCTOR MICROCAVITY AT THE ELECTRICAL

EXCITATION

Theoretical treatment of electrically pumped microcavities is

important for various applications (see Introduction). In the

framework of our formalism, at the electrical pumping the

exciton density is determined by electron and hole concentrations

in the active region (QW). In this case the exciton formation is

described by the Boltzmann equation [118]:

2( | | )RR R

nWnp R n

t

, (5.6)

where W is the exciton formation rate, n and p are electrons and

holes concentrations. The spatial-temporal evolution of the

carriers’ concentrations can be described in the framework of the

drift diffusion equations model (DDE), which is based on the

following assumptions [136,137]:

- all dopants are assumed to be ionized

- all variables are independent of time

- the temperature is constant throughout the device.

This approach can be used to calculate carriers’ concentrations for

the heterostructure, shown in Fig. 21. below.



Figure 21. Quantum-well microcavity comprising of the emitter layers, cladding layers,

DBR layers, n and p contacts under electrical excitation.

The distribution of the electric potential in any semiconductor

structure can be described by the well-known Poisson equation:

2

0

2

d= =

dz

, (5.7)

with the charge density +

D A= q(N N + p n) , where +

DN , AN ,

n and p are concentrations of ionized donors, ionized acceptors,

electrons and holes, respectively. Since the carriers are governed

by the Fermi statistics, the concentrations of electrons and holes

are given by the formulas:

1/2n C

C

F E +qn= N F

kT

, (5.8a)

1/2

V p

V

E F qp = N F

kT

, (5.8b)

where 1/2

0

2

1 exp

xdxF (ξ)=

+ (x ξ)

is the Fermi integral of the order , Fn and Fp are the Fermi

quasi-energies of electrons and holes, NC and NV are the

densities of states in the conduction and valence bands,

respectively, ЕС and EV are the conduction band bottom and the

valence band top energies. The densities of states in Eq. (5.8) are:



3/2

22

2

nC

m kTN =

π

, (5.9a)

3/2 3/2

2 22 2

hh lhV

m kT m kTN = +

π π

, (5.9b)

where mn, mhh and mlh are effective electron mass, the light and

heavy holes masses, respectively. The concentrations of ionized

impurities are also given by the Fermi-Dirac distribution:

1 exp

+ DD

n C DD

NN =

F E +E +q+ g

kT

(5.10a)

1 exp

AA

V A p

A

NN =

E + E F q+ g

kT

. (5.10b)

Here gD = 2 for the donor states, and gA = 4 for the acceptor ones,

ED and EA are the ionization potentials [136].

Fermi quasi-energies can be found from the following

generalized Poisson equations:

n nμ n F = q (G R) (5.11a)

=+p pμ p F q (G R) (5.11b)

where nμ and pμ are the mobilities of electrons and holes. G

and R are the carriers generation and recombination rates,

respectively.

The recombination rate can also be found [139] as

11 exp

n p

p n

F FR = np +γ

kT τ n+ τ p

, (5.12)

where nτ and pτ are the non-radiative lifetimes of the electrons

and holes on a point-like defects (the Shockley-Read mechanism),

and is the radiative recombination rate.

The e-h pairs generation rate, G, in a semiconductor is given by



+

0

dEG(z)= P(E) α(E) [ 1- r(E)]exp [-α(E)z]

E

, (5.13)

where r(E) is the reflectivity, α(E) is the light absorption

coefficient and P(E) is spectral radiance function, E is energy. In

the 2D semiconductor nanostructure, the z-dependence can be

ignored. Optical pumping can be considered as black body

radiation at the temperature T. In the such a case the spectral

radiance function is given by is the black body spectral radiance

[140].

Equations (5.8-12) constitute a consistent system of equations,

which should be solved to obtain electric potential and Fermi

levels distribution along a heterostructure. The bias (applied

voltage) can be introduced through the boundary condition:

0n pF ( ) F (L)= qU . (5.14)

We apply the theory presented above for a realistic nitride-

based heterostructure. An InGaAlN alloy-based microcavity is

presented in Fig. 21. The active region of the heterostructure

consists of 5 nm In0.06Ga0.94N QW. It is located between

n+Al0.15Ga0.85N and p+Al0.15Ga0.85N highly doped DBR emitters.

More details about material and parameters can be found in

[paper I], the scheme of heterostructure is presented in Fig. 21.

Fig. 22 shows the distribution of the carriers density through

the active region when a voltage U is applied. One can see that

with the increase of U, the carriers concentrations inside the

active region increases (a-c). The band diagram shown in the left-

hand inset in Fig. 22(a) clearly demonstrates that the Fermi quasi-

energies are quite far from the band edges in the quantum-well

region, i.e. the carrier concentration is low. The spatial

distribution of the scalar electric potential is presented in the

right-hand inset in Fig. 22(a). Figure 22(b) shows the spatial

distribution of the carrier concentrations at the polariton BEC

threshold at approximately U = 2.3 V (See Fig. 23) when the

concentration of the e-h pairs is high enough to achieve the

polariton BEC.



Figure 22. Distribution of electrons and holes along the z-axis of the heterostructure

presented in Fig. 21 under forward bias of (a) 1 V, (b) 2.3V (с) 3 V. Left side insets show

conduction and valence bands and Fermi energies for electrons and holes for each bias.

Right side insets dependence of the scalar potential on z. DDE were solved using

COMSOL for heterostructure consisting of 5nm In0.06Ga0.94N QW and 150nm

Al0.15Ga0.85N high doping emitters. For more details see paper I.



Using the GPE (5.5), nonequilibrium Bolzmann equation (5.6),

and the DDE (5.8-12) one can simulate the evolution of the system

at any temperature, including the room temperature. Applying

this approach to heterostructure presented in Fig. 21 one can

observe the threshold like behavior, which corresponds to the

condensation. Fig. 23(a) shows the ground state occupation as a

function of applied voltage. For low voltages, EPs occupy states

corresponding to their thermal distribution. However when the

applied voltage exceeds U = 2.3 V, the occupation drastically

increases (up to 10 orders in magnitude for less then 0.1 V step)

indicating the polariton condensation threshold. Further increase

of the voltage does not result in a significant change of the particle

number.

Figure 23. Exciton-polariton density in the vicinity of k = 0 for the InGaN quantum-well

diode presented in Fig. 21 as a function of forward bias U. Insets show a color map of

the quasi-particles distribution in momentum space for different voltages at U = 2.2 V

(under threshold, left) and at U = 2.3 V (above threshold, right).

5.7 SATURABLE ABSORPTION

In a medium with strong and fast nonlinearity, the resonant

absorption coefficient can decrease at high light intensities. This

phenomenon is called the absorption saturation [124,125]. It may



occur in the case of relatively long relaxation time of the excited

state that is when its population increases during the excitation

pulse. The absorption saturation can be observed in various

materials, including atomic gases, ions embedded into a

crystalline matrix and others. In semiconductors, saturation of

the interband absorption (i.e. increase of the transmission) can

take place when electrons are promoted to the conduction band

faster than they relax down to the valence band.

In the presence of the saturation, the dependence of the

absorption coefficient, on the light intensity, I , can be described

by the following equation:

0

01

sn

I I

(5.15)

where αn and αs are non-saturable and saturable parts of the linear

absorption coefficient, respectively, and 0I is the saturation

intensity. In our theory, we adapt this formula for description of

the dynamics of the photon part of EPs in a SESAM microcavity

[126,127].

5.8 DISSIPATIVE SOLITONS IN MICROCAVITY

A soliton is a structurally stable solitary wave propagating in a

nonlinear medium. There exist two types of solitons: conservative

(in Hamiltonian systems) and dissipative solitons (in open

systems). Solitons, as essentially nonlinear wave objects, have

common features, such as shape stability in space, time or space

and time simultaneously.

Conservative solitons constitute a family of solutions with a

continuously varying parameter, e.g. width of the soliton or a

maximum intensity. Dissipative solitons impose set of basic

parameters are not continuous, but discrete. This fact leads to

increased stability of dissipative optical solitons with respect to

conservative ones. It makes them promising for various

applications, including those for optical processing information.



Appearance of a soliton in SA-structures results from the

increase of effective lifetime of the cavity photon in the regions of

high EPs density and can be introduced by adding a relevant

nonlinear term to the equation of motion for the order parameter.

This fact provides spatially-dependent lifetime-enhanced

formation of localized solutions. The bright soliton is the trade off

between gain and dispersion on the one hand, and nonlinear

losses, on the other hand [128-132]. Mathematically, SA in

semiconductor microcavities can be described by replacing in Eq.

(5.5) c with 21 1c

, where and σ describe

saturable absorption and saturation intensity. In this case, the

Gross-Pitaevskii equation yields

2

2( ( ) ( (1 )) )

2 1LP R c ci E k i Rn V g P

t

(5.16)

Figure 24. Semiconductor microcavity with saturable absorber embedded in one of the

DBRs. Electrical (or light CW) incoherent pumping (P0) is used for maintaining of the

DS, coherent (Pc) and incoherent (Pi) light pulses are used to switch the DS on and off.

The cavity photons and polaritons are localized between two Bragg mirrors and in the

QW, respectively. In a GaAs/AlGaAs QW microcavity, InGaAs layer can be used as a

saturable absorber.

2( | | ) R iR

R

nR n P

t

, (5.17)



where R and iP are exciton decay rate and incoherent pumping

power, respectively.

An important feature of EPs in a microcavity is strong

interaction with acoustic phonons of the crystal lattice. Thus,

phonons tend to convert a soliton-like propagation into a

diffusion-like motion. This leads to detuning and destruction of

the soliton. In order to model the interaction with acoustic

phonons, we use the Fröhlich Hamiltonian [133-135].

Figure 25. (Upper panel) DS density in QW as a function of transversal coordinate x

and time t at temperature T = 15 K. (Lower panel) The phase of the condensate particles

as a function of x and t.

5.9 DISSIPATIVE SOLITON PROTOCOL

Formation and lifetime of a DS critically depend on the relation

between the pumping and losses. An excitation realized by two

Gaussian in space beams is created in the center of the sample by

a short strong laser pulse thus preparing two spatial regions with

different particle concentration, and incoherent pumping (using



a light diode or electric current). Further, the system evolves only

under background homogeneous nonresonant excitation. We

present the DS protocol in Fig. 26.

Figure 26. DS protocol at T = 0 for heterostructure presented on Fig. 24. When the

exciton density is saturated, coherent and incoherent light pulses excite the system at t

=1500 ps. The coherent pulse with duration of 100 fs increases the polariton population,

while incoherent pulse with duration of 1 ps increase exciton density. The switch-on

time is τon = 600 ps. At t = 2500 ps, CW pumping is switched off (switch-off time is τoff

≈ 100 ps) and then switched on. The exciton population growth time is τr = 900 ps.

Figure is taken from paper II, which contain details of the numerical simulation.


Excitons in a semiconductor microcavity can resonantly interact

with cavity photons to form microcavity polaritons. Features of

polaritons are determined by the microscopic resonance

corresponding with electronic structure of an active medium, and

macroscopic optical resonance of light waves in the microcavity.

If the microcavity contains layers with SA, optical dissipative

polariton solitons can be created. Properties of the SA are defined

by layer thickness. Formation of solitons is due to the bistability

of states. The polariton condensation under the electrical

pumping can be described in terms of the DDE approach and

Bolzman equation describing evolution of the exciton reservoir.


75

6 Summary

In this Thesis, the results of the experimental and theoretical

study of the resonance optical phenomena occurring in nonlinear

and active photonics nanostructures are presented. This work is

dedicated to the fundamental aspects of the nanophotonics as

well as to the modeling and fabrication of advanced photonic

nanostructures using clean room technologies.

Resonance phenomena in nonlinear nanophotonics based on

glass-metal nanocomposite media are governed by the size,

shape and nature of metal nanoinclusions embedded in a

dielectric matrix. Optical response of the GMN is determined by

the collective oscillation of conduction electrons (surface

plasmons), which strongly influenced by the shape of the metal

particle. By solving Maxwell equations using eigenmodes

expansion method we described the optical absorption spectra of

the Maxwell Garnett GMN comprised of spheroidal

nanoparticles. In experiments, we observed the evolution of these

spectra under irradiation of the GMN with intense femtosecond

laser pulses. We demonstrate that elongation of the initially

spherical nanoparticles results in the lifting polarization

degeneracy of the surface plasmon leading to the linear

dichroism of the GMN. This allows one to deduce shape

parameters of the spheroidal particles via measuring absorption

coefficient of the GMN for light waves polarized parallel and

perpendicular to the spheroids’ symmetry axis.

Nonlinear effects in the media with metallic nanoinclusions,

such as second harmonic generation, also depend on the position

of the SPR. The shape of a nanoparticle, and, more specifically, its

asymmetry determines the magnitude of the dipole moment at

the second harmonic frequency. Using numerical simulation of

the electric field distribution in the metallic hemisphere placed on

a dielectric surface we deduced contribution of its sharp edges to



the linear and nonlinear response. In particular, plasmon

resonance of the hemisphere manifests itself in the enhancement

of the second harmonics signal generated when the hemisphere

is irradiated with an intense light beam. This can be visualized by

studying the dependence of the SHG signal on the thickness of

the dielectric shell covering the metal nanoparticle.

Optical and electronic properties of semiconductor

nanostructures, such as quantum wells or quantum dots, are

determined by electron confinement, i.e. by the size and shape of

the nanostructure. Using a semiconductor nanostructure as an

active laser medium enables smooth adjustment of the radiative

transition strength and the laser cavity modes. In particular, the

disc-shaped semicondutor microcavity allows one to employ

WGM, which have extremely high Q-factor, for lasing. We

fabricated and studied the semiconductor ring/disc microcavities

with diameter of a few microns that possess a Q-factor as high as

20000. In such ultra small cavities the modes are well separated

from one another allowing us to achieve single-mode lasing. In

the Thesis, the stimulated emission of InAs quantum dots

embedded in the ultra small microcavities is studied by

microphotoluminescence measurements. Since the WGM are

localized at the cavity surface, the demonstrated fabrication

techniques employed to obtained micro-and nanostructures with

improved smoothness is crucial for applications. The threshold

and efficiency of the microlasers are also determined by the size

and shape of the cavity. They can vary in a wide range as a

function of ring/disc thickness and diameter.

Excitons in semiconductors can resonantly interact with light

to form polaritons. Properties of these bosonic quasi-particles are

determined by the strength of coupling between excitons in

quantum wells and microcavity photons. We performed

theoretical investigation of the exciton polaritons dynamics in a

semiconductor microcavity with an incorporated saturable

absorber. In particular, we demonstrate that nonlinear

dissipation in the microcavity results in a bistable behavior of the

polariton condensate. We developed protocols of soliton

formation and destruction in such structures. A microscopic



theory of the lasing in the bias-controlled heterostructure was

developed. In particular, we simulated the dynamics of the

exciton-polariton ensemble and revealed threshold dependence

of the number of quasiparticles on the applied bias.

It is important to stress that recent advances in the

nanofabrication and nanotechnology lead to merging

phenomena described in the Thesis. Specifically, by coating metal

nanoparticles with J-aggregates one may create hybrid plasmon-

exciton quasiparticles also referred to as plexitons [141,142].

These quaiparticles can form condensate [143] similarly to cavity

polaritons described in Chapter 5, and can be used for design of

a spaser [144-146] and other novel devices. The cavity polaritons

have been observed in ring microcavities, and polariton

condensation has been demonstrated [147-149]. From the other

hand, coating of the microcavity surface by a metal layer makes

possible excitation of Tamm polaritons [150].


79

7 References

1. S. A. Maier, Plasmonics: Fundamentals and Applications

(Springer, New York, 2007).

2. L. Novotny and B. Hecht, Principles of Nano-Optics

(Cambridge University Press, New York, 2006).

3. V. Klimov, Nanoplasmonics (Pan Stanford Publishing Pte. Ltd,

2014).

4. U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters

(Springer Series in Materials Science, Vol. 25, Springer Berlin

1995).

5. A. Campion and P. Kambhampati, “Surface-enhanced Raman

scattering”, Chemical Society Reviews 27, 241–250 (1998).

6. A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, “Surface

enhanced Raman scattering”, Journal of Physics: Condensed

Matter 4, 1143 (1992).

7. P. L. Stiles, J. A. Dieringer, N. C. Shah, and R. P. Van Duyne,

“Surface-Enhanced Raman Spectroscopy”, Annual Review of

Analytical Chemistry 1, 601–626 (2008).

8. V.V. Zhurikhina, P.N. Brunkov, V.G. Melehin, T. Kaplas, Y.

Svirko, V.V. Rutckaia and A.A. Lipovskii, “Formation of

metal island films for SERS by reactive diffusion”, Nanoscale

Res. Lett. 7(676), (2012).



9. H. Tuovinen et al., “Linear and second-order nonlinear

optical properties of arrays of noncentrosymmetric gold

nanoparticles”, J. Nonlin. Opt. Phys. Mater. 11, 421–432 (2002).

10. B. K. Canfieldet et al., “Linear and nonlinear optical responses

influenced by broken symmetry in an array of gold

nanoparticles”, Opt. Express 12, 5418–5423 (2004).

11. C. Flytzanis, F. Hache, M. C. Klein, D. Ricard, and P.

Roussignol, ‘‘Nonlinear optics in composite materials,’’ Prog.

Opt. 29, 321–411 (1991).

12. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics”,

Nature Photonics 6, 737–748 (2012).

13. C Flytzanis, “Nonlinear optics in mesoscopic composite

materials”, J. Phys. B 38(9), 661-679 (2005).

14. J. B. Khurgin and G. Sun, “Plasmonic enhancement of the

third order nonlinear optical phenomena: Figures of merit”,

Optics Express 21(22), 27460-27480 (2013).

15. C. Anceau, S. Brasselet, J. Zyss, and P. Gadenne, “Local

second-harmonic generation enhancement on gold

nanostructures probed by two-photon microscopy,” Optics

Letters 28, 713–715 (2003).

16. A. Wokaun, et al., “Surface second-harmonic generation from

metal island films and microlithographic structures”, Phys.

Rev. B 24, 849–856 (1981).

17. S. Linden et al., “Collective effects in second-harmonic

generation from split-ring-resonator arrays”, Phys. Rev. Lett.

109, 015502 (2012).

References


18. M. D. McMahon, R. Lopez, R. F. Haglund Jr, E. A. Ray and P.

H. Bunton, “Second-harmonic generation from arrays of

symmetric gold nanoparticles”, Phys. Rev. B 73, 041401 (2006).

19. Stalmashonak, A., Seifert, G. & Abdolvand, A., “Ultra-Short

Pulsed Laser Engineered Metal-Glass Nanocomposites”,

Springer Briefs in Physics, 59–67 (Springer, 2013).

20. G. Seifert, A. Stalmashonak, H. Hofmeister, J. Haug, M.

Dubiel, Laser-Induced, Polarization Dependent Shape

Transformation of Au/Ag Nanoparticles in Glass, Nanoscale

Res Lett. 4 (11), 1380–1383 (2009).

21. H. Kettunen, H. Wallén and A. Sihvola, “Polarizability of a

dielectric hemisphere”, J. Appl. Phys. 102, 044105 (2007).

22. H. Kettunen, H. Wallén and A. Sihvola, “Electrostatic

resonances of a negative-permittivity hemisphere”, J. Appl.

Phys. 103, 094112 (2008).

23. S. A. Scherbak, O. V. Shustova, V. V. Zhurikhina, A. A.

Lipovskii, “Electric Properties of Hemispherical Metal

Nanoparticles: Influence of the Dielectric Cover and

Substrate”, Plasmonics 10(3), 519-527 (2014).

24. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A.

Logan, “Whispering gallery mode mirodisk lasers”, Applied

Physics Letters 60, 289-291 (1992).

25. N. Matsumoto, and K. Kumabe, ”AlGaAs-GaAs

Semiconductor Ring Laser ”, Jpn. J. Appl. Phys. 16, 1395-1398

(1977).

26. A.S.H. Liao, and S. Wang, “Semiconductor injection lasers

with a circular resonators”, Applied physics Letters 36, 801

(1980).



27. Ledentsov N.N., Ustinov V.M., Egorov A.Yu., Zhukov A.E.,

Maksimov M.V., Tabatadze I.G., Kop’ev P.S., Fiz. Tekh.

Poluprovodn. 28, 1483 (1994).

28. N.V. Kryzhanovskaya, M.V. Maximov, A.E. Zhukov,

“Whispering-gallery mode microcavity quantum-dot lasers”,

Quantum Electronics 44(3), 189 – 200 (2014)

29. Gayral B., Gerard J.M., Lemaitre A., Dupuis C., Manin L.,

Pelouard J.L., “High-Q wet-etched GaAs microdisks

containing InAs quantum boxes”, Appl. Phys. Lett. 75, 1908

(1999).

30. Levi A.F.J., Slusher R.E., McСall S.L., Tanbun-Ek T., Coblentz

D.L., Pearton S.J., “Room temperature operation of microdisc

lasers with submilliamp threshold current”, Electron. Lett. 28,

1010 (1992).

31. Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, “Micrometre-scale

silicon electro-optic modulator”, Nature 435, 325 (2007).

32. A.W. Poon, X.S. Luo, F. Xu, H. Chen, “Cascaded

microresonator-based matrix switch for silicon on-chip

optical interconnection”, Proc. IEEE 97, 1216 (2009).

33. D. Hah, J. Bordelon, D. Zhang., “Mechanically tunable optical

filters with a microring resonator”, Appl. Optics 50, 4320 (2011).

34. Fujita M., Inoshita K., T. Baba, “Room temperature

continuous wave lasing characteristics of GaInAsP/InP

microdisk injection laser”, Electron. Lett. 34, 278 (1998).

35. A.F.J. Levi, R.E. Slusher, S.L. McCall, T. Tanbuk-Ek, D.L.

Coblentz, S.J. Perton., “Room temperature operation of

microdisc lasers with submilliamp threshold current”,

Electron. Lett. 28, 1010 (1992)

References


36. T. Baba, D. Sano., “Low threshold lasing and Purcell effect in

microdisk lasers at room temperature”, IEEE J. Select. Top.

Quant. Electron. 9, 1340 (2003).

37. N. Antoine-Vincent, F. Natali, D. Byrne, A. Vasson, P. Disseix,

J. Leymarie, M. Leroux, F. Semond, J. Massies, “Observation

of Rabi splitting in a bulk GaN microcavity grown on silicon”,

Phys Rev B. 68, 153313 (2003).

38. F. Semond, I.R. Sellers, F. Natali, D. Byrne, M. Leroux, J.

Massies, N. Ollier, J. Leymarie, P. Disseix, A. Vasson, “Strong

light-matter coupling at room temperature in simple

geometry GaN microcavities grown on silicon”, Appl Phys Lett.

87, 021102–3 (2005).

39. R. Butté, G. Christmann, E. Feltin, J. Carlin, M. Mosca, M.

Ilegems, N. Grandjean, “Room-temperature polariton

luminescence from a bulk GaN microcavity”, Phys Rev B. 73,

033315 (2006).

40. R. Shimada, J. Xie, V. Avrutin, U. Özgür, H. Morkoc, “Cavity

polaritons in ZnO-based hybrid microcavities”, Appl Phys Lett.

92, 011127 (2008).

41. S. Faure, C. Brimont, T. Guillet, T. Bretagnon, B. Gil, F.

Médard, D. Lagarde, P. Disseix, J. Leymarie, J. Zuniga-Perez,

M. Leroux, E. Frayssinet, J.C. Moreno, F. Semond, S.

Bouchoule, Relaxation and emission of Bragg-mode and

cavity-mode polaritons in a ZnO microcavity at room

temperature, Appl Phys Lett. 95, 121102 (2009)

42. M. Nakayama, S. Komura, T. Kawase, DaeGwi Kim,

“Observation of Exciton Polaritons in a ZnO Microcavity with

HfO2/SiO2 Distributed Bragg Reflectors”, J Phys Soc Jpn. 77,

093705 (2008)



43. R. Schmidt-Grund, B. Rheinländer, C. Czekalla, G. Benndorf,

H. Hochmuth, M. Lorenz, M. Grundmann, “Exciton-

polariton formation at room temperature in a planar ZnO

resonator structure”, Appl Phys B. 93, 331–337 (2008)

44. S. Halm, S. Kalusniak, S. Sadofev, H.-J. Wunsche, F.

Henneberger, “Strong exciton-photon coupling in a

monolithic ZnO/(Zn,Mg)O multiple quantum well

microcavity”, Appl Phys Lett. 99, 181121 (2011).

45. O. Jamadi, F. Réveret, E. Mallet, P. Disseix, F. Médard, M.

Mihailovic, D. Solnyshkov, G. Malpuech, J. Leymarie, X.

Lafosse, S. Bouchoule, F. Li, M. Leroux, F. Semond, J. Zuniga-

Perez, “Polariton condensation phase diagram in wide-band-

gap planar microcavities: GaN versus ZnO”, Phys. Rev. B. 93

(2016).

46. A. Brehier, R. Parashkov, J.S. Lauret, E. Deleporte, “Strong

exciton-photon coupling in a microcavity containing layered

perovskite semiconductors”, Appl. Phys. Lett. 89, 171110 (2006)

47. D.G. Lidzey, D.D.C. Bradley, T. Virgili, A. Armitage, M.S.

Skolnick, S. Walker, “Room Temperature Polariton Emission

from Strongly Coupled Organic Semiconductor

Microcavities”, Phys Rev Lett. 82, 3316–3319 (1999)

48. S. Kéna-Cohen, M. Davanço, S.R. Forrest, “Strong Exciton-

Photon Coupling in an Organic Single Crystal Microcavity”,

Phys Rev Lett. 101, 116401 (2008)

49. K.S. Daskalakis, S.A. Maier, R. Murray, S. Kéna-Cohen,

“Nonlinear interactions in an organic polariton condensate”,

Nature Materials. 13, 271–278 (2014).

50. J.D. Plumhof, T. Stöferle, L. Mai, U. Scherf, R.F. Mahrt,

“Room-temperature Bose–Einstein condensation of cavity

References


exciton–polaritons in a polymer”, Nature Materials. 13, 247–

252 (2013)

51. Paul E. Philipson and Peter Schuster, Modeling by Nonlinear

Differential Equations: Dissipative and Conservative Processes

(World Scientific Publishing Company, 2009).

52. B. Brogliato, R. Lozano, B. Maschke, O. Egeland, Dissipative

Systems Analysis and Control. Theory and Applications. (Springer

Verlag, London, 2nd Ed., 2007)

53. J.C. Willems. Dissipative dynamical systems, part I: General

theory; part II: Linear systems with quadratic supply rates.

Archive for Rationale mechanics Analysis, vol.45, (1972).

54. F. Marsault, H. S. Nguyen, D.Tanese, A. Lemaître, E. Galopin,

I. Sagnes, A. Amo1 and J. Bloch, “Realization of an all optical

exciton-polariton router”, Appl. Phys. Lett. 107, 201115 (2015)

55. H. Flayac and I. G. Savenko ,“An exciton-polariton mediated

all-optical router”, Appl. Phys. Lett. 103, 201105 (2013)

56. T. Gao, P. S. Eldridge, T. C. H. Liew, S. I. Tsintzos, G.

Stavrinidis, G. Deligeorgis, Z. Hatzopoulos, and P. G.

Savvidis, “Polariton condensate transistor switch”, Phys. Rev.

B 85, 235102 (2012)

57. K. V. Kavokin, M. A. Kaliteevski, R. A. Abram, A. V. Kavokin,

S. Sharkova and I. A. Shelykh, “Stimulated emission of

terahertz radiation by exciton-polariton lasers”, Appl. Phys.

Lett. 97, 201111 (2010)

58. I. G. Savenko, I. A. Shelykh, and M. A. Kaliteevski,

“Nonlinear Terahertz Emission in Semiconductor

Microcavities”, Phys. Rev. Lett. 107, 027401 (2011)



59. T. C. H. Liew, A. V. Kavokin, and I. A. Shelykh, “Optical

Circuits Based on Polariton Neurons in Semiconductor

Microcavities”, Phys. Rev. Lett. 101, 016402 (2008)

60. A. Amo, T. C. H. Liew, C. Adrados, R. Houdré, E. Giacobino,

A. V. Kavokin & A. Bramati, “Exciton–polariton spin

switches”, Nat. Photon. 4, p. 361-366 (2010)

61. Gabriel Christmann, Raphaël Butté, Eric Feltin, Jean-François

Carlin and Nicolas Grandjean, “Room temperature polariton

lasing in a GaN∕AlGaNGaN∕AlGaN multiple quantum

well microcavity”, Appl. Phys. Lett. 93, 2966369 (2008)

62. Tien-Chang Lu, Jun-Rong Chen, Shiang-Chi Lin, Si-Wei

Huang, Shing-Chung Wang, and Yoshihisa Yamamoto,

“Room Temperature Current Injection Polariton Light

Emitting Diode with a Hybrid Microcavity”, Nano Lett. 11

2791–2795, (2011)

63. Le Ru E., Etchegoin P., Principles of SERS (The Netherland

Linacre house, 2009, 663)

64. J. D. Jackson, Classical Electrodynamics (Wiley, New York,

1999).

65. Agranovich M S, Katsenelenbaum B Z, Sivov A N and

Voitovich N N, Generalized Method of Eigenoscillation in

Diffraction Theory (Berlin: Wiley, 1999)

66. H. Du, J. Gong, C. Sun, R. Huang, and L. Wen, “Carrier

Density and Plasma Frequency of Aluminum Nanofilms,”

Journal of Material Science & Technology 19, 365–367 (2003).

67. M. I. Markovic and A. D. Rakic, “Determination of the

reflection coefficients of laser light of wavelengths λǫ(0.22

μm,200μm) from the surface of aluminum using the Lorentz-

Drude model,” Applied Optics 29, 3479–3483 (1990).

References


68. A. Kawabata an.d R Kubo, “Electronic Properties of Fine

Metallic Particles. II. Plasma Resonance Absorption”, J. Phys.

Soc. Jpn. 21, 1765-1772 (1966)

69. L. Genzel, U. Kreibig, Dielectric function and infrared

absorption of small metal particles, Zeitschrift für Physik B

Condensed Matter 37(2), 93-101 (1980)

70. Morse PM, Feshbach H, Methods of Theoretical Physics, Part I.

(New York: McGraw-Hill., 665–666, 1953).

71. Moon PH, Spencer DE, "Bispherical Coordinates (η, θ, ψ). Field

Theory Handbook, Including Coordinate Systems, Differential

Equations, and Their Solutions” (3rd print ed., New York:

Springer Verlag.,110–112, 1988)

72. Guzatov D V and Klimov V V, “Optical properties of a two-

nanospheroid cluster: analytical approach”, (2010),

arXiv:1010.5760v1

73. W.Rechberger, A.Hohenau, A.Leitner, J.R.Krenn,

B.Lamprecht, F.R.Aussenegg, Optical properties of two

interacting gold nanoparticles, Opt.Comm. 220, p.137-141

(2003).

74. L.D. Landau & E.M. Lifshitz, “Electrodynamics of Continuous

Media, Volume 8 of A Course of Theoretical Physics” (Pergamon

Press, 1960).

75. J.C. M. Garnett, “Colours in metal glasses and in metallic

films”, Philos. Trans. R. Soc. Lond. A 203, 385–420 (1904).

76. A. D. Buckinghaamnd j. A. Pople, “The dielectric constant of

an imperfectnon-polar gas”, Trans. Faraday Soc. 51, 1029-1035,

(1955)



77. John G. Kirkwood, “On the Theory of Dielectric Polarization”,

J. Chem. Phys. 4, 592 (1936)

78. A.Sihvola, J.A.Kong, “Effective Permittivity of Dielectric

Mixtures”, IEEE Trans. Geosciences and Remote Sensing 26 (4)

420-429 (1988).

79. P. B. Johnson and R. W. Christy. Optical constants of the noble

metals. Phys. Rev. B 6, 4370–4379 (1972).

80. J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis

of second harmonic generation at metal surfaces”, Phys. Rev.

B 21, 4389–4402 (1980).

81. N. Blombergen, Y.R. Shen, “Optical Nonlinearities of a

Plasma”, Phys. Rev. 141, 298-305 (1966).

82. C.S. Wang, J.M. Chen And J.R. Bower, “Second harmonic

generation from alkali metals”, Optics communications 8(4),

275-279 (1973).

83. P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in

surface second-harmonic generation”, Phys. Rev. 38, 7985,

(1988)

84. N. Blombergen, Y.R. Shen, “Optical Nonlinearities of a

Plasma”, Phys. Rev. B 141, 298-305 (1966).

85. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee,

“Optical Second-Harmonic Generation in Reflection from

Media with Inversion Symmetry”, Phys. Rev. 174, (1968)

86. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York,

1984).

87. R. W. Boyd, “Nonlinear Optics” (3rd edition, Academic, 2008).

References


88. Rayleigh L. Philos. Mag. 20, 1001 (1910).

89. Garrett C.G.B., Kaiser W., Bond W.L., “Stimulated Emission

into Optical Whispering Modes of Spheres”, Phys. Rev. 124,

1807 (1961).

90. Grudinin I.S., Ilchenko V.S., Maleki L., “Ultrahigh optical Q

factors of crystalline resonators in the linear regime”, Phys.

Rev. A 74, 063806 (2006).

91. Borselli M., Srinivasan K., Barclay P.E., Painter O., “Rayleigh

scattering, mode coupling, and optical loss in silicon

microdisks”, Appl. Phys. Lett. 85, 3693 (2004).

92. Borselli M., Johnson T.J., Painter O., “Beyond the Rayleigh

scattering limit in high-Q silicon microdisks: theory and

experiment”, Opt. Express 13, 1515 (2005).

93. D. Bimberg, M. Grundmann, N.N. Ledentsov, Quantum Dot

Heterostructures (Wiley, New York, 1998)

94. McCord M. A., Handbook of Microlithography, Micromachining

and Microfabrication (SPIE Press, 2000)

95. Principles of Lithography, Electron-beam lithography and mask

writers «For two decades, the MEBES systems were the primary

beam writers used to make photomasks», Third Edition (SPIE Press,

2011)

96. R.L. Puurunen, “Surface chemistry of atomic layer deposition:

A case study for the trimethylaluminum/water process”, J.

Appl. Phys. 97, 121301 (2005).

97. A.A. Malygin, J. Ind. Eng. Chem. 12(1), 1-11 (2006).

98. T. Suntola, J. Antson, U.S. Patent 4,058,430, (1977).



99. Puurunen, Riikka L., "Surface chemistry of atomic layer

deposition: A case study for the trimethylaluminum/water

process",. Journal of Applied Physics 97 (12), 121301 (2005)

100. W. M. Moreau, Semiconductor Lithography: Principles,

Practices, and Materials (Plenum Press, New York, 1988).

101. M. J. Madou, Fundamentals of Microfabrication: The Science

of Miniturization, Second ed. (CRC Press LLC, Boca Raton, 2002).

102. P. Harrison., Quantum Wells, Wires and Dots. 3rd edition

(Wiley, 2009)

103. C. Weisbuch and B. Vinter, Quantum semiconductor

structures (Academic Press, 1991).

104. P. Y. Yu and M. Cardona., Fundamentals of Semiconductors.

Third edition ,(Springer, 2005).

105. N. W. Ashcroft and N. D. Mermin., Solid State Physics

(Harcourt College Publishers, New York, 1976).

106. H. Haug and S. W. Koch., Quantum theory of the optical and

electronic properties of semiconductors (World Scientific, 2004).

107. Wannier, Gregory, "The Structure of Electronic Excitation

Levels in Insulating Crystals", Physical Review. 52(3), 191 (1937)

108. R. L. Greene, K. K. Bajaj, and D. E. Phelps, “Energy levels

of Wannier excitons in GaAs − Ga1 − xAlxAs quantum-well

structures”, Phys. Rev. B 29(4), 1807–1812 (1984).

109. L. C. Andreani and A. Pasquarello, “Accurate theory of

excitons in GaAsGa1−xAlxAs quantum wells”, Phys. Rev. B

42(14), 8928–8938 (1990).

110. Frenkel, J., "On the Transformation of light into Heat in

Solids. I", Physical Review. 37(17), (1931).

References


111. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P.

Jeambrun, J. M. J. Keel-ing, F. M. Marchetti, M. H. Szymaska,

R. André, J. L. Staehli, V. Savona, P. B. Lit-tlewood, B.

Deveaud, and Le Si Dang, “Bose–Einstein condensation of

exciton polar-itons”, Nature 443, 409-414 (2006)

112. H. Bruus and K. Flensberg, “Many-Body Quantum Theory

in Condensed Matter Physics: An Introduction” (Oxford

Graduate Texts, 2004)

113. M. S. Skolnick, T. A. Fisher, and D. M. Whittaker, “Strong

coupling phenomena in quantum microcavity structures”,

Semiconductor Science and Technology 13(7), 645 (1998)

114. M. Kuwata-Gonokami, S. Inouye, H. Suzuura, M. Shirane,

R. Shimano, T. Someya, and H. Sakaki, “Parametric Scattering

of Cavity Polaritons”, Phys. Rev. Lett. 79, 1341 (1997)

115. Y. P. Svirko, M. Shirane, H. Suzuura, and M. Kuwata-

Gonokami, “Four-Wave Mixing Theory at the Excitonic

Resonance: Weakly Interacting Boson Model”, J. Phys. Soc. Jpn.

68, 674-682 (1999)

116. D. Gerace and L. Claudio Andreani, “Quantum theory of

exciton-photon coupling in photonic crystal slabs with

embedded quantum wells”, Phys. Rev. B 75, 235325 (2007)

117. D. Bajoni, D. Gerace, M. Galli, J. Bloch, R. Braive, I. Sagnes,

A. Miard, A. Lemaître, M. Patrini, and L. C. Andreani,

“Exciton polaritons in two-dimensional photonic crystals”,

Phys. Rev. B 80, 201308 (2009)

118. L. P. Pitaevskii and S. Stringari, Bose–Einstein Condensation

(Clarendon Press, Oxford, 2003).

119. M. Steger, G. Liu, B. Nelsen, C. Gautham, D. W. Snoke, R.

Balili, L., Pfeiffer, and K. West, “Long-range ballistic motion



and coherent flow of long-lifetime polaritons”, Phys. Rev. B 88,

235314 (2013).

120. M. Steger, C. Gautham, D. W. Snoke, L. Pfeiffer, and K.

West, “Slow reflection and two-photon generation of

microcavity exciton–polaritons”, Optica 2(1), 1–5 (2015).

121. J. Scheuer, M. Orenstein, and D.Arbel, “Nonlinear

switching and modulational instability of wave patterns in

ring-shaped vertical-cavity surface-emitting lasers”, JOSAB

19, 2384 (2002).

122. M.Wouters and I. Carusotto, “Excitations in a

Nonequilibrium Bose-Einstein Condensate of Exciton

Polaritons”, Phys. Rev. Lett. 99, 140402 (2007).

123. E. A. Ostrovskaya, J. Abdullaev, M. D. Fraser, A. S.

Desyatnikov, and Yu. S. Kivshar, “Self-Localization of

Polariton Condensates in Periodic Potentials”, Phys. Rev. Lett.

110, 170407, (2013).

124. S. Colin, E. Contesse, P. Le Boudec, G. Stephan, F. Sanchez,

“Evidence of a saturable-absorption effect in heavily erbium-

doped fibers”, Optics Letters 21(24), 1987-1989 (1996)

125. Thomas L. Paoli, “Saturable absorption effects in the self -

pulsing (AlGa)As junction laser”, Appl. Phys. Lett. 34, 652

(1979)

126. U.Keller, D. A. B. Miller, G. D. Boyd, T. H. Chiu, J. F.

Ferguson, and M. T. Asom, “Solid-state low-loss intracavity

saturable absorber for Nd:YLF lasers: an antiresonant

semiconductor Fabry–Perot saturable absorber”, Opt. Lett. 17,

505 (1992).

127. F. Trager, Handbook of Lasers and Optics (Springer, New

York, 2007).

References


128. S. Schmitt-Rink, D. S. Chemla, and D. A. B. Miller,

“Theory of transient excitonic optical nonlinearities in

semiconductor quantum-well structures”, Phys. Rev. B 32(10),

6601–6609 (1985).

129. A. N. Oraevskii, “Features of the dynamics of lasers with

a saturable absorber”, Quantum Electonics 33, 849 (2003).

130. S. V. Fedorov, A. G. Vladimirov, G. V. Khodova, and N.

N. Rosanov, “Effect of frequency detunings and finite

relaxation rates on laser localized structures”, Phys. Rev. E 61,

5814 (2000).

131. N. Akhmediev and A. Ankiewicz (eds.), Dissipative

Solitons, Lecture Notes Phys. Vol. 661 (Springer, Berlin, 2005).

132. N. Akhmediev and A. Ankiewicz (eds.), Dissipative

Solitons: From Optics to Biology and Medicine, Lecture Notes

Phys. Vol. 751 (Springer, Berlin, 2008).

133. F. Tassone, C. Piermarocchi, V. Savona, A. Quattropani,

and P. Schwendimann, “Bottleneck effects in the relaxation

and photoluminescence of microcavity polaritons”, Phys. Rev.

B 56, 7554 (1997).

134. V. E. Hartwell and D.W. Snoke, “Numerical simulations

of the polariton kinetic energy distribution in GaAs quantum-

well microcavity structures”, Phys. Rev. B 82, 075307 (2010).

135. I. G. Savenko, T. C. H. Liew, and I. A. Shelykh,” Stochastic

Gross-Pitaevskii Equation for the Dynamical Thermalization

of Bose-Einstein Condensates”, Phys. Rev. Lett. 110, 127402

(2013).

136. S. Selberherr, Simulation of Semiconductor Devices and

Processes (Springer-Verlag, Wien, New York)



137. K. Tomizawa, Numerical Simulation of Submicron

Semiconductor Devices (The Artech House Materials Science

Library).

138. H. Morkoc, Handbook of Nitride Semiconductors and Devices,

Electronic and Optical Processes in Nitrides Vol. 2 (Wiley,

Weinheim, 2008)

139. W. Shockley, W. T. Read, “Statistics of the Recombinations

of Holes and Electrons”, Physical Review 87(5), 835-842 (1952)

140. Planck, M., The Theory of Heat Radiation., Masius, M. (transl.)

2nd ed. (P. Blakiston's Son & Co, 1914)

141. Taeho Shin, Kyung-SangCho, Dong-JinYun, Jinwoo Kim,

Xiang-Shu Li, Eui-Seong Moon, Chan-Wook Baik, Sun I, Kim,

Miyoung Kim, Jun HeeChoi, Gyeong-Su Park, Jai-Kwang

Shin, Sungwoo Hwang & Tae-Sung Jung, ”Exciton

Recombination, Energy-, and Charge Transfer in Single- and

Multilayer Quantum-Dot Films on Silver Plasmonic

Resonators”, Scientific Reports 6, 26204 (2016)

142. Wenjing Liu, Bumsu Lee, Carl H. Naylor, Ho-Seok Ee,

Joohee Park, A. T. Charlie Johnson, and Ritesh Agarwal,

“Strong Exciton–Plasmon Coupling in MoS2 Coupled with

Plasmonic Lattice”, Nano Lett. 16 (2), 1262–126 ( 2016)

143. S. R. K. Rodriguez, J. Feist, M. A. Verschuuren, F. J. Garcia

Vidal, and J. Gómez Rivas, ”Thermalization and Cooling of

Plasmon-Exciton Polaritons: Towards Quantum

Condensation”, Phys. Rev. Lett. 111, 166802 (2013)

144. M. Stockman, “Plasmonic Lasers: On the Fast Track”, Nat.

Phys. 10, 799-800 (2014)

145. Dabing Li and Mark I. Stockman, “Electric Spaser in the

Extreme Quantum Limit”, Phys. Rev. Lett. 110, 106803 (2013)

References


146. M. I. Stockman, Quantum Nanoplasmonics, in: Photonics,

Volume II: Scientific Foundations, Technology and Applications,

edited by D. L. Andrews (John Wiley & Sons, Inc., Hoboken, NJ,

USA., 2015)

147. L. Sun, Z. Chen, Q. Ren, Ke Yu, L. Bai, W. Zhou, H. Xiong,

Z. Q. Zhu, and X. Shen, “Direct Observation of Whispering

Gallery Mode Polaritons and their Dispersion in a ZnO

Tapered Microcavity”, Phys. Rev. Lett. 100, 156403 (2008)

148. L. Sun, H. Dong, W. Xie, Z. An, X. Shen, and Z. Chen,

“Quasi-whispering gallery modes of exciton-polaritons in a

ZnO microrod”, Optics Express 18(15), 15371-15376 (2010)

149. Qingqing Duan, Dan Xu, Wenhui Liu, Jian Lu, Long

Zhang, Jian Wang, Yinglei Wang, Jie Gu, Tao Hu, Wei Xie,

Xuechu Shen and Zhanghai Chen “Polariton lasing of quasi-

whispering gallery modes in a ZnO microwire”, Appl. Phys.

Lett. 103, 022103 (2013)

150. R. Brückner, A. A. Zakhidov, R. Scholz, M. Sudzius, S. I.

Hintschich, H. Fröb, V. G. Lyssenko & K. Leo, “Phase-locked

coherent modes in a patterned metal–organic microcavity”,

Nature Photonics 6, 322–326 (2012)

uef.fi



ISBN 978-952-61-2350-9ISSN 1798-5668


DIS

SE

RT

AT

ION

S | D

EN

IS K

AR

PO

V | R

ES

ON

AN

CE

PH

EN

OM

EN

A IN

NO

NL

INE

AR

AN

D A

CT

IVE

NA

NO

PH

OT

ON

ICS

| No

251

DENIS KARPOV

RESONANCE PHENOMENA IN NONLINEAR AND ACTIVE NANOPHOTONICS


This work is dedicated to theoretical and

experimental investigation of the resonance optical phenomena occurring in nonlinear and active photonics nanostructures. By

using a wide range of theoretical and experimental techniques we studied glass-metal nanocomposites, whispering gallery mode semiconductor quantum dots lasers and exciton-polaritons lasing in the bias-

controlled heterostructures.

DENIS KARPOV

Dissertations in Forestry and Natural Sciences...Dissertations in Forestry and Natural Sciences No...

Documents

Transcript of Dissertations in Forestry and Natural Sciences...Dissertations in Forestry and Natural Sciences No...