Simulation of Light-Mediated Coupling
in Planar Photonic Resonators
vorgelegt vom Diplom-Physiker
Stefan Declair
Fakultät III - Naturwissenschaften
der Universität Paderborn
Department Physik
Computational Nanophotonics
Dissertation zur Erlangung des akademischen Doktorgrades Dr. rer. nat.
genehmigte Dissertation
Datum der mündlichen Prüfung: 31.05.2011
Promotionsausschuss:
Vorsitzender: Prof. Dr. rer. nat. Artur Zrenner
1. Gutachter: Dr. rer. nat. Jens Förstner
2. Gutachter: Prof. Dr. rer. nat. Torsten Meier
Simulation of Light-Mediated Coupling
in Planar Photonic Resonators
vorgelegt vom Diplom-Physiker
Stefan Declair
Fakultät III - Naturwissenschaften
der Universität Paderborn
Department Physik
Computational Nanophotonics
Dissertation zur Erlangung des akademischen Doktorgrades Dr. rer. nat.
genehmigte Dissertation
Datum der mündlichen Prüfung: 31.05.2011
Promotionsausschuss:
Vorsitzender: Prof. Dr. rer. nat. Artur Zrenner
1. Gutachter: Dr. rer. nat. Jens Förstner
2. Gutachter: Prof. Dr. rer. nat. Torsten Meier
Für meinen Großvater.
7
Zusammenfassung
Das Hauptaugenmerk dieser Arbeit ist die lichtvermittelnde Kopplung in planaren photonis-
chen Resonatoren unter der Benutzung der zeitaufgelösten Finite-Differenzen Methode (FDTD).
Mit dieser Methode werden die dreidimensionalen Maxwellgleichungen numerisch ausgewertet.
Zwei verschiedene Arten photonischer Resonatoren werden untersucht: Mikrodisk Resonatoren
und photonische Kristallkavitäten. Das Ziel ist die Untersuchung der Kopplung zwischen
Quantenpunkten und optischen Resonatoren über das Lichtfeld, eingesperrt in der Kavität.
Zur Beschreibung der Quantenpunkte werden dynamische Bewegungsgleichungen für die Ko-
härenz und für die Besetzungsdichte auf einer quantenmechanischen Basis verwendet, um das
optische Polarisationsfeld zu berechnen. Dieses Polarisationsfeld ist selbstkonsistent an die
Maxwellgleichungen gekoppelt. Außerdem wird die Kopplung zwischen Mikrodisks und zwis-
chen photonischen Kristallkavitäten über das Lichtfeld untersucht.
Als erstes werden Eigenmoden von Mikrodisk Resonatoren berechnet, die das Licht inner-
halb eines kleinen räumlichen Gebietes nur über Totalreflektion an der Grenzfläche zwis-
chen dem Resonator und dem umgebenden Material einsperren. Der Effekt der Form auf die
Resonanz der Flüstergalleriemoden wird untersucht. Die Verwendung eines optisch uniaxial
anisotropen Umgebungsmaterials läßt vermeidete Kreuzung zwischen verschiedenen Mode-
nordnungen Flüstergalleriemoden auftreten. Zur Benutzung von uniaxialer Anisotropie als
Einstellungsmechanismus für spektrale Eigenschaften eines Resonators werden numerische
Daten von dreidimensionalen Simulationen eines Mikrodisk Resonators, eingebettet in eine
Flüssigkristallumgebung, mit experimentellen Daten verglichen.
Als zweites werden Resonatoren basierend auf photonischen Bandlücken numerisch unter-
sucht. Ein zweidimensionaler photonischer Kristall mit einem Liniendefekt, bestehend aus 3
fehlenden Luftlöchern, wird verwendet. Modifizierung des Defektgebietes mit der Methode
der behutsamen Einschränkung erhöht die Abklingzeit der Kavitätmoden um eine Größenord-
nung. Für einen Liniendefekt, bestehend aus 7 fehlenden Luftlöchern wird ein Mechanismus
zur permanenten Einstellung spektraler Eigenschaften durch Aufdampfen einer zusätzlichen
dünneren Schicht eines zweiten Materials numerisch berechnet. Daten aus einem Experiment
werden mit den numerischen Daten verglichen und diskutiert.
Starke Resonator-Resonator-Wechselwirkung zwischen Mikrodisk Resonatoren und zwischen
photonischen Kristallkavitäten werden als drittes gezeigt. Hoch asymmetrische Linienaufspal-
tung bei kurzen Abständen ist von der Formierung von Supermoden mit ultraniedrigen und
ultrahohen Q-Faktoren in jedem der beiden Systeme begleitet. Ersteres ist durch einen tight-
binding-ähnlichen Ansatz erklärt, während zweiteres das Resultat von Interferenzeffekten ist.
In dreifach gekoppelten Kavitäten in einem hexagonalen Gitter wird eine geeignete Anordnung
vorgeschlagen, um die Kopplung effizienter zu machen. Es wird außerdem die Möglichkeit
gezeigt, Licht in gekoppelten optischen Resonatorwellenleitern entlang verschiedener Symme-
trierichtungen des Gitters effizient zu führen. Ein einzelner Quantenpunkt, gekoppelt an eine
Kavität und mit einem intensiven Laserfeld angeregt, zeigt symmetrische Seitenbandübergänge
um den Hauptübergang, ein Mollow-Triplet. Intensitäts- und Q-faktor-abhängige Rechnungen
werden dazu durchgeführt.
8
Abstract
The focus of this thesis is the light-mediated coupling in planar photonic resonators using the
Finite-Difference Time-Domain method. With this method, the three-dimensional Maxwells
equations are evaluated numerically. Two different kinds of optical resonators are investigated:
microdisk resonators and photonic crystal cavities. The goal is the investigation of the cou-
pling between quantum dots and optical resonators via the light field, confined in the cavity.
For the description of the quantum dots, dynamic equations of motion for the coherence and
population on a quantum mechanical level are used to calculate the generated optical polariza-
tion field. This macroscopic polarization field is coupled back self-consistently to the Maxwell
equations to accurately describe the coupled system. Also, coupling between microdisks and
between photonic crystal cavities via the electromagnetic field is investigated.
First, eigenmodes in microdisk resonators, which confine light inside a small spatial volume
only by total internal reflection at the interface between the microdisk and the surrounding en-
vironment, are calculated. The effect of the shape on the resonance of whispering gallery modes
is investigated. Applying an optical, uniaxial anisotropic environment, anticrossings between
different mode orders of the whispering gallery modes occur. Using uniaxial anisotropy as a
tuning mechanism for spectral properties of a resonator, numerical data of a three-dimensional
simulation of a microdisk, embedded in a liquid crystal environment, are compared with ex-
perimental data.
Second, photonic gad gap-based resonators are investigated numerically. A two-dimensional
photonic crystal with a line defect of three missing air holes is used. Modifying the defect area
with the method of gentle confinement, the cavity decay time of the defect modes is increased
by one order of magnitude. For a line defect with 7 missing air holes, a permanent tuning
mechanism for the spectral properties, induced by evaporation of a thin additional layer of a
second material, is also evaluated numerically. Data from an experiment are compared with
the numerical results and discussed.
Strong resonator-resonator interaction between microdisk resonators and between photonic
crystal cavities is shown. Strongly asymmetric line splitting at short gap sizes is accompanied
by the formation of ultra-low- and ultra-high-Qsuper-modes in each of the coupled systems.
The first is explained in a tight-binding like approach, while the latter is shown to be the result
from interference effects. In three-fold coupled cavities in a hexagonal lattice, a proper align-
ment is proposed to efficiently coupled multiple cavities. Also, the possibility to guide light
in coupled resonator optical waveguides along non-symmetry directions of the lattice is pro-
posed. A single quantum dot, coupled to a microcavity and excited by an intense laser pulse,
shows symmetric side band peaks around the main transition, a Mollow triplet. Intensity- and
Q-factor-dependent calculations are performed.
Contents
1 Introduction 17
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2 Structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Fundamentals 23
2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 FDTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Update Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 Accuracy and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.3 !-Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.4 CPML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 PhC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Optical Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 MD Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 PhCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 QD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5.1 QDs in the FDTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5.2 Rabi Oscillations in QDs . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Coupling Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.7 LC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.8 Analysis Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Validation of Maexle 57
3.1 Validation of the FDTD Implementation . . . . . . . . . . . . . . . . . . . . . . 58
3.1.1 H1-type PhCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1.2 L3-type PhCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Microdisk Resonator 71
4.1 Sub-µMicrodisk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.1.1 Geometrical Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.1.2 Mode Tuning with Uniaxial Anisotropy . . . . . . . . . . . . . . . . . . 81
4.2 3µMicrodisk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2.1 Mode Tuning with a LC: Comparison with the Experiment . . . . . . . 85
9
CONTENTS 10
5 Photonic Crystal Cavities 89
5.1 PhCC: Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 Permanent Mode Tuning of a L3-type PhCC . . . . . . . . . . . . . . . . . . . 91
5.3 Comparison with the Experiment: L7-type PhCC . . . . . . . . . . . . . . . . . 101
6 Coupled Systems 111
6.1 Coupled Sub-µMDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.1.1 Strong Interaction Between Two Sub-µMDs . . . . . . . . . . . . . . . . 112
6.2 Single QD Strongly Coupled to a WGM of a Sub-µMD . . . . . . . . . . . . . 117
6.3 Coupled PhCCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3.1 Strong Interaction between 2 PhCCs . . . . . . . . . . . . . . . . . . . . 118
6.3.2 Strong Interaction between 3 PhCCs . . . . . . . . . . . . . . . . . . . . 124
6.4 A Single QD Coupled to a PhCC . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4.1 Strong Excitation of QDs: Mollow Triplets . . . . . . . . . . . . . . . . . 126
6.4.2 Strong Excitation of a QD in a PhCC . . . . . . . . . . . . . . . . . . . 129
7 Conclusion and Outlook 133
A Numerical Methods 137
A.1 RK4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A.2 Lagrange Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
B Parameters and Constants for the Numerical Simulations 139
B.1 Constants of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.2 Parameter List for the Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . 139
B.3 Parameter List for the FDTD Simulations . . . . . . . . . . . . . . . . . . . . . 140
Bibliography 141
List of Figures
2.1 Yee-cube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Update scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Staircasing effect on frequency in a circular resonator. . . . . . . . . . . . . . . 30
2.4 Staircasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Performance for a subpixel-shift of a dielectric object. . . . . . . . . . . . . . . 32
2.6 Gap map for a hexagonal lattice of air holes in a dielectric slab. . . . . . . . . . 36
2.7 Photonic density of states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.8 Graphical solution of transcendental equation. . . . . . . . . . . . . . . . . . . . 39
2.9 In-plane field distribution calculated analytically. . . . . . . . . . . . . . . . . . 40
2.10 In-plane field distribution calculated with FDTD. . . . . . . . . . . . . . . . . . 41
2.11 DBR cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.12 Bandstructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.13 Mode patterns for the H1-, H2- and L3-type cavity. . . . . . . . . . . . . . . . 44
2.14 Comparison of the extrapolation schemes in the RK4 integrator. . . . . . . . . 46
2.15 The QD in the FDTD grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.16 Rabi-Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.17 Light-mediated coupling between 1D-cavities . . . . . . . . . . . . . . . . . . . 52
2.18 Characteristics of the LC 5CB. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1 Validation: H1PhCC, after [1]. Calculated spectral response. . . . . . . . . . . 59
3.2 Validation: H1PhCC after [1]. Calculated spatiospectral response. . . . . . . . 60
3.3 Validation: H1PhCC, after [2]. Calculated spectral response. . . . . . . . . . . 61
3.4 Validation: H1PhCC after [2]. Calculated spatiospectral response. . . . . . . . 62
3.5 Validation: L3PhCC after [3]. Calculated spectral response. . . . . . . . . . . . 65
3.6 Validation: L3PhCC after [3]. Calculated spatiospectral response. . . . . . . . 66
3.7 Validation: L3PhCC after [3]. Calculated spatiospectral response. . . . . . . . 67
3.8 Validation: H2PhCC after [4]. Calculated spectral response. . . . . . . . . . . . 68
3.9 Validation: L3PhCC after [4]. Calculated spatiospectral response. . . . . . . . 69
3.10 Validation: L3PhCC after [4]. Calculated spatiospectral response. . . . . . . . 70
4.1 Geometrical Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Spectral response from broadband excited sub-µmicrodisk. . . . . . . . . . . . 73
4.3 Real- and reciprocal space field distributions. . . . . . . . . . . . . . . . . . . . 74
4.4 Radius-dependency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5 Elliptic shaped MD resonators. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.6 Mode patterns in an elliptically shaped MD. . . . . . . . . . . . . . . . . . . . . 78
11
LIST OF FIGURES 12
4.7 Edge profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.8 3D spatial field distributions of TE1,10,1. . . . . . . . . . . . . . . . . . . . . . . 80
4.9 3D spatial field distributions of TE1,7,3. . . . . . . . . . . . . . . . . . . . . . . 80
4.10 2D spatial field distributions of TE1,7,3(y-z-plane), iso-intensity surface . . . . 81
4.11 Anticrossing of WGMs in a uniaxial anisotropic environment. . . . . . . . . . . 82
4.12 Comparison between (an-)isotropic environment. . . . . . . . . . . . . . . . . . 83
4.13 Spectrum and mode pattern (TE3,30,1) of a 3µm MD in vacuum. . . . . . . . . 84
4.14 SEM image of a 3µm MD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.15 Orientation parameter and Comparison with the experiment. . . . . . . . . . . 87
4.16 Detailed view to the resonance shift. . . . . . . . . . . . . . . . . . . . . . . . . 88
5.1 Determination of Simulation domain parameters. . . . . . . . . . . . . . . . . . 91
5.2 Method of gentle confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 (Un-)modified L3-type cavity in a hexagonal PhC-lattice. . . . . . . . . . . . . 93
5.4 Band structure and spectral response of the cavity. . . . . . . . . . . . . . . . . 93
5.5 Field distribution of the fundamental mode for the unmodified PhCC. . . . . . 94
5.6 Intensity in reciprocal space without modification. . . . . . . . . . . . . . . . . 94
5.7 Frequency- and Q-shift: modification of the 1st air hole. . . . . . . . . . . . . . 95
5.8 Frequency- and Q-shift: modification of the 2nd air hole. . . . . . . . . . . . . . 96
5.9 Field distribution for the modified PhCC. . . . . . . . . . . . . . . . . . . . . . 97
5.10 Intensity in reciprocal space with modification. . . . . . . . . . . . . . . . . . . 97
5.11 Field distribution in real- and k-space with and without modification. . . . . . 99
5.12 Spectral localization of the fundamental mode in the 1st band gap. . . . . . . . 100
5.13 Spectral response of the (un-)modified cavity. . . . . . . . . . . . . . . . . . . . 100
5.14 L7-type cavity: SEM image and computational parameters. . . . . . . . . . . . 102
5.15 Band structure and spectral response of the cavity. . . . . . . . . . . . . . . . . 103
5.16 Spatiotemporal response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.17 Accumulated energy distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.18 Effect on Frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.19 Effect on amplitude and Q-factor. . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.20 Comparison with the experiment: spectral response. . . . . . . . . . . . . . . . 110
5.21 Comparison with the experiment: frequency shift and Q-factor. . . . . . . . . . 110
6.1 Beating between two coupled MDs. . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Line and loss splitting in two coupled MDs. . . . . . . . . . . . . . . . . . . . . 114
6.3 Mode patterns for two coupled MDs. . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4 Radiation patterns for two coupled MDs. . . . . . . . . . . . . . . . . . . . . . . 116
6.5 A single QD strongly coupled to MD. . . . . . . . . . . . . . . . . . . . . . . . . 117
6.6 Geometry and spectral response for linear alignment. . . . . . . . . . . . . . . . 119
6.7 Mode patterns for linear alignment. . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.8 Geometry and spectral response for angular alignment. . . . . . . . . . . . . . . 120
6.9 Mode patterns for angled alignment. . . . . . . . . . . . . . . . . . . . . . . . . 120
6.10 Geometry and spectral response for lateral alignment. . . . . . . . . . . . . . . 121
6.11 Mode patterns for lateral alignment. . . . . . . . . . . . . . . . . . . . . . . . . 121
6.12 Beating between the cavities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.13 Avoided crossing of two coupled PhCCs. . . . . . . . . . . . . . . . . . . . . . . 123
6.14 Distant-dependent line splitting of different alignments. . . . . . . . . . . . . . 124
13 LIST OF FIGURES
6.15 Alignment schemes for three cavities. . . . . . . . . . . . . . . . . . . . . . . . . 125
6.16 Spectral response from three coupled cavities. . . . . . . . . . . . . . . . . . . . 125
6.17 Spatiospectral response from three laterally aligned cavities. . . . . . . . . . . . 127
6.18 Mollow triplet, schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.19 Frequency- and Q-factor modification. . . . . . . . . . . . . . . . . . . . . . . . 129
6.20 Intensity-dependent Mollow triplets for different Q-factors. . . . . . . . . . . . . 130
6.21 Q-dependent Mollow triplets for different Excitation amplitudes. . . . . . . . . 131
LIST OF FIGURES 14
List of Tables
2.1 Quantum Mechanics vs. Electrodynamics: A comparison. . . . . . . . . . . . . . 34
B.1 List of Constants of Nature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.2 List of Parameters for the Quantum Dot. . . . . . . . . . . . . . . . . . . . . . 139
B.3 List of Parameters for the PhC Cavities and Microdisks. . . . . . . . . . . . . . 140
B.4 List of CPML Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
15
LIST OF TABLES 16
Chapter 1
Introduction
1.1. MOTIVATION 18
1.1 Motivation
In the last years, optoelectronics and photonics became a new and important field in physics.
In general, optoelectronics deals with the transfer of electronically coded information into
information coded in the basic quantum of the electromagnetic interaction, the photon, and
vice versa. But still, this can be explained and expressed in a semiclassical way where the
photons are described via electromagnetic waves according to Maxwells equations. However,
more specifically, photonics deals with the nonclassical part of the optoelectronics.
To do optoelectronics, semiconductor heterostructures can be used. In general, semiconduc-
tors can have two different electronic properties: they can be either conducting or metallic,
depending on several parameters like temperature, pressure, doping density, etc. In the con-
ducting regime, valence band (VB) and conduction band (CB) are separated by a gap. Charge
carriers like electrons cannot be permanently located in this gap energetically. To transfer
the charge carrier from one band to the other by bridging the energetically forbidden gap, an
energy carrying particle is needed. In optoelectronics, this energy is provided by the photon.
Electrons falling from higher energy levels to lower energy levels of the semiconductor band
structure cause the emission of light, and hence coding information of the electronic system
before the electronic decay into the photon. The other way around, if a photon is absorbed
by a semiconductor, an electron is excited from a lower energy state into a higher one, and
information coded in the photon is then coded in the electronic system of the conducting
semiconductor.
With the today available technology, semiconductor structures can be made smaller in size
than the wavelength of the light, which is used to transfer the energy of the light to the
electronic system or manipulate the electronic system in any way. This gives the possibility
for compound structures consisting of many different semiconductor materials to use all the
different properties at once and use the combined electronic band structure properties. For
example, the quantum well (QW) is a thin layer of a semiconductor material sandwiched
between thick layers of a semiconductor material with a larger band gap. Molecular beam
epitaxy (MBE) or metal organic chemical vapor deposition (MOCVD) are exemplarily men-
tioned for the growing procedure. This forms a potential well, in which the charge carriers can
only move within the thin layer, which can be treated as a two-dimensional electron gas. Per-
pendicular to the plane of the thin layer, the movement of the electrons is not allowed due to
the potential walls. This phenomenon is called confinement (the effect of confinement occurs,
when the confinement length is in the range of the de Broglie wavelength of the particle). The
energetically allowed region for charge carriers is then no longer a continuum, but discrete
energy levels. For example, the QW is used in lasing systems like the quantum cascade laser.
Many layers of QWs are stacked behind each other and splitting of the energy bands occurs,
hence so-called minibands are formed. The carriers fall, after being injected on one side, down
the energy levels from this multiple QW structure in a cascade, emitting a photon at every
change of energy. Also, QWs are used in several diode laser systems as well in Distributed
Bragg Reflectors (DBR). Another representative of a semiconductor heterostructure is the
quantum dot (QD). One epitaxial procedure to grow quantum dots is the Stranski-Krastanov
(SK) procedure using the MBE, which results in self-assembled QDs with measures from tens
to some hundreds nanometers scaling in each direction with a density of up to 104atoms or
molecules. Those quantum dots look like flat pyramidal structures, which confine the charge
carriers in all three spatial directions. The charge carriers have then discrete energy levels
in all dimensions like in in atomic systems or a particle in a box. That is why a QD is also
19 1.1. MOTIVATION
called an artificial atom. Hence, this effectively leads to a zero-dimensional structure. QDs
are already widely used, e.g. in QD lasers, diodes, single electron transistors. Also, they are
promising candidates for quantum information processing [5], e.g. for storage or manipulation
of quantum information, since dephasing mechanisms for the charge and spin degree of free-
dom are already investigated and understood well, see for example [6–8].
Both, the QW and the QD can be manipulated by a light field via direct illumination, or by
placing the active media in an optical resonator, a cavity, like a microdisk resonator (MD) or
a photonic crystal cavity (PhCC). There, the light field is concentrated within a small spatial
region. When the electronic system is placed in a high-field position in the resonator, the light
can interact temporally in the range of the cavity decay time. Different cases can occur: First,
when the cavity decay time is larger than the dephasing time of the QD, the spontaneous
emission rate of the QD is enhanced by the Purcell factor [9–12]. This is called the weak-
coupling regime. Second, the cavity decay time can be tuned by several mechanisms up to the
nanosecond regime [13], even on a chip [14], which equals the dephasing time of excitons in
QDs. In this case, after emission of the photon from the QD, the photon is reabsorbed again
before it can leave the cavity and the so-called strong-coupling regime is reached. Hence, this
results in new eigenstates, the cavity polaritons, manifested in a line splitting in the frequency
domain, the Vacuum Rabi Splitting (VRS) [15]. Third, this can be pushed further, when the
splitting energy of the cavity polaritons is in the same order of magnitude as the uncoupled
energies and is referred as the ultrastrong coupling regime [16]. Besides coupling between the
light field and a quantum mechanical oscillator like a QD, strong coupling is also observed
in systems consisting of a light field and a macroscopic oscillator (e.g. in a one-dimensional
cavity, one of the mirrors is fixed while the other is movable due to a small spring [17–19]).
It reveals also quantum mechanical properties of macroscopic systems, like the ground state
of the mechanical oscillator [20]. Further investigations on strong optomechanical coupling in
photonic crystal cavity systems are described in [21–23].
Liquid crystals (LC) are know since the end of the 19th century [24] (however the designa-
tion liquid crystal was introduced by a German physicist [25]), and till now still fascinate
researchers all over the world. Actually, research with LCs started about 40 years ago in the
1970’s. Not only because LCs can be utilized for fundamental research, but also because LCs
found their way into every days life. The most popular usage is the LC display, which makes
use of the electro-optical switching properties, present in calculators, watches, cell phones,
navigation systems, automatic teller machines, just to mention some of the merely infinite
possibilities of usage. Also, medical and security applications are already employed. In gen-
eral, LCs can exhibit special reversible or irreversible optical properties: optical anisotropy.
The change from isotropic to anisotropic behavior can be induced by several external param-
eters like temperature, pressure or electric fields. Possible future applications can make use
of the switchability between isotropy and birefringence to e.g. tune resonances [26, 27]. Also,
in the new field of transformation optics [28, 29], anisotropy is used often to guide the light
around small-sized objects to camouflage them. There are many different kinds of LCs, how-
ever, in this thesis only thermotropic crystals in the isotropic or nematic phase are considered.
All these mechanisms can be utilized for manipulating electronic properties of the semicon-
ductor heterostructures or the photons itself. The range of applications is broad: those sys-
tems are already used for probing, investigating and manipulating the fundamental physics of
light-matter interaction. Devices for single photon generation, high finesse filters, slow light
generation, etc. are also realized. Still, this is a current research topic in modern physics in
the area of nanooptics, nanophotonics and optoelectronics. But there is still a problem: if one
1.2. STRUCTURE OF THIS THESIS 20
wants to build devices which are operating in a specific coupling regime, then this device has to
be tested. One problem is to place the QDs in high-field positions, and additionally to this, to
have the gap energy of the QD match exactly (or within a decay rate) the energy of the cavity
mode. To overcome trial-and-error procedures, numerical analysis became a crucial tool in de-
signing optoelectronic devices. Nowadays, large-scaled computers and computer clusters allow
simulations of various problems of optoelectronics and photonics and treat the electromag-
netic together with the material equations for large simulation volumes with nanoscale-leveled
spatial discretization for reliable results in acceptable times.
For near-field resolved numerical analysis of optoelectronic devices, several methods can be
used. Common methods are the Finite-Difference Time-Domain method (FDTD) [30–32] and
the Finite Element Method (FEM) [33, 34]. In general, these two methods differ in the working
domain. As the name already reveals, FDTD is a time domain method, so with one calculation
one can get information over a whole frequency range. On the other hand, FEM is a frequency
domain method, thus to have a spectrum, one needs to calculate the response for every single
frequency. On the other hand, FEM is more flexible in the choice of the spatial grid. Both
methods have their own advantages and disadvantages, and the method of choice is highly
problem-dependent. Within the framework of this thesis, the FDTD method is used.
At the University of Paderborn, methodical development in this research area is done. Used
methods are the finite integration method, based on the Finite Integration Technique (FIT)
[35, 36] and is widely used for large-sized systems. This method solves Maxwells equations on a
primary and dual grid, similar to FDTD, but the differential operators are not expressed with
finite differences, but treated with a matrix formulation, resulting in (sparse) matrices with
entries 0,±1. FIT can be used for time-domain simulations as well as for frequency domain
simulations simply by replacing the time derivative ∂t−→ iω. Also, non-orthogonal grids are
supported to save a huge amount of computational effort [37]. Especially for one-dimensional
(linear) problems, the Transfer Matrix Method (TMM) is the method of choice [38, 39]. Here,
a two-dimensional input vector (containing the incident and reflected field) is multiplied with
2×2transfer matrices for every interface between different isotropic dielectric/magnetic ma-
terials to get the output vector (containing the transmitted and incident field from the other
side). This method can also be generalized for anisotropic materials [40, 41]. In fact, the lat-
ter is for acoustic waves in anisotropic media, but can be directly transferred to propagation
of electromagnetic waves in anisotropic dielectric (dispersive) media. Since some years, also
the Discontinuous Galerkin Time Domain method DGTD is widely used, which combines the
advantages of FDTD and FEM [42–46].
1.2 Structure of this thesis
The structure of this thesis is divided into 6 chapters. The fundamentals are shown in Chap.
2. Starting from the basis of the whole thesis, in Sec. 2.1, Maxwells equations, the reader is
guided to the numerical evaluation with the FDTD method in Sec. 2.2. The principles for
the derivation of the update equations is depicted and stability, accuracy and problems of the
algorithm are discussed. The need of numerical methods for averaging optical properties in a
spatially discretized grid and of a finite simulation volume calls attention in this section, too.
Different kinds of optical resonator, photonic crystal cavities and microdisk resonators, taken
into account in the framework of this thesis, are shown in Sec. 2.3 and 2.4, respectively. As a
quantum-mechanical resonator, the QD model is introduced in Sec. 2.5, including nonlinear-
21 1.2. STRUCTURE OF THIS THESIS
ities and the embedding into the FDTD grid. Fundamentals of coupling between oscillators,
going from the weak coupling regime to the ultra-strong coupling regime, is found in Sec. 2.6.
For simulations including optical anisotropy, Sec. 2.7 gives basic information about LCs. The
last Sec. 2.8 depicts the methods used for data analysis and special features, which can be
accessed in the used FDTD implementation.
Chapter 3 validates the implementation of the FDTD method. Here, numerical results from
H1- and L3-type photonic crystal cavities are compared with experimental results from rel-
evant publications. The spectral response for different polarization are compared and mode
patterns of the resonances are shown.
Chapter 4 guides the reader from two-dimensional to three-dimensional calculations of mi-
crodisk devices. A sub-µmicrodisk, providing highly confined modes with a vacuum wave-
length smaller than the device diameter, is used to numerically investigate the effect of im-
perfections, occurring e.g. in the fabrication process, in Sec. 4.1.1. Effects of a non-circular
shape (Sec. 4.1.1) and edge profiles (Sec. 4.1.1) are taken into account. The subsequent Sec.
4.1.2 shows, how highly confined modes are affected in an uniaxial anisotropic environment.
Anti-crossing features occur due to the applied anisotropy, especially in an experimentally
accessible range for the ordinary and extraordinary refractive indices. After the study of dif-
ferent effects on resonances in a circular-shaped structure, a microdisk device embedded in a
LC environment is simulated using uniaxial anisotropy in Sec. 4.2.1 and numerical results are
compared with experimental data, which agree nicely.
The numerical results for PhCCs are given in Chap. 5. Different possibilities for manipulating
resonances in line-defect PhCC in a hexagonal lattice of air holes are shown. Permanent tuning
of the L3-type PhCC with the method of gentle confinement [3] in Sec. 5.2 and the L7-type
PhCC with a one-side evaporated additional dielectric layer of lower permittivity in Sec. 5.3
exhibit the behavior of the defect modes and the effect on the spectral properties. The latter
PhCC modification is shown to agree very well with experimental data.
Coupling of different resonators is calculated in Chap. 6. Section 6.1 shows numerical data
for coupled sub-µMDs to investigate the distant-dependent line and loss splittings. Strong
resonator-resonator interaction is observed. Section 6.3 deals with the coupling between de-
fects in photonic band gap materials, say photonic crystal cavities, where the coupling is
investigated distant- and alignment-dependent. For applications, i.e. coupled resonator opti-
cal waveguides (CROWs), Sec. 6.3.2, shows, that the coupling between the defect cavities can
be enhanced with a proper alignment. The last Sec. 6.4 shows the coupling between a photonic
crystal cavity mode and a single quantum dot, placed at a high-field position. Effects of the
nonlinearity of the QD on the QD emission are studied, depending on the exciting intensity
and the cavity decay time.
The last Chap. 7 concludes the content of this thesis and gives a short outlook on possible
future works.
In the appendix, numerical methods for the integration of ordinary differential equations and
an interpolation scheme are shortly depicted in Sec. A and constants for the numerical inves-
tigations are listed in Sec. B.
1.2. STRUCTURE OF THIS THESIS 22
Chapter 2
Fundamentals
2.1. MAXWELL’S EQUATIONS 24
2.1 Maxwell’s Equations
In 1865, James Clark Maxwell published the basic set of equations for electric and magnetic
fields Eand H, respectively, today famously called Maxwell’s Equations [47]. These equations
connect the electric and magnetic properties of the fields and show, that the sources of the
electric and magnetic fields are the scalar function ρ, the charge density, and the vector function
j, the current density. Since Maxwell’s equations are form-invariant under unit transformation,
such transformations change definitions and constants [39, 48]. Throughout this thesis, SI units
are used, if not mentioned otherwise.
The equations in their differential form can be written as [49]:
∇×E(r,t)=−˙
B(r,t)(2.1)
∇×B(r,t)=µ0j(r,t)+µ0!0˙
E(r,t)(2.2)
∇·B(r,t) = 0 (2.3)
∇·E(r,t)=ρ(r,t)
!0
,(2.4)
where E(r,t)is the electric field strength, B(r,t)the magnetic flux density, ρ(r,t)the charge
density, j(r,t)the current density at position rand time t,!0and µ0are the free space permit-
tivity (vacuum permittivity) and free space permeability (vacuum permeability), respectively.
Maxwell’s equations can also be written in integral form [49]:
!
∂S
E(r,t)·dl=−∂tΦB,S (2.5)
!
∂S
B(r,t)·dl=µ0IS+µ0!0!r(r)∂tΦE,S (2.6)
!
∂V
B(r,t)·dA=0 (2.7)
!
∂V
E(r,t)·dA=1
!0!r(r)ρ∂V.(2.8)
Here, ΦB,S and ΦE,S are the magnetic and electric flux through a surface S,ISis the electric
current through surface Sand ρ∂Vis the charge density inside the boundary of volume V.
The vectorial length and area elements are dland dA, respectively.
The differential forms of Maxwell’s equation (2.1)-(2.4) can be transformed into the integral
forms (2.5)-(2.8) with vector-analytical methods using Kelvin-Stokes’ and Gauss’ theorem, see
[48, 49]. Equations (2.1)-(2.4) have to be extended to obey optical and magnetic properties
of material on a non-atomic level. Calculations on the atomic level are utterly impossible
to perform. Therefore, an averaged quantity of the electric and magnetic response of the
considered material is introduced, which flattens possible fluctuations in the fields due to
atomic-sized objects. Thus, the electric field is modified with the dielectric constant !rand
the magnetic field is modified with magnetic permeability µr, which typically depend on
position r. Actually, these two quantities are tensors of second second rank and frequency-
dependent. The space dependency and tensor characteristic will be mentioned in the later
chapters in more detail. Since the interest lies on nonmagnetic materials, µris always equal
to 1.µr(r)only differs weakly from unity, so that approximation to 1is justified. For the sake
25 2.2. FDTD
of completeness, µr(r)will still appear in the following equations. Therefore, eqs. (2.1)-(2.4)
extend to the macroscopic Maxwell’s Equations via the constitutive relationships [48]:
B(r,t)=µ0µr(r)H(r,t)+M(r,t),D(r,t)=!0!r(r)E(r,t)+P(r,t),(2.9)
where Hand Dare the magnetic field strength and dielectric displacement field at position
rat time t. Magnetization M(r,t)and polarization P(r,t)are not taken into account here.
With this, eqs. (2.1)-(2.4) transform to the macroscopic Maxwell equations
∇×E(r,t)=−˙
B(r,t)(2.10)
∇×H(r,t)=j(r,t)+ ˙
D(r,t)(2.11)
∇·B(r,t) = 0 (2.12)
∇·D(r,t)=ρ(r,t),(2.13)
where ρ(r,t)and j(r,t)are now macroscopic charge and current densities, respectively. The
time-dependent Maxwell’s curl equations (2.1) and (2.2) together with the constitutive re-
lationships (2.9), and assuming that the fields are known at an initial position in time and
space, form a hyperbolic partial differential equations (PDE). This set of hyperbolic PDEs
has a unique solution, uniquely depending on the initial values. For numerical evaluation, one
uses the curl equations with E(r,t)and H(r,t)for reasons which can be found in Sec. 2.2:
∇×E(r,t)=−1
µ0µr(r)˙
H(r,t)(2.14)
∇×H(r,t)=j(r,t)+!0!r(r)˙
E(r,t).(2.15)
Discretization of eqs. (2.14) and (2.15) is fundamentally shown in Sec. 2.2.1 for numerical
evaluation. For further information, see [30, 32].
2.2 Numerics With Maxwell’s Equations: The Finite-Difference
Time-Domain Method
To numerically evaluate eqs. (2.10)-(2.13), one needs to discretize time and space, since con-
tinuous problems cannot be solved on computers due to limited computational resources like
memory. The discretization scheme in space is based on the idea of Yee [50], where the different
field components of different field quantities are placed at different positions on a staggered
grid, the so-called Yee-Cube (see Fig. 2.1, left). Electric field components are centered on the
edges of the electric grid, the primary grid, whereas magnetic field components are centered
on the edges of the magnetic grid, the secondary grid. The structure of eqs. (2.5)-(2.8) gives
rise, that the magnetic field has to be on a grid, which has its corner in the center of the grid
for the electric field. For example, in eq. (2.5) one has to calculate the line integral of the
electric field along a closed curve to obtain the magnetic flux through area bounded by this
closed curve (see also Fig. 2.1, right). The proposal from Yee exactly fulfills the conditions,
given by the integral forms of Maxwell’s equations. Namely, the secondary grid is shifted half
a grid edge length in each direction, so that the magnetic field components centered on the
surfaces of the primary grid and vice versa. This section only provides a short sketch of how
to discretize Maxwell’s equations and follows the work of [30, 32]. For further information and
reading, the interested reader is referred to these references.
2.2. FDTD 26
Figure 2.1: Left: Primary and secondary grid of the Yee-cube at node (i, j, k). Electric field com-
ponents live on the edges of the primary grid (blue), while magnetic fields live on the edges of the
secondary grid (red). The arrangement of both grids, staggered half a cell length in all directions,
let electric field be localized face-centered on the secondary grid, and magnetic field be located face-
centered on the primary grid. Right: Scheme of the line integral at node (i, j, k). Due to the staggered
arrangement, the line integral, exemplarily for eq. (2.5) can be evaluated easily.
2.2.1 Update Equations of the FDTD
Spatially the computational volume is separated into voxels, the three-dimensional analogue
to the two-dimensional pixel, which are indexed with natural number triples (i, j, k)for the
x, y, z-direction. Neighboring cells have incremented or decremented indices depending on the
direction. An edge length of a grid cell is ∆µ,µ=x, y, z, so that neighboring field quantities
are separated by one discretization length. Now, to address the field components in a numeric
scheme, the following nomenclature is used:
(E(r))µ=Eµ ijk,t =Eµ"(i+1
2δµx)∆x, (j+1
2δµy)∆y,(k+1
2δµz)∆z#(2.16)
(H(r))µ=Hµ ijk =Hµ"(i+1
2(1 −δµx),t)∆x, (j+1
2(1 −δµy))∆y,(k+1
2(1 −δµz))∆z)#,
(2.17)
where δµνis the Kronecker-symbol, which is equal to 1 if µ=νand 0otherwise. Required
field components on positions on the Yee-cube, where they are not defined, like Exi+1
2j+1
2k,
have to be averaged arithmetically from known field components, e.g.
Exi+1
2j+1
2k=1
2$Exi+1
2jk +Exi+1
2j+1k%(2.18)
Hxi+1
2j+1
2k=1
2$Hx ij+1
2k+Hxi+1j+1
2k%
=1
4$Hx ij+1
2k−1
2+Hxi+1j+1
2k+1
2%.(2.19)
With this spatial discretization scheme and the possibility to average field components at
positions on the Yee-Cube, where the desired field component is not defined, one can now
27 2.2. FDTD
Figure 2.2: Update scheme for the fields (leapfrog scheme). Time stepping is in horizontal direction,
while spatial stepping is in vertical direction (direction according to arrows). To update the magnetic
field Hn+1
2
i−1
2
, the field one full time step ∆tearlier, Hn−1
2
i−1
2
, and the electric fields half a time step before
and half a space step back and forth each, En
i−1and En
i+1, is needed. Update of the electric field is
similar, but fields are exchanged and half a step has to be added to time and space (Here, two half
steps in time and space is shown to have the update scheme separated).
discretize the spatial differential operator ∂µ. Since the derivative is a quantity obtained by
a limes, discretization is needed to let the differential operator fit on the spatial grid. So
the derivative is now written like in its derivation, but with central differences. Exemplarily,
∂x(H(r,t))ythen yield
∂xHyi+1
2jk+1
2=1
∆x(Hyi+1jk+1
2−Hy ijk+1
2).(2.20)
The macroscopic Maxwell’s equations (2.10)-(2.13) now have to be discretized in time with
central differences, too, due to the partial derivatives in time. Temporal integration is then done
with the so-called Leap-Frog-algorithm, where electric field quantities at time tare calculated
from neighboring field quantities from the magnetic quantities at t+1
2∆tand t−1
2∆t, where
∆tis the discretization in time. See Fig. 2.2 for a coarse scheme. Therefore, the numerical
scheme for all the field components is extended with another natural numbered index, n, for
the time, i.e.
(E(r,t))x=En
xi+1
2jk =Ex((i+1
2)∆x, j∆y, k∆z, n∆t)(2.21)
(H(r,t))x=Hn+1
2
x ijk =Hx(i∆x, (j+1
2)∆y, (k+1
2)∆z, (n+1
2)∆t).(2.22)
A generic discretization of the partial temporal derivative with central differences of (E(r,t))x
and (H(r,t))zthen yields:
∂tEn
xi+1
2jk =1
∆t"En+1
2
xi+1
2jk −En−1
2
xi+1
2jk#(2.23)
∂tHn+1
2
zi+1
2j+1
2k=1
∆t"Hn+1
zi+1
2j+1
2k−Hn
zi+1
2j+1
2k#.(2.24)
With these tools one can now derive update equations to calculate the components of electric
and magnetic field on all grid points at the next unknown time step. To do so, one uses the
2.2. FDTD 28
curl equations (2.12)-(2.13) and replaces the derivatives in time and space like eq. (2.20) and
(2.23)-(2.24). This yields the following update equations exemplarily for the x-component of
the electric and magnetic field:
En+1
xi+1
2jk =En
xi+1
2jk −∆t
!0!i+1
2jk
×"1
∆y(Hn+1
2
zi+1
2j+1
2k−Hn+1
2
zi+1
2j−1
2k)−1
∆z(Hn+1
2
yi+1
2jk+1
2−Hn+1
2
yi+1
2jk−1
2
)−jn+1
2
xi+1
2jk#
(2.25)
Hn+1
2
x ij+1
2k+1
2
=Hn−1
2
x ij+1
2k+1
2−∆t
µ0
×"1
∆y(En
z ij+1k+1
2−En
z ijk+1
2
)−1
∆z(En
y ij+1
2k+1 −En
y ij+1
2k)#,(2.26)
from which one can calculate the unknown (E)µat time t+∆tusing (E)µat time tat the
same position in the Yee cube and Hon the surrounding positions.
These update equations are now based on a centered difference scheme in time as well as in
space. As one may have already noticed, the spatial derivatives ∂µ(E(r,t))µdid not emerge.
The reason is, that this update scheme based on the Yee-cube naturally fulfills the diver-
gence equations (2.12)-(2.13). Hence, the full set of electrodynamic equations (2.10)-(2.13) is
complete with temporally and spatially update eqs. (2.25)-(2.26) of the FDTD.
2.2.2 Accuracy and Stability
Discretization of the Maxwell eqs. (2.14) and (2.15) induces mistakes in the temporal and
spatial derivatives, since they are continuous, but the FDTD grid in time and space is discrete.
A Taylor expansion shows the order of the error in terms of the spatial discretization. Assume
a function f(x)probed by a grid with discretization ∆xand use central differences to express
a derivative in x. For the left-aligned (right-aligned) function values, f&x−∆x
2'(f&x+∆x
2'),
the Taylor expansion equals
f"x+∆x
2#=f(x)+∆x
2f(1)(x)+1
2"∆x
2#2
f(2)(x)+1
6"∆x
2#3
f(3)(χ+)(2.27)
f"x−∆x
2#=f(x)−∆x
2f(1)(x)+1
2"∆x
2#2
f(2)(x)−1
6"∆x
2#3
f(3)(χ−),(2.28)
(2.29)
where f(n)is the nth derivative of fand χ±∈[x, x ±∆x
2]. Now, subtracting (2.28) from (2.27)
and dividing by ∆x, one gets
f(1)(x)=f&x+∆x
2'−f&x−∆x
2'
∆x−1
3"∆x
2#3
f(3)(χ),(2.30)
with χ∈[x−∆x
2,x+∆x
2]. So the centered first derivative approximation eq. (2.30) is of
O(∆x2).
The time integration to get E(r,t)and H(r,t)for the new time steps in eqs. (2.25) and
(2.26) is done according to Fig. 2.2. Here, E(r,t)(H(r,t)) at time t=(n+ 1)∆t(for H(r,t):
29 2.2. FDTD
t=(n+1
2)∆t) is calculated with the previous value at t=(n−1)∆t(for H(r,t):t=(n−1
2)∆t)
and the surrounding H(r,t)(E(r,t)) at time t=(n+1
2)∆t(for E(r,t):t=n∆t). Expressing
this procedure with the functions f(t)and g(t), which are defined at temporal positions
t=n∆tand t=(n+1
2)∆t, respectively, and assuming that F(·)is some functional, the
update scheme looks like
f((n+ 1)∆t)=f(n∆t)+F"g"(n+1
2)∆t## (2.31)
g"(n+1
2)∆t#=g"(n−1
2)∆t#+F(f(n∆t)) .(2.32)
Since the time integration is of second order in the discretization, too, the stability criterium for
the FDTD implementation, the so-called Courant-criterium, can be exploited to the maximum.
The Courant-criterium says, that
∆t⩽s·1
c0(∆x−2+∆y−2+∆z−2(2.33)
ensures stability of the implementation. In the case of different orders of accuracy for the
time and space derivative approximations, the factor shas to be s << 1, resulting in a small
temporal discretization. In the FDTD case, sis mostly chosen to be ≈1.
If sequals 1, (2.33) is called the magic time step. In a one-dimensional FDTD implementation,
the simulation equals the analytic solution exactly. The reason is the numerical dispersion,
which is induced by the spatial grid. Numerical dispersion in general means, that a wave
propagating in the spatial grid will be slowed down compared to its natural speed, c0. In a
one-dimensional case, one can choose the time step according to the magic time step, ∆t=∆x
c0.
Hence, the time, which is needed for a propagating wave to go c0∆tequals exactly eq. (2.33)
with s=1, and the speed of the wave is exactly c0, so no numerical dispersion occurs. In
higher-dimensional FDTD simulations, the magic time step would only ensures no numerical
dispersion along the principle axes of the simulation volume (if a uniform grid is provided).
The other directions then carry a numerical dispersion, which are calculated via
)1
c0∆tsin "ω∆t
2#*2
=+
µ=x,y,z ,1
∆µsin -˜
kµ∆µ
2./2
,(2.34)
where ˜
kµare the components of the numerical wave vector (see [30]).
In other words, the temporal discretization is determined by the spatial discretization and
s. Since eq. (2.34) is of the same order of accuracy as eq. (2.32), the numerical dispersion
can be reduced drastically when a smaller spatial discretization is used. The problem is, that
when reducing the the spatial discretization by ∆µcoarse =m·∆µfine, memory raises with
O(m3)and computational time raises with O(m4). Accordingly, a blind reduction of ∆µhas
no use, in fact one has to think about the problem one wants to solve. A good value to
start with is ∆µ=0.1λ0, with λ0being the smallest wavelength in the material with the
highest dielectric permittivity and the highest conductivity one wants to detect. However,
this is of course no rule, but a rule of thumb. Another point to think of is, what is the
smallest geometrical shape one wants to resolute. Usually, curvatures need a higher resolution
to let the waves in the computational domain interpret the structure as a curved structure.
In cartesian grids, curvatures always result in staircasing, Fig. 2.4, which affects resonant
2.2. FDTD 30
Figure 2.3: Staircasing effect on the resonance of an eigenmode in a circular resonator (two-
dimensional calculation, [51]). A cartesian (uniform) grid is used to map the perfectly circular struc-
ture. The spatial cell size is decreased from 10 nm down to 1.7nm in 0.1nm steps. The vertical axes
shows the resonant frequency for a certain eigenmode. Red: numerically calculated resonance. Green:
analytical value.
frequencies drastically [52, 53]. Non-cartesian grids can be applied to solve the problem. For
cylindrical structures, the best choice is a cylindrical coordinate system (if one has point
symmetry to the origin). Therefore, transform eqs. (2.12)-(2.13) into cylindrical coordinates
and then perform the discretization in the new coordinate system, like in the free available
software package MEEP [31]. Another method is to use a polygonal mesh without orthogonal
principal axis like in the frequency-domain method FEM [33, 34] or the DGTD method [42–
46]. To overcome a lack in accuracy due to staircasing effects, one can use an averaging method
for the permittivity within the cells (Sec. 2.2.3).
Figure 2.3 shows the dependency of the numerical frequency of a perfectly circular resonator
structure in a metallic environment on the size of the spatial grid, used in a two-dimensional
FDTD [51]. The radius of the resonator structure is set to R= 361 nm and the permittivity
is set to the GaAs value of != 11.56. The usage of a cartesian grid yields the mapping of the
dielectric structure to be staircased at the boundary. Therefore, an increase or decrease of the
cell size changes the effective resonator shape, hence the frequency of the resonance changes.
The analytically calculated value of the 4th resonance is f= 295 THz (green). As can be
seen, the numerically calculated values for the resonances (red) approach asymptotically the
analytical value, in fact it stays below the exact value, even for cell sizes much smaller than
the expected wavelength. Thus, convergence for a curved structure in a cartesian grid is not
expected along this thesis, but sufficiently small spatial probing will be chosen to e.g. ensure
experimentally unmeasurable frequency changes.
31 2.2. FDTD
Figure 2.4: Two-dimensional scheme of the staircasing. The black line shows the interface between
two media with different dielectric permittivity (white and grey). Spatial cells, which contain the
interface, are filled with only one of the two materials, if no averaging routine is provided. The shape
of the originally curvatures object gets staircased.
2.2.3 Subpixel-Averaging for Isotropic Dielectric Media
In the setup of the spatial grid and the implantation of dielectric material in arbitrary shape,
it happens that Yee-cells are not filled totally with one and the same dielectric material, hence
one cell has two or even more different values for the dielectric permittivity. The question
arising is, what value is assigned to this cell. Obviously, one can, e.g. in the case of two
dielectric materials in one cell, calculate the amount of each material filling this cell, and
then fill the whole cell with the most frequently present. This results, especially for circular
structures, in a staircasing, see Fig. 2.4. Also, this reduces the accuracy of the algorithm to
1st order. To overcome this problem, one uses an averaging method for the permittivity, based
on the filling factor of the different permittivities in the corresponding cell. In general, such
averaging schemes can drop the accuracy down to 1st order in the spatial discretization. For
two reasons, the method of choice is the subpixel averaging scheme developed in [54]. First,
the averaging scheme is chosen to be in a way that, treating the problem in a pertubative
approach with pertubative parameter s, for s→0the 1st error vanishes, providing 2nd order
accuracy. In fact, the result is then an enhanced accuracy, up to 2nd order for special cases and
specific fields, but in general a higher accuracy than without averaging. Second, this averaging
scheme can be extended to full anisotropic materials, too [55].
The performance of the averaging process is shown in Fig. 2.5. Here, a two-dimensional FDTD
calculation with a cylindrical dielectric object with a radius of r=3µm, embedded in vacuum,
is done. The behavior of a specific mode around ε=0.806 eV is chosen to be tracked, while
the cylindrical object is shifted in subpixel steps over 4 cells (right part of Fig. 2.5, shown
schematically). The spatial discretization correspond to ∆x=∆y= 40 nm. Spectral data
are extracted with the Harmonic Inversion tool (see Sec. 2.8). In the top case, no !-averaging
routine is used in the simulation. A periodic shifting up to ∆ε=1meV in the resonant energy
of the system with a periodicity of the cell length, as well as a modification in the intensity of
the resonance is observed. In fact, shifting the dielectric object in subpixel steps changes the
geometry, since cells, which were filled before, are no longer filled (or vice versa) after the shift.
The lower case, where the !-averaging procedure is applied, shows no shift in the resonance.
2.2. FDTD 32
Figure 2.5: Left: Comparison of a simulation of a cylindrical object with a radius of r=3µm. Top:
no !-averaging. Bottom: with !-averaging. Chasing the resonance occurring at ε=0.806 eV, one can
clearly see blue-shift of the resonance up to ∆ε=1meV with a periodicity of 40 nm, equal to the
spatial discretization, resulting from an efficiently smaller object due to staircasing. Right: scheme of
the subpixel shift of the dielectric cylindrical object. As the true boundary passes a cell, the cell itself
is either full filled or half filled.
Still, the intensity in the resonance changes with a periodicity as before without the averaging
process, but the deviation is of lower order.
2.2.4 CPML: Truncate the Computational Volume
Since the computational resources are limited, one has to, next to temporal and spatial dis-
cretization, truncate the computational volume in space. Setting all the field values out of the
simulation volume hard to zero would end up as a perfect electric (magnetic) conductor, PEC
(PMC). Hence, all fields inside the computational volume are then reflected from the bound-
ary, since no fields would be able to penetrate the outer regions. The result then is that the
implanted energy will stay inside the simulation volume forever (despite usage of lossy mate-
rials). Since this is unphysical, one at best needs absorbing boundary conditions (ABC) which
totally absorb the energy of incident waves, independent of the polarization, frequency and
incident angle, in fact are 100% transmittive. Furthermore, the absorbing boundary should be
effective when scattering objects or sources, especially point sources, are close. Additionally,
since the ABC area also consists of cells which need to update E(r,t)and H(r,t)in every time
step with additional effort, the absorbing area should not be too large to not unnecessarily
enlarge the computational domain and hence increase the computation time.
The first widely used numerically stable, second-order accurate ABC was formulated by Mur
[56], but the disadvantage is that it only absorbs specific polarization well. Berenger then
introduced a highly efficient ABC [57], the so-called perfectly matched layer (PML), based
on formulation of split fields, which are orthogonal to each other, resulting in 12 coupled 1st
PDEs with loss parameters. The big advantage of the PML is the universal applicability to
cut the simulation volume consisting not only of isotropic lossless dielectric material, but also
33 2.3. PHC
anisotropic, nonlinear and dispersive materials, just to mention some. Later, a formulation
with stretched coordinates was developed. This gave the possibility to also apply the PML
concept to other orthogonal coordinate systems than the cartesian one. The split-field PML
proposed by Berenger and its stretched analogue is based on a mathematical model. The
uniaxial PML (UPML) based on an anisotropic PML yields a physical formulation [58]. The
performance of Berengers split-field PML and the UPML is the same, but they differ in Gauss’
law. Since the use of a classical tensor coefficient can lead to large reflection errors in the low
frequency range, the complex-frequency shifted (CFS) PML overcomes this problem [59, 60].
However, Roden and Gedney implemented a more efficient version using strechted coordinates
from Berenger based on a recursive-convolution formulation, the convolutional PML (CPML).
The result is more efficient, more accurate than the UPML, and provides better applicability
to terminate regions with generalized media.
The strechted coordinates are built with the CFS tensor coefficient sµ=κµ+σµ
αµ+iω&0, where
the parameter ranges are κµ⩾1,σµ⩾0and αµ⩾0and µ=x, y, z. Berengers strechted
coordinates metrics is obtained for κµ=1and αµ=0. The update equation within the CPML
material exemplarily for the x-component of the electric field yields
En+1
xi+1
2jk =En
xi+1
2jk
−∆t
!0!i+1
2jkµ0"1
κyj∆y"Hn+1
2
zi+1
2j+1
2k−Hn+1
2
zi+1
2j−1
2k#−1
κzk∆z"Hn+1
2
yi+1
2jk+1
2−Hn+1
2
yi+1
2jk−1
2##
−∆t
!0!i+1
2jkµ0"ΨEx,y
n+1
2
xi+1
2jk −ΨEx,z
n+1
2
xi+1
2jk#(2.35)
(2.36)
with
ΨEx,y
n+1
2
xi+1
2jk =byjΨEx,y
n−1
2
xi+1
2jk +cyj
∆y"Hn+1
2
zi+1
2j+1
2k−Hn+1
2
zi+1
2j−1
2k#(2.37)
αµ=αµ, max $1−x
d%mα
bµ=e−“σµ
κµ+αµ”∆t
$0
cµ=σµ
κµ(σµ+κµαµ)(bµ−1) ,(2.38)
where 0⩽x⩽d, the boundary to the CPML is located at 0,dis the thickness of the CPML
and mαdetermines the scaling order. A detailed derivation can be found in [60].
Within this thesis, open boundary conditions are used, utilizing a CPML with parameters as
given in Tab. B.4, if not mentioned otherwise.
2.3 The Photonic Crystal
In nature, the concept of photonic crystals is not a new feature. One of the most famous rep-
resentatives is the butterfly Morphus rhodopteron with its shiny blue wings or the opal, which
can please every womans’ temper. The effect is no biological or chemical pigmentation, rather
a physical one. Multiple reflections of incident light on successive layers lead to constructive
and destructive interference, depending on wavelength and angle of incidence.
2.3. PHC 34
Quantum mechanics Electrodynamics
Field Ψ(r)e−iEt
!H(r)·e−iωt
Eigenvalue Problem ˆ
HΨ=EΨˆ
ΘH=$ω
c0%2H
Hermetian Operator −!2
2m∇2+V(r)∇×!r(r)−1∇×
Potential V(r)!r(r)
Table 2.1: Quantum Mechanics vs. Electrodynamics: A comparison. After [62].
Within the last decades, PhCs have inspired great interest not only due to their applications in
research, but also in various commercial areas accessible in the future. The most used phrase
in this context for sure is optical information processing. There, electronic manipulation of
data is then substituted by optical or optoelectronical manipulation [61]. Also, since speed
of light (in vacuum) is the highest velocity known by mankind, data transfer speed can be
drastically increased. Scientists all over the world deal with the experimental realization of
optical and optoelectronical devices, miniaturization, integrability and usability. Therefore,
PhC is a promising candidate, since, once a device is characterized, the characteristics can be
scaled up and down in size, because Maxwell’s equations (2.14)-(2.15) scale with length [62].
What is a photonic crystal? A crystal is a periodic arrangement of atoms (molecules) within a
certain pattern of repetition, forming the crystal lattice, which provides a potential. Thereby it
does not matter, if this potential is of electronic or optical kind. Charge carriers, i.e. electrons
in a lattice of a conducting crystal can propagate resistance-free (if no defects like impurities
are present), because electrons do not behave just like particles, which would be scattered
by the atomic bodies, but also like waves. When those waves match particular criteria, they
can propagate through the crystal without being scattered. On the other hand, they can also
be prohibited from propagating through the crystal lattice. Normally, there is not only one
criterium for a prohibition of propagation. In this way, different energetically forbidden areas
emerge, called the electronic band gap, which divides the valence band from the conduction
band and is basis for many important inventions in semiconductor physics [63–71]. In a PhC,
the optical analogue to a conductive crystal for charge carriers, photons behave in the same
manner [72–74]. The atomic (molecular) lattice is then substituted by different dielectric ma-
terials, which form the crystal, hence the potential for the photons. The compound material
can then form a photonic band gap for the light for a specific frequency range. The prohibition
of propagation can be manipulated to be in all directions, for all polarization, for smaller or
larger frequency ranges, depending on the designing rules applied to the compound material
[62, 75, 76] (see Fig. 2.6 as an example). Thus, manipulation of light can be performed on the
nanoscale, which is the basic key for small-sized photonic devices for future quantum infor-
mation processing computers.
To derive an equation to access the fields in arbitrarily shaped and arbitrarily composited
dielectric structures, one explicitly needs spatial variation of the dielectric constant even on
small length scales (in the range of the material wavelength, or even smaller). Therefore,
!r≡!r(r). Attention needs to be paid in eq. (2.13), since the divergence is taken from the
dielectric displacement field D(r,t)to get the charge density. Now, inserting eq. (2.9) results
in
∇·D(r,t)=∇·(!r(r)E(r,t)) = ρ(r,t),(2.39)
35 2.3. PHC
while eqs. (2.12), (2.14) and (2.15) remain unchanged. If one wants now to derive the wave
equation for the time-harmonic fields H(r,t)=H(r)·e−iωtand E(r,t)=E(r)·e−iωtand
without any sources of electromagnetic radiation in the standard way [48, 77, 78] one ends up
with [62]
∇×&!r(r)−1∇×H(r)'=ˆ
ΘHH(r)="ω
c0#2
H(r),(2.40)
!r(r)−1∇×∇×!r(r)−1D(r)=ˆ
ΘDD(r)="ω
c0#2
!r(r)−1D(r).(2.41)
Maxwell’s equations are now rewritten as an eigenvalue problem with the linear operator ˆ
ΘH
(ˆ
ΘD), corresponding eigenvalue $ω
c0%2and eigenvector H(r)(D(r)). Equation (2.41) was di-
vided by an extra !r(r)to keep the ˆ
ΘDhermitian. This is an important point, because the
eigenvalues are then real positive semi-definite, say larger or equal to zero. Additionally, the
fact that ˆ
ΘXis a linear operator, cannot be underrated, since (linear) superpositions of found
solutions are also solutions. For mathematical convenience, eq. (2.40) is normally used with
ˆ
ΘH=ˆ
Θ. Access to D(r,t)is gained by solving eq. (2.40), and subsequently using eq. (2.11)
in its time-harmonic form. The solutions of eq. (2.40) have some more interesting and useful
properties. For example, the analogy to quantum mechanics, Schrödinger’s equation, which
allows to transfer some properties from quantum mechanics directly to classical electrody-
namics (find a comparison in Tab. 2.1), like symmetries and orthogonality of the solutions of
eq. (2.40). The latter one is the consequence of the hermitian property of ˆ
ΘH. For two different
solutions labeled with 1and 2, one gets
0 = (ω2
1−ω2
2)(H2(r),H2(r)) ,(2.42)
where (·,·)denotes the inner product as defined in a Hilbert-space. So either the inner prod-
uct has to be zero, then the two solutions are orthogonal, or the eigenvalues have to be the
same, then the two solutions are degenerate. In real systems, degeneracy normally is lifted
due to e.g. broken symmetries.
Due to the astonishing similarities between the electronic and electromagnetic eigenvalue prob-
lem, methods known from electronic system can be used for the electromagnetic problem, too,
like a tight-binding formalism [79, 80]. This answers directly the question for the origin of the
electromagnetic band gap.
Another way to understand the origin of the band gap is to locate the points of high energy of
the harmonic modes H(r)·e−iωt[62]. A variational principle ansatz shows, that the harmonic
mode with the lowest frequency in the band gap concentrates its energy density mostly in the
material with the highest permittivity (often called a dielectric band), while others locate less
energy density in these regions, but in the material with lower permittivity (often called air
band). This is why the modes (band) at the low side of the first band are called dielectric
modes (band), while the modes locates above the band gap are called air modes (band). A
collection of different band structures is depicted in Sec. 2.4.2 for two- and three-dimensional
systems.
Depending on the application, for which a photonic band gap material is designed, one wants
to modify the size of the band gap and/or the spectral position of the band gap. This can
be done by various modifications, for example change of the periodicity, modification of the
permittivities, combine more than two materials with different permittivities, etc. [62]. Figure
2.3. PHC 36
Figure 2.6: Band gap size vs. hole diameter for different lattice constants a(see right inset) in a
triangular lattice of air holes in a dielectric slab (!= 11.56) of fixed thickness (t= 240 nm). Data are
extracted from MPB [82], the upper lines denote the upper band edge of the 1st photonic band, the
lower lines denote the lower band edge of the 2nd photonic band [81].
2.6 shows the variation of the band gap size with the hole diameter dof the air holes in a
triangular lattice in a dielectric slab with != 11.56 of fixed thickness (t= 240 nm) and
different lattice constants a[81]. The wavelength in nm is given on the vertical axis, the hole
diameter is on the horizontal axis. Lines of the same color belong to a setup of the PhC with
the lattice constant shown in the inset. Upper and lower lines are extracted from numerical
data, calculated with MPB [82], and denote the band edge of the 1st photonic band and the
low band edge of the 2nd photonic band. One can see, that the width of the photonic band
gap changes with changing lattice constant. To be more precise, the band gap increases with
increasing lattice constant. The hole diameter also affects the band gap size. A maximum of
the gap size emerges for every lattice constant, close to the maximal size of the hole diameter
for a fixed lattice constant (compare the gap size for a= 280 nm (black) and a= 380 nm
(purple) for large hole diameters). So the width of the gap size is primary controlled by the
hole diameter, while the spectral position is controlled primary by the lattice constant.
Last but not least one has to mention the photonic density of states (PhDoS), which gives the
number of allowed photonic modes per frequency interval δω. Inside a PBG (yellow region in
Fig. 2.7), the density of states is zero, because no propagating states are allowed within this
frequency range. Photonic states which would be in the PBG region are pushed away and are
localized above and below the PBG (blue areas in Fig. 2.7). But modes with an imaginary
wave vector are allowed in the PBG region, since these modes are not propagating modes, but
evanescent ones. Localization of evanescent modes in the PBG can be done with a defect in
the periodicity of the crystal, as is depicted as the red line in Fig. 2.7. Section 2.4.2 shows
how one can make use of the photonic band gap and the enlarged PhDoS, to store light in a
small volume (comparable to the cubic material wavelength of the confined photons) over a
long time, when introducing a defect in the PhC, a PhC cavity (PhCC).
37 2.4. OPTICAL RESONATORS
Figure 2.7: Photonic density of states (PDoS) vs. frequency in arbitrary units. The blue region shows
propagating modes with a real wave vector k. Near the photonic band gaps (yellow parts), the photonic
density of states raises, since the states from the band are pushed outside, hence increasing the PDoS.
Here, only modes with an imaginary wave vector are allowed, since they represent evanescent modes,
which only decay and show no propagate. In these regions, one can confine the light introducing a
defect, like a point or line defect, thus increasing the density of states for a certain frequency matching
the resonance of the defect. After [62].
Nowadays, this concept has already been applied to various kinds of PhCs. Just to mention
some, examples are the two-dimensional dielectric slabs with periodic arrangements of air
holes exhibiting a hexagonal symmetry, the three-dimensional Yablonovite, a woodpile stack
of layers of parallel oriented dielectric columns, alternating in height in a 90◦angle or the
inverse opal structure.
2.4 Nanoscale Optical Resonators
There are various kinds of optical resonators. The focus is on microcavities, so this thesis
concentrates on optical resonators in the (sub-) micrometer range. In general, there are two
different kinds of optical microresonators: first, a material without a PBG like a micro-toroid
or a microdisk (Sec. 2.4.1), purely relying on the principle total internal reflection (TIR), and
second, a PBG material (Sec. 2.4.2) with a defect to confine the light, like a PHCC.
2.4.1 The Microdisk Resonator
The MD resonator is a good example for a resonant structure without a PBG, because pe-
riodicity is not provided. As will be seen in this chapter, the electromagnetic problem of a
(perfectly) cylindrical dielectric object can be solved analytically, if one omits the post on
which the MD is standing to separate the dielectric layer forming the disk from the substrate
material. As will be seen later, the post will not affect the field distribution in the microdisk.
The ratio between thickness of the microdisks membrane and radius can thereby vary over
some orders of magnitude [26, 83].
Regarding Maxwell’s equations in their time harmonic form, utilizing the curl equations and
2.4. OPTICAL RESONATORS 38
assume, that no free charges or currents are present (j(r,t)=ρ(r,t)≡0) , one can rewrite
eqs. (2.1) and (2.2) with usage of (2.9) to
∇×E(r,ω)=−iω
!0c2
0
H(r)·e−iωt(2.43)
∇×H(r,ω)=−iω!0!r(r)E(r)·e−iωt.(2.44)
Decoupling (2.43) and (2.44) results in wave equations for Eand H
∇×∇×E(r)=!r(r)"ω
c0#2
E(r),(2.45)
∇×!r(r)−1∇×H(r)="ω
c0#2
H(r),(2.46)
which can be used to derive the Helmholtz-equation, exemplarily for the electric field E(in
areas of constant !r(r)) using standard vector calculus:
((+n2k2)E(r) = 0 .(2.47)
The dispersion relation k2=&ω
c'2, is used here, kis the wave vector, (=0
µ=x,y,z
∂2
µdenotes
the Laplacian operator and it is assumed, that !r(r)=n2, since the focus lies on investigating
pure dielectric materials without resonances.
Without loss of generality, one now assumes an infinitely extended dielectric slab in the x−y-
plane, whose permittivity is piecewise constant at the interfaces, a finite thickness tin z-
direction and propagating fields in the x-direction. The thickness is an important quantity
if one is interested in single-mode operating devices, whose basis is a planar waveguiding
system. Total internal reflection keeps the light within the thin dielectric slab. Using the fields
propagating in x-direction (with wave vector kx) and eq. (2.47), one gets for the TE-case
(Ey=˜
Ey(z)·eikxx)
∂2
z+-!r"ω
c0#2
−k2
x.
456 7
(∗)
˜
Ey=0.(2.48)
For the permittivity, !r=!for |z|<t
2and !r=1for |z|>t
2applies. So for both spatial
domains, one can rewrite the (∗)-marked part in eq. (2.48) into
(∗)=κ2=!"ω
c0#2
−k2
x,for |z|<t
2(2.49)
(∗) = (iγ)2=i2-k2
x−"ω
c0#2.for |z|>t
2.(2.50)
Since this problem has spatial symmetry in z-direction, symmetric as well as antisymmetric
solution can be derived. To obtain exponentially decaying solutions outside and standing
waves inside the slab, κand γneed to be larger than 0. Using the boundary conditions at
39 2.4. OPTICAL RESONATORS
Figure 2.8: Graphical solution of eqs. (2.51) and (2.52). The intersections show the valid values
for κ, hence for γ. The RHS is shown in red. The used parameters are: != 11.56,λ0= 1000 nm,
t= 250 nm (green), 500 nm (blue), 1000 nm (magenta). For increasing thickness, more modes (higher
harmonics)are occupied.
the interface of the two different dielectric media (the field and its derivative with respect to
space have to be continuous), one derives the transcendental equations in order to calculate
the wave vector
tan ξ=γd
2ξ,for even modes and (2.51)
cot ξ=−γd
2ξ,for odd modes .(2.52)
where ξ=κd
2. Rewriting γd=d2"$ω
c0%2(!−1) −κ2#using some algebraic transformations,
the graphical solution for eqs. (2.51) and (2.52) result in Fig. (2.8), respectively, where every
intersection is a valid value for κand γfor the given system (the used parameters are given
in the caption).
For the in-plane solution one considers a dielectric cylinder with permittivity !r=!and radius
R, infinitively extended in z-direction, to calculate the field distribution, hence to understand
the in-plane confinement of electromagnetic radiation. Since the slab is assumed to be in the
x−y-plane, one uses now the Helmholtz equation (2.47) and transforms the problem into
cylindrical coordinates (x, y)−→ (r, φ)yielding
-1
r∂r(r∂r)+ 1
r2∂2
φ+!"ω
c0#2.Ey(r, φ) = 0 .(2.53)
Here, one uses a product ansatz for (E)y(r, φ)=R(r)·Φ(φ)=R(r)·e±iMφ(φ-harmonic with
azimuthal order M; the ±occurs because of double degeneracy for propagation clockwise
or counterclockwise) to separate the variables rand φ. Utilizing the product ansatz and
the common substitution to a unitless, effective radius ρ,r−→ c0
nωρ, one finally finds the
differential equation for the radial component of (E)yto be a Bessel differential equation
-∂2
ρ+1
ρ∂ρ+-1−"M
ρ#2..R(ρ) = 0 ,(2.54)
2.4. OPTICAL RESONATORS 40
Figure 2.9: Analytic solution of eq. (2.54) for the case of a metallic environment or different cases
of Nand M:N=1,M =2(top left), N=6,M =2(top right), N=1,M =6(bottom left),
N=2,M=6(bottom right). The radius of the disk is 1.
whose solutions are the Mth order Bessel function of first kind JM(ρ)inside the cylinder
(ρ⩽R) and a (complex) superposition of the Mth order Bessel functions of first kind JM(ρ)
and second kind YM(ρ), the Mth order Hankel-function HM(ρ)=JM(ρ)+iYM(ρ), outside
the cylinder (ρ> R). The radial mode order is coded in the effective radius ρ, where the
wave vector enters as kN=nωN
c0. The order N now depends on which solution one chooses
from the equation, resulting from the continuity at the interface between the disk and the
environment. Figure 2.9 shows the analytical field distributions for a cylinder with metallic
environment ((E)y≡0outside) and different combinations of M, N. As can be seen, the field
is mostly concentrated in the rim of the cylinder (except for large radial orders and / or small
azimuthal mode orders). That motivates the assumption that the post, on which the microdisk
is standing freely, has (almost) no influence on the field distribution.
Figure 2.10 shows the absolute value of the field (in the disk plane, normalized to 1) for the
fundamental modes (radial mode order N=1) in a sub-µmicrodisk resonator in a metallic
41 2.4. OPTICAL RESONATORS
Figure 2.10: Numerical solution, calculated with a two-dimensional FDTD code (implemented in
MATLAB [51]) on a cartesian grid (cell size ∆x=∆y= 10 nm), for a dielectric microdisk (dielectric
permittivity != 11.56 and radius R= 361 nm) in a metallic environment (the red line denotes the
interface between dielectric disk and the PEC) for the first fundamental (radial mode order N=1)
modes: 1⩽M⩽3in the top line and 4⩽M⩽6in the bottom line (from left to right). The intensity
is color-coded (red: 1, black: 0).
environment (the red circle denotes the boundary of the microdisk), calculated with the two-
dimensional FDTD code implemented in MATLAB [51]. A cartesian grid with a spatial cell
size of ∆x=∆y= 10 nm is used. The permittivity and the radius of the disk are chosen to
be != 11.56 and R= 361 nm, respectively. In the top line, the first 3 modes (azimuthal mode
orders 1⩽M⩽3) are depicted. Note, that the absolute value is plotted here, therefore the
double amount of maxima are visible, compared to the azimuthal mode order. The subsequent
azimuthal mode orders 4⩽M⩽6are shown in the bottom line.
2.4.2 The Photonic Crystal Cavity
In a Photonic Crystal Cavity, PhCC, one makes use of the increased density of states around
the PBG. Evanescent modes, so harmonic modes with a purely complex wave vector, are
allowed to penetrate the PBG region. In a defect, like a missing hole in a periodic arrangement
of air holes in a two-dimensional dielectric slab, this evanescent mode can manifest itself and
become a highly confined mode, a cavity mode. Cavity modes in general reflect the symmetry
of the PhC, for example a rotational symmetry of the lattice leads to degenerate modes, say
these modes have the same frequencies, but are not necessarily orthogonal. This degeneracy
can be lifted by destroying a symmetry property, for example the air holes are not perfectly
circular shaped, but more of elliptical shape [2]. Then, the lifted degeneracy manifests itself
2.4. OPTICAL RESONATORS 42
Figure 2.11: λ/2-defect in the center of a distributed Bragg reflector (material 1 (green): low ref.
index, material 2 (blue): high ref. index). The layer thickness of the DBR is determined via di=λ/4ni,
i=1,2and the electric field (red) is concentrated in the defect area. After [62].
in polarization properties of the harmonic modes. The degenerate spectral line splits into two
polarization-dependent modes.
The most simple model of a PhCC is a cavity in a one-dimensional DBR (Fig. 2.11). Here,
the periodicity is broken with a λ/2layer (center) in the low dielectric medium instead of the
λ/4thickness (λis the wavelength in the material), acting as a defect which is able to confine
light (as described in Sec. 2.3).
As a two-dimensional example for a PhCC, the H-, Land hybrid-type cavities are mentioned.
The basis for all kinds can be for example a square, hexagonal or honey-comb lattice of circular
air holes in a dielectric slab, extended infinitively perpendicular to the whole plane. But one
can also imagine a lattice of square holes in a certain lattice. As long as the periodicity is
provided, all kinds can be combined. In these systems, confinement is based on fulfilling the
Bragg condition of reflection, like in the DBR case, but here in all directions. Figure 2.12
(top) shows the band structures for two different kinds of lattices [62]. In a two-dimensional
simulation case, there is always a TE and a TM solution. Here, the square lattice of dielectric
rods in air show a TM gap, while the air holes in a dielectric slab show a TE gap [62]. Also,
simultaneous occurrence (meaning in the same frequency range) of both gaps can be realized,
too, when using a hexagonal lattice of air holes in a dielectric slab.
In reality, one always deals with a three-dimensional device. The confinement in the third
dimension is provided by total internal reflection (TIR). The thickness tof the device is
normally chosen to be in the range of t≈λ
2,λis the vacuum wavelength, divided by the
refractive index of the desired material, to have only the lowest optical mode in this direction
occupied (single mode operation, see Sec. 2.4.1). Thus, all components mix and one cannot
uncouple the different polarization. Hence, only TE- and TM-like band gaps occur, see Fig.
2.12, bottom. Since hexagonal lattices provide larger band gaps, they are often preferred and
for the rest of this thesis, the hexagonal lattice is taken into account for PhCs. A defect is now
introduced by leaving away an air hole in a hole-array, or a rod in a rod-array. Typically, in the
H-case, one leaves for example a single air hole , yielding a H1-cavity or every next neighbors
of this single defect to get a H2-cavity. The light is confined in the defect region, where it can
be concentrated in the material part, referring to modes located energetically near the material
band (band below gap) or in the air part, referring to modes located energetically close to the
air band (band above gap) [62]. Its evanescent parts then leak out into the non-cavity region,
where the Bragg condition of reflection is fulfilled, hence they are reflected back. Exemplarily,
mode patterns for material modes for a H1-, H2-cavity and a L3-cavity, meaning leaving
away a line of three air holes in a hexagonal lattice, in the x-y-plane are shown in Fig. 2.13
43 2.4. OPTICAL RESONATORS
Figure 2.12: Band structures for a two- and three-dimensional PhCC with a square lattice of dielectric
rods in air (left) and with a hexagonal lattice of air hole in a dielectric slab (right). The insets show the
assumed real-space geometry as well as the irreducible Brillioun zone (blue) in the reciprocal lattice.
Points of high symmetry are marked. In the band structure (frequencies, normalized to the lattice
constant, over the wave vector). Top: No photonic band gap occurs in left geometry, while for the
right, a full band gap (TE- and TM-band at the same frequencies). Bottom: For the left geometry, a
TM-like band gap occurs, while the right geometry provides a TE-like band gap. After [62].
(from left to right). The black thin lines represent the air hole lattice in the dielectric slab
and color-coded one finds the amplitude of the y-component of the electric field. The largest
contributions are found to be inside the defect, hence in the material. The used parameters
for the simulations can be found in [1–3]. The L3-cavity will gain more interest in the later
Chaps. 5 and 6, since tuning mechanisms are present to enhance the life time of the mode.
Strong cavity-cavity interaction (strong interaction means, the line splitting exceeds the line
width) between resonant and detuned L3-cavities is investigated in Chap. 6.
2.5. QD MODEL 44
Figure 2.13: Mode patterns for the H1-, H2- and L3-type cavity (from left to right). In each plot,
the plane shows the x-y-plane and color-coded the amplitude of the electric field (y-component). Used
Parameters are taken from [1–3], including modifications of the first outer air holes for the L3-type
defect (see Sec. 5.2 for details).
2.5 The Quantum Dot Model
A QD can be considered as a two-level system (TLS) like an atom with only two energetically
allowed levels: a ground state and an excited state. The excited state normally is due to an
exciton resonance. An exciton, in fact, is a bound electron-hole pair, which can be excited
optically. The electron from the valence band is excited into the conduction band via the inci-
dent optical field and a hole is left where the electron was before. Now, since the two particles
have opposite charge (and momentum), the Coulomb interaction bounds them together. Like
in an atom, a Bohr-radius a0Xis defined by a0X=4π&0&r!2
µe2,eis the elementary charge (defined
positive). The excited state will be denoted with |1*while the ground state will be denoted
by |0*. The main reason for this notation is the possible usage of quantum dot exciton states
for quantum information, where the logical bits 0and 1are mapped to the quantum-logical
qubits |0*and |1*on the Bloch sphere [61] (the quantum-logical states |0*and |1*are referred
as the poles of the sphere).
For the derivation of the equations of motions for the microscopic polarization and the conduc-
tion band density of the QD, the formalism of second quantization is used [66, 68]. Therefore,
fermionic ladder operators (creation operator a†
µ, annihilation operator aµ) are defined. The
subscript µlabels the bands. Since no dispersion of the bands or spin for the electrons is
assumed, labels for the wave vector or spin are omitted. A coherence, or microscopic polar-
ization, between the ground and excited state is then given by a†
vbacb, where an electron in
the valence band is destroyed and and electron is created in the conduction band at the same
time. The two bands are coupled via a dipole matrix element dµµ!. Generally, for a n-level
resonator with normalized wave functions Ψµfor the carriers, dµµ!is defined as
dµµ!=−e;
R3
drΨ∗
µ(r)rΨµ+(r).(2.55)
where ris the spatial coordinate and ∗means complex conjugation.
To calculate the time dynamics of such a quantum mechanical system, one needs the Hamilton
operator ˆ
Htot, which lets the system evolve in time according to the Heisenberg equation of
45 2.5. QD MODEL
motion of a time-dependent operator ˆ
A
d
dtA=−1
i![ˆ
Htot,ˆ
A]−+∂tˆ
A, (2.56)
where [A, B]−=AB −BA is the commutator. The last term vanishes for not explicitly time-
depending operators (which is the case for the coherence and density). For the QD system,
the Hamilton operator with only the free part and light-matter interaction is given by
ˆ
Htot =ˆ
Hfree +ˆ
Hlight-matter
=674 5
!+
µ
ωµa†
µaµ+674 5
i
!E(t)·+
µµ$
dµµ$a†
µaµ$.(2.57)
E(t)is the exciting light field. For the coherence pand conduction band density none gets
with eq. (2.57)
˙p=iωgapp+iΩR(1 −2n)−γpp(2.58)
˙n=−2,(ΩRp∗)−γnnwith (2.59)
with Θ(t)=
t
;
−∞
dt$ΩR(t$)and ΩR(t$)=dµµ$·E(t$)
!.(2.60)
The quantity Θ(t$)is the pulse area at a time t$,ΩRis the Rabi frequency, Eis the envelope
function of the exciting electric field (defined via E=E·cos(ωt+φ),φis the phase), ωgap
is the gap frequency of the QD material, γpand γnare the phenomenologically introduced
dephasing rates for the coherence pand the conduction band density n, respectively. In the
calculations, γn=2γp(coherent limit).
2.5.1 Quantum Dots in the FDTD: Back-Coupling to the Electromagnetic
Field and Limits
A QD provides a microscopic polarization after optical or electronic excitation. This micro-
scopic polarization couples back to the surrounding light field via the macroscopic polarization,
which enters Maxwell’s equation (2.11) via an additional term in the dielectric displacement
field D(r,t)=!0!r(r)E(r,t)+P(r,t):
∇×H(r,t)=µ0µr(r)j(r,t)+µr(r)
c2
0
˙
D(r,t)(2.61)
=µ0µr(r)j(r,t)+µr(r)
c2
0
(!r(r)˙
E(r,t)+ ˙
P(r,t)) (2.62)
where the macroscopic polarization P(r,t)is calculated by the sum of all microscopic polar-
izations, say coherences, of the QDs, weighted with the corresponding dipole matrix element:
P(r,t)=+
µµ!
dµµ!-a†
µaµ!*Ψ∗
µ(r)Ψµ!(r).(2.63)
This macroscopic polarization feeds the light field which itself again excites the QD.
2.5. QD MODEL 46
-0.03105
-0.031
-0.03095
-0.0309
-0.03085
494.85 494.875 494.9 494.925 494.95 494.975 495
coherence
time [fs]
no
linear
Lagrange
Maxwell
Figure 2.14: Comparison of different extrapolation schemes for calculating future values for the
electric field for the 4th Runge-Kutta integrator (Sec. A.1, [84]). Coherence of the QD vs. time on
a very short time window in fs. Red: no extrapolation, En+1
2=En+1. Green: linear extrapolation.
Blue: Lagrange extrapolation of 3rd order. Magenta: Extrapolation via FDTD update with Maxwell’s
equations. The Lagrange and Maxwell extrapolation scheme agree nicely, while no and only linear
extrapolation show comparably large deviations.
Since the QD is embedded in the spatial and temporal FDTD grid, the stepping for the
update equation needs to be matched. The spatial stepping is done straight forward. For
the temporal stepping an extrapolation method needs to be used. Exemplarily, for a one-
dimensional case, the modified update equation for E(r,t)|iat Yee-cell iand time (n+1
2)∆t,
taking the macroscopic polarization P(r,t)|ifrom the QD into account, is
En+1
i=En
i−∆t
!0!i∆x"Hn+1
2
i+1
2−Hn+1
2
i−1
2#−&Pn+1
i−Pn
i'.(2.64)
As can be seen from eq. (2.64), the values for the polarization at t=(n+1)∆tare necessary to
update the electric field, because both quantities are defined on the same temporal grid. Using
a standard 4th order Runge-Kutta integrator (see Sec. A.1) [84] to integrate the equations of
motion for the QD, the electric field at times t=(n+1
2)∆tand t=(n+ 1)∆tare needed,
but not provided by the FDTD method. The missing values can be obtained by standard
extrapolation methods, like linear extrapolation or nth polynomial Lagrange extrapolation,
shortly depicted in Sec. A.2 [84]. In the used code, a field update utilizing the FDTD update
scheme is performed to calculate the electric field at the unknown time steps. A comparison of
the performance of different extrapolation schemes is shown in Fig. 2.14. The plot shows the
coherence of the QD on a very short time scale only covering some time steps, but nicely shows
the differences between the schemes. No extrapolation (red), meaning a constant electric field
is used, and linear extrapolation (green) show comparably strong deviations from the more
accurately working 3rd order Lagrange extrapolation scheme (blue). The extrapolation scheme
taken from the FDTD update (magenta) is consistent with the Lagrange scheme.
As described in the Sec. 2.5.2, the coherence and conduction band density have to be damped
to describe a physically senseful picture. In the presence of the fields, occupying the spatial
FDTD grid, the QD starts to decay radiatively once excited. The decay constant can be
calculated analytically [85], assuming an infinitely thin QW structure and linear excitation,
47 2.5. QD MODEL
0
10
20
30
40
50
60
0 0.25 0.5 0.75
relaxation time [ns]
distance [λ0]
0
0.1
0.2
0.3
0.4
0.5
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5
dephasing [GHz]
distance [λ0]
Figure 2.15: Left: Relaxation time in ns vs. QD distance in units of the resonant wavelength λ0.
Green: QW limit, calculated via (2.65). Red: numerical data. Clear deviations from the QW-limit can
be seen with the bare eye to emerge at QD distances larger than 0.3λ0. Right: Radiative dephasing
in GHz of the QD array vs. QD distance in λ0(red). Clear resonance features are visible due to the
radiative coupling between the QDs. The non-integer resonances emerge from coupling in diagonal
directions (√3,√5...). Green: QW limit [85]. Blue: analytical value for an isolated QD [86].
yielding
ΓQW
rad =ωd2
VC
2!0c0!a2
0x
,(2.65)
where, ωis the angular frequency. This result is valid for a dense grid of QDs in two dimen-
sions, simulating a quantum well like structure. If a delta-like QD is assumed, the radiative
decay constant can be derived analytically [86]. There, Maxwell equations for the full electric
field are solved in the reciprocal space with a Greens tensor approach and using the Lippmann-
Schwinger equation to determine the frequency-dependent electric field in real space. Calcu-
lating the averaged electric field at the QD position and inserting it into the macroscopic
polarization with a renormalized susceptibility (renormalized to the exciting electric field at
the QD position), the radiative decay constant of the quantum dot yields
ΓQD
rad =2ω3d2
VC
3c3
0
.(2.66)
Figure 2.15 depicts the numerical performance of a QD array in the three-dimensional FDTD
grid, depending on the distance between the QDs. In x-direction, the computational domain
is set to 800 nm with CMPL boundary (10 cells). Periodic boundary condition are applied in
y- and z-direction and the edge length of the computational domain was gradually increased
in the same manner (horizontal axis, in units of the (vacuum) emission wave length λ0of the
quantum dot) with the QD placed in the center to get an infinitively extended square lattice of
QDs. On the left side, the relaxation time of the QD array in ns is depicted over the resonant
wavelength λ0. Green shows the QW limit, see eq. (2.65), while red shows the numerical data.
One can clearly see the deviation from the QW limit, starting above 0.3λ0. On the right side,
the dephasing in GHz is plotted vs. the distance in λ0. The blue line shows the analytical
value for an isolated QD, in green the QW limit is shown and red shows the numerical data,
extracted from the QD. At all integer distances, clear dips and resonant features occur. The
2.5. QD MODEL 48
radiative coupling (constructive interference of the emitted spherical waves) between the QDs
suppress the emission, hence the dephasing drops. The resonant features emerging at non-
integer distances arise from the radiative coupling between QDs which are not next-neighbored
[85].
2.5.2 Nonlinearities in Quantum Dots: Rabi Oscillations
Eqs. (2.58)-(2.59) are also called the Optical Bloch Equations. The Bloch equations are often
discussed in a somehow more intuitive and graphical way: as a dynamic three-dimensional
vector sin the Bloch sphere. As components of s, the inversion of the QD −w=1−2nand
absorptive part u(dispersive part v) of the coherence p=u−iv appear:
s=
u
v
w
=
u
v
2n−1
.(2.67)
As can be seen, |s|=1for all possible u, v , w. So the possible orbit for the Bloch vector is a
unit sphere, on which it rotates with angular velocity ωaround the w-axis (can be eliminated
with the rotating wave approximation, RWA).
Under certain assumptions, the coupled eqs. (2.58) and (2.59) can be solved analytically.
First assumption is, that one can write eqs. (2.58) and (2.59) in RWA, eliminating the fast
oscillations of the exciting laser field and the coherence and just leaves the slowly varying
envelopes (denoted with a bar). This yields a new set of differential equations:
˙
p=iΩR(1 −2n)(2.68)
˙
(1 −2n)=−2,&ΩRp∗'.(2.69)
The Rabi frequency ΩRis defined as ΩRin Sec. 2.5 (since there is used already the envelope of
the field). As a second condition for solvability, the phenomenologically introduced dephasing
rates are set to zero and ΩR∈R(for resonant excitation). This set of equations is now solvable
analytically with
p=i
2sin Θ(t)
2,1−2n= cos Θ(t)
2(2.70)
resulting in
n= sin2Θ(t)
2.(2.71)
If the pulse area Θ(t→∞) = 2 mπ, the system performs mfull flops of the upper density,
means inverting the QD and destroying occupancy again mtimes. In the picture of the Bloch
sphere this means, that the Bloch vector srotates mtimes on the edge of the sphere and
returns to his originating position where it started.
49 2.5. QD MODEL
0
0.25
0.5
0.75
1
0 100 200 300 400 500
coherence / occupation
without phenomenological dephasing
time [fs]
Θ=1π
0
0.25
0.5
0.75
1
0 100 200 300 400 500
coherence / occupation
with phenomenological dephasing
time [fs]
Θ=1π
0
0.25
0.5
0.75
1
coherence / occupation
Θ=2π
0
0.25
0.5
0.75
1
coherence / occupation
Θ=2π
0
0.25
0.5
0.75
1
coherence / occupation
Θ=4π
0
0.25
0.5
0.75
1
coherence / occupation
Θ=4π
0
0.25
0.5
0.75
1
0 100 200 300 400 500
coherence / occupation
time [fs]
Θ=10π
0 100 200 300 400 500
0
0.25
0.5
0.75
1
coherence / occupation
time [fs]
Θ=10π
Figure 2.16: Rabi Oscillations in a TLS, calculated with the FDTD method and self-consistent
backcoupling. From top to bottom: densities of conduction (red) band, valence band (green), coherence
between conduction and valence band (blue) and the exciting pulse (magenta). Parameters for the TLS
are: dipole matrix element dcv =3e˚
A, gap energy εgap =1.5eV, phenomenological dephasing γ= 10
THz (right). Radius of Gaussian wave function for the carriers r= 30 nm. Parameters for the exciting
pulse are: center time t0= 250 fs, pulse width τ= 50 fs, energy ε=1.5eV. The slight oscillations,
covering the whole dynamics of pand n, are a no-RWA effect.
2.6. COUPLING REGIMES 50
Introducing a phenomenological dephasing rate γaffects the Rabi oscillations. The flops of
the conduction band density is not complete and after the exciting light pulse vanishes, the n
stays at a finite value. In the Bloch sphere picture, swould remain somewhere on the Bloch
sphere and not return to his starting pole. Dephasing always occurs in such systems, because
the QDs are never isolated, but coupled to e.g. phonons of the substrate. As discussed in the
last Section 2.5.1, a QD will decay radiatively in the FDTD without a phenomenologically
introduced dephasing. Hence, a physical picture is obtained. Figure 2.16 depicts the dynamics
of eqs. (2.58) and (2.59), calculated in a three-dimensional simulation. Red (green) show the
CB density (VB density), blue the coherence between VB and CB and the envelope of the
exciting light field is shown in magenta to show the temporal presence of the field. The left
row is calculated without a phenomenological damping for the QD, while the right row has a
damping rate of γ= 10 THz, which is chosen to be quite large to show the effect on a short
time scale. The inset denotes the pulse area, accordingly, the system performs the expected
number of density flops of the QD in the undamped case. In the damped case, the density
flopping is suppressed.
2.6 Coupling of Oscillators
Considering a system of two oscillators in the states |Ψ1*and |Ψ2*, coupled with the coupling
constant Jand eigenenergies ε1and ε2, the result is a beating between those two oscillators
when the eigenfrequencies are equal. The problem can be formulated as
"ε1J
Jε2#"Ψ1
Ψ2#="ε+J0
0ε−J#"Ψ+
Ψ−#(2.72)
for ε1=ε2=ε. Diagonalizing the matrix, thus uncoupling the oscillations into two orthogonal
normal oscillations Ψ+and Ψ−, yields an energy splitting ∆, depending on the coupling
strength, ∆=2 J.
In quantum optics, the Jaynes-Cummings ladder (JC ladder) describes the interaction between
a single mode field with frequency in a cavity ω0with an embedded TLS with frequency ω21
between upper (|2*) and lower level (|1*) [87]. The JC Hamilton operator reads
ˆ
HJC =!ω0c†c
456 7
field
+!ω21
2σz
456 7
TLS
+!(gcσ++g∗c†σ−)
456 7
interaction
,(2.73)
where c†(c) are the bosonic creation (annihilation) operators for the single mode field,σzis the
occupancy difference of the two levels |1*and |2*and σ+(σ−) are the creation (annihilation)
operators of the TLS. The interaction constant is denoted by g. Neglecting the interaction part
of eq. (2.73) and assuming that ω0=ω21 =ωand nphotons in the system, the eigenenergies
of the two possible configurations of the system are degenerate: the state with n−1photons
in the system and an excited TLS has the same energy as a system with nphotons and a TLS
in the ground state. Switching on the interaction part between, TLS and single mode field,
this degeneracy is lifted, resulting in the new eigenenergies
εn,±="n+1
2#!ω±1
2!Ωn,Ωn=2
√ng . (2.74)
51 2.6. COUPLING REGIMES
For n≡1, this is called Vacuum Rabi Splitting (VRS). With the semi-classical FDTD ap-
proach, where the field is treated classically and a TLS is treated quantum-mechanically, the
normal mode splitting, which equals the line splitting for the limit n=1in the quantum-
optical picture, can be calculated, too (see Sec. 6.2).
A detuning ∆between the single mode frequency ω0and the TLS frequency ω21 enters eq.
(2.74) in Ωn=(4ng2+∆2. Now, two important cases are distinguished:
First, the weak coupling regime, or Stark regime. Here, 2√n|g|0|∆|. That means, that even
with a strong field, hence large number of photons, the coupling regime can still be weak.
Rewriting and tailoring Ωnyields
Ωn=|∆|<4ng2
∆2+1≈|∆|(1 + 4ng2
∆2),(2.75)
one can calculate the new eigenenergies of the coupled system to be
εn,±="n+1
2#!ω±1
2!ω21 +!ng2
|∆|,(2.76)
which shows a red shift for both states, called the 2nd order Stark effect, since eq. (2.75) is
quadratic in the electric field.
Second, the strong coupling regime. Here, 2√n|g|1|∆|is fulfilled. Neglecting the detuning
in eq. (2.74), the new eigenenergies in the coupled system read
εn,±="n+1
2#!ω±!√n|g|.(2.77)
So the new eigenstates, the dressed states, are first non-degenerate due to the light -matter
coupling, and second are split by ∆ε=2
√n|g|, which means, that the coupling can be
enhanced via the photon number n. This splitting is linear in the field, hence referred to the
linear or dynamical Stark effect [88].
Above the strong coupling regime, there also exists a ultrastrong coupling regime. This regime
is reached, when the splitting ∆ε≈!ω. In this regime, the RWA, often used in the weak and
strong coupling regime, breaks down. Also, additional terms in the Hamiltonian have to be
considered, which are proportional to (a†+a)2, the creation and annihilation operators for
the TLS, yielding a violation in energy conservation. Since this topic is not discussed in this
thesis, the interested reader is referred to [89–91].
As an example of coupling between two photonic resonators, Fig. 2.17 shows coupled dielec-
tric cavities, sandwiched between DBRs, the corresponding field patterns of the electric field
(normalized to 1) and the spectral response (transmission spectrum) of the coupled systems
[93] (calculated with the FDTD method [94], taken from [92]). At the top, as a reference,
a single cavity surrounded by 15 layers of alternating lambda-quarter plates with different
permittivities is shown. The left side shows the spectral response, the right side shows the
spatiotemporal response in a short time window. The cavity length corresponds to a lambda-
half plate for 1.5eV (λ0≈826 nm). Deviations from the exact value of 1.5eV originate
from rounding errors of the layer thicknesses due to spatial discretization, since Yee-cells are
filled completely with a permittivity. The incident wave travels from left to right (on the left
side the reflections from the first DBR layers are clearly visible). The middle part of Fig.
2.17 shows two coupled cavities. The eigenstates of the coupled system can be regarded as
in atomic physics, like a bonding and an anti-bonding state, where the bonding state has a
2.6. COUPLING REGIMES 52
0
0.25
0.5
0.75
1
1.2 1.3 1.4 1.5 1.6 1.7 1.8
Transmission / Einc(ω)
Energy [eV]
0
0.25
0.5
0.75
1
1.2 1.3 1.4 1.5 1.6 1.7 1.8
Transmission / Einc(ω)
Energy [eV]
AB
0
0.25
0.5
0.75
1
1.2 1.3 1.4 1.5 1.6 1.7 1.8
Transmission / Einc(ω)
Energy [eV]
A
B
C
Figure 2.17: Coupling between two and three cavities. Left: Spectral response (transmission). Red
shows the spectral width of the incident pulse. Right: Spatiotemporal response. The DBR structure
is shown as alternating red/white bars (red: !r= 13, white !r=8.7). Top: Reference spectrum from
a single cavity (resonance at ε=1.5eV). Middle: 2 coupled (identical) cavities. The transmission
spectrum shows a clear line splitting (strong cavity-cavity interaction). Bottom: 3 coupled (identical)
cavities. Three distinct peaks appear in the transmission. The corresponding spatiotemporal responses
are labeled with capital letters. From [92].
decreased energy due to the binding energy, while the anti-bonding state has an increased
energy compared to the uncoupled state. This is reflected in the field patterns: the formation
of anti-symmetric (bonding) and symmetric (anti-bonding) modes, see labels A and B. At the
bottom side of Fig. 2.17 three coupled dielectric cavities with their corresponding spatiotem-
53 2.7. LC MODEL
poral response are depicted. The spectral response exhibits a third resonance, corresponding
to field pattern with zero amplitude in the central cavity (exactly in the center), while the
other two resonances correspond to the bonding and anti-bonding field profiles.
2.7 The Liquid Crystal Model: Uniaxial Anisotropy
Optical anisotropy is a common tool in current research (resonance tuning, transformation
optics, etc. [26, 27, 95–97]) and used already in a broad range of commercial applications (i.e.
LC-displays [98, 99]). LCs exhibit different phases for different environmental parameters,
like temperature, pressure or applied electric fields [26, 95] which can be applied. In the
scope of this thesis, LCs with the following two phases are taken into account: an isotropic
phase and an anisotropic phase. Phase transition between a liquid crystalline and an isotropic
phase of a liquid crystal can be observed even with the naked eye. Figure 2.18 [100] shows
the wavelength-dependent refractive indices (left) for a temperature of T= 25.1◦C. Open
and closed circles (extraordinary and ordinary refractive index, respectively) are experimental
data, the solid and dashed lines are results from different theories proposed in [100], which
agree both very well. For large wavelengths, both curves go into saturation. Therefore, for
the wavelength range used in Sec. 4.2.1, the same values can be used. In the middle of Fig.
2.18, the dependency of the refractive indices from the temperature are shown. Again, open
and closed circles (extraordinary and ordinary refractive index, respectively) are experimental
data for different wavelengths (triangle: 450 nm, circle: 550 nm, square: 650 nm), the solid
line is a result from theory, proposed in [100]. Again, both curves agree very well. Both
branches approach a saturation value for increasing wavelengths. The situation is summed up
in the right part of Fig. 2.18 [101]. The diagram shows the temperature-dependency of the
refractive indices for λ=4.45 µm. Therefore, extrapolation of the curves from the diagram on
the right from high wavelengths up the wavelength range of interest is satisfied. In the high
temperature regime, the LC behaves optically isotropic (refractive index n). Below the clearing
temperature TCis the nematic phase. The optical properties of liquid crystals depend strongly
on the confinement due to the anchoring conditions on surfaces and the form (geometry) of
the surrounding. In this theoretical work, the liquid crystal is treated as a medium, where
uniaxial birefringence can be induced by homogeneous electric fields. This model is somewhat
inspired by solid media like LiNbO3, but assumes a considerably higher electric response by
means of the absolute value of the induced birefringence. This simple approach leads to a
qualitative understanding of the optical phenomena, investigated in this thesis, although LC
filled specimen can have a highly complex spatial dependence of the refractive index tensor,
especially near walls in dielectric systems. Applying the electric field results in a splitting of
the isotropic curve into two branches, the ordinary noand the extraordinary branch neo. As
temperature decreases, the level of birefringence, say the difference between the ordinary and
the extraordinary refractive index, increases until a saturation regime is reached. At this point,
the maximal alignment of the LC molecules is reached.
Calculations performed within this thesis concerning LCs are using a uniaxial anisotropic
dielectric medium without subpixel averaging routine and without magnetic properties. Hence,
the dielectric permittivity is treated as a second rank tensor !with diagonal elements, while
magnetic permeability µis still treated as a scalar equal to 1. The off-diagonal elements
are responsible for the component mixture and have to be taken into account if one wants
to use biaxial anisotropic dielectric materials. Liquid crystal properties like molecule shape,
2.8. ANALYSIS TOOLS 54
Figure 2.18: Characteristics of the liquid crystal 5CB [100, 101]. Left: wavelength dependency of
the refractive indices (ordinary: close circles, extraordinary: open circles) at a constant temperature of
T= 25.1◦C. Solid and dashed lines are from different theoretical models in [100]. Middle: temperature
dependency of the refractive indices (ordinary: close symbols, extraordinary: open symbols) for different
wavelengths (triangle: 450 nm, circle: 550 nm, square: 650 nm). The solid lines are fitting curve
according to [100]. Right: Used temperature profile of the refractive indices [101]. For the isotropic
case below the clearing temperature TC≈35 ◦C, ¯nis the averaged refractive index in the nematic
phase without external field.
polarizability, special anchoring mechanisms of the molecules on a surface, loss mechanisms,
local orientation features like twist, splay or bend and coherence lengths of the LC molecules
are not taken into account.
2.8 Analysis Tools, Methods and Special Features
All the simulations shown in this thesis are performed with an in-house FDTD code, Maexle,
implemented in the Junior Research group Computational Nanophotonics.Maexle numerically
evaluates the three-dimensional Maxwell equations with the methods explained in Section 2.2.
For data analysis and post-processing of data, quite a lot of tools are available directly within
the program. Next to the standard discrete Fourier Transform (DFT), spectral analysis is also
possible with the Filter Diagonalisation Method (FDM) Harmonic Inversion [102–105] or the
Fast Fourier Transform (FFT) [106]. The first method relies on the fact, that one can transform
the spectral analysis of a time-evolving dataset f(t)of finite, exponentially decaying sum of
oscillations into a matrix equation, which can be diagonalized to efficiently extract spectral
properties from time data like resonant frequencies f, phases φand decay rates γ:
f(t)= +
k<∞
Ake−i(2πfk−iγk)t+φk,(2.78)
where Akis the (complex) amplitude of the oscillation with index k. In general, the computa-
tional performance of the Harmonic Inversion is essentially better than of the DFT, but one
has to be careful with large sets of oscillations. The great advantage of the FDM is, that the
time signal does not need to decay to zero to get a fully converged spectrum. The complexity
of the FDM, for Ntime points and Jfrequencies, i O(NJ +J3). Compared, the DFT uses
orthogonality relations of oscillations of different frequency. The complexity here is O(N2).
The latter method, the FFT, is a quite complicated algorithm and should be left for those
who invented it and work with it every day. But it should be mentioned, that compared again
with the DFT, the performance is greatly increased. The complexity here is O(Nlog2N).
55 2.8. ANALYSIS TOOLS
Additionally, a uniaxial anisotropic material is provided, which is used to model the LC ma-
terial described in the Section 2.7. Uniaxial anisotropy is easy to implement, because the field
components do not mix as they do when one wants to implement full anisotropy. This mixture
of field components results in needed fields at positions, where they are not defined. Also, the
subpixel averaging procedure described in Section 2.2.3 cannot be easily applied to both, the
uniaxial and full anisotropic materials.
2.8. ANALYSIS TOOLS 56
Chapter 3
Validation of Maexle
3.1. VALIDATION OF THE FDTD IMPLEMENTATION 58
3.1 Validation of the FDTD Implementation
Validation of the implementation is a crucial point before using a program for research. In
this case, the validation is done by comparing numerical results with results from four high-
impact publications with the PhCC topic. For this purpose, three-dimensional calculations are
performed, investigating the spectral and spatiospectral response of the two-dimensional H1-
type cavity from [1] and [2], and L3-type from PhCC [3] and [4]. The planar photonic structures
are located in the x-y-plane of the computational volume. Computational details about the
simulation volume and time are: Vsim = 19a×9.5√3a×2a,∆x=∆z=a
16,∆y=√3
2∆xand
tsim =5ps for all cases. These settings are sufficiently enough to gain reliable results from the
simulation. Also, the extension of the first photonic band gap of the referenced structures are
calculated with the MPB package [82]. For every investigated structure, the general form of
the graphics are the same: the spectral properties are collected in the first figure, containing
the band structure, spectral information from the publication and the spectrum calculated by
Maexle. The second (third) figure shows the spatiospectral response of the structure in the
TE-plane for the different resonances within the first band gap for x- and y-polarization.
3.1.1 H1-type PhCC
At first, the H1-type cavity from [1] is rebuild. A full investigation of the resonances located
within the first band gap can be found in [1] in dependence of the outward shift sof the outer
air holes away from the defect and modulating the radius ∆rof the holes. The parametric set
for the H1-cavity is a lattice constant of a= 350 nm, slab thickness t=0.71 ×a, air hole
radius r=0.31×aand a dielectric permittivity for a GaAs slab, !r= 11.56. Modifications, as
shown in the publication, are not applied. The top left of Fig. 3.1 shows the band structure,
the right side shows the experimental data from [1]. As can be seen, the 1st band gap extends
from roughly fmin = 220 THz to fmin = 290 THz and contains ten modes, depending on s
and ∆r. The focus for the validation purpose lies on the case for no modification, therefore
∆r=s=0. Calculations with MPB [82] confirm the spectral extension of the band gap
(middle). The figures below depict the spectral response for the x-component (middle) and
y-component (bottom) of the electric field, calculated with Maexle, in a broader range to also
show that outside of the band gap no other resonances occur. The intensity in arbitrary units
is plotted over the frequency (the upper horizontal axis shows in frequencies normalized to
the lattice constant for better comparison with the upper figures, the lower axes shows the
frequency in THz). The grey-shaded region (1st band gap) contains three resonances A, B
and C. Resonance A is identified with the dipole-like mode, as shown in [1]. Resonances B
and C are higher order modes, pushed into the band gap (grey-shaded region). Deviations in
the spatial grid and possible differences in the !-averaging process cause the differences to [1].
Nevertheless, the additionally visible modes are no artifacts, since depending on the radius-
to-lattice constant more modes move into the photonic band gap (see next paragraph). Figure
3.2 shows the spatiospectral response of the defect in-plane(rip) of the PhCC for the peaks
A, B and C. The electric field pattern Ex/y(rip,ω)(left), the absolute square |Ex/y(rip,ω)|2
(middle) and the absolute square of the spatial Fourier transform |Ex/y(kip,ω)|2(right), where
kip is the in-plane component of the wave vector. The grey lines in the field patterns show the
boundaries of the air-to-material interfaces. All patterns are normalized to 1. The electric field
concentrates in the defect region in different patterns, depending on symmetry and frequency.
59 3.1. VALIDATION OF THE FDTD IMPLEMENTATION
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
normalized frequency [a/λ]
wave vector k
ΓMK Γ
Figure 3.1: Spectral response for the H1PhCC, after [1]. Top left: Band structure of the PhC,
containing the first 5 photonic bands and the light cone (red line). Normalized frequencies are plotted
vs. the wave vector. High-symmetry points of the reciprocal space are marked. Note the 1st band gap
between the first two bands. Calculated with MPB [82]. Top right: data from [1]. The spectral region
around the 1st band gap allows 10 different modes. The focus lies on the modes for no modification
of the PhCC, thus ∆r=s=0. The normalized frequencies are plotted over the modification. Middle
and bottom: Spectral response (middle: x-component, bottom: y-component), calculated with Maexle.
Intensity in arbitrary units over the frequency in THz is shown (the top horizontal axes also provides
normalized frequencies for better comparison). The 1st band gap is marked with a grey shade. The
dipole-like resonance A is clearly identified. B and C are higher-order modes, pushed into the bandgap.
3.1. VALIDATION OF THE FDTD IMPLEMENTATION 60
Figure 3.2: Spatiospectral response, calculated with Maexle. From top to bottom: mode patterns
belonging to the peaks A-C in Fig. 3.1, middle and bottom. From left to right: electric field pattern
Eµ(rip,ω), absolute square |Eµ(rip,ω)|2and spatial Fourier transform |Eµ(kip,ω)|2in-plane of the slab
(µ=x, y).
61 3.1. VALIDATION OF THE FDTD IMPLEMENTATION
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
normalized frequency [a/λ]
wave vector k
ΓMK Γ
Figure 3.3: Spectral response for the H1PhCC, after [2]. Top left: Band structure of the PhC,
containing the first 5 photonic bands and the light cone (red line). Normalized frequencies are plotted
vs. the wave vector. High-symmetry points of the reciprocal space are marked. Note the 1st band gap
between the first two bands. Calculated with MPB [82]. Top right: Experimental and numerical data
from [2]. Two modes are shown in the 1st band gap (grey-shaded area). Normalized frequencies are
plotted vs. r/a. Middle and bottom: Spectral response (middle: x-component, bottom: y-component),
calculated with Maexle. Intensity in arbitrary units over the frequency in THz is shown (the top
horizontal axes also provides normalized frequencies for better comparison). The 1st band gap is
marked with a grey shade. The dipole-like resonances A (1st-order) and B (2nd-order) are clearly
identified.
3.1. VALIDATION OF THE FDTD IMPLEMENTATION 62
Figure 3.4: Spatiospectral response, calculated with Maexle. From top to bottom: mode patterns
belonging to the peaks A and B in Fig. 3.3, middle and bottom. From left to right: electric field
pattern Eµ(rip,ω), absolute square |Eµ(rip,ω)|2and spatial Fourier transform |Eµ(kip,ω)|2in-plane
of the slab (µ=x, y).
63 3.1. VALIDATION OF THE FDTD IMPLEMENTATION
Next, the H1-type cavity from [2] is calculated with Maexle. The PhCC parameters are a
lattice constant of a= 366 nm, slab thickness t=0.71 ×a, air hole radius r=0.32 ×a
and a dielectric permittivity for GaAs, !r= 11.56. A comparison of the numerical results
and the experimental data is depicted in Fig. 3.3. The only slightly different radius-to-lattice
constant ratio compared with the previous case give rise for a similar extension of the band
gap, as it is confirmed via a MPB calculation (left). On the right side data from [2] are shown.
Normalized frequencies are shown on the vertical axes over the radius-to-lattice constant ratio
r/a. The grey-shaded areas distinguish between the 1st band gap for quasi-TE-like modes
(thin boundaries) and quasi-TM-modes (thick boundaries). Half-filled diamonds are results
from the experiment for the 1st-order dipole mode, all other points are calculated. In the
insets the electric energy is depicted for the 1st- and 2nd-order dipole modes. Since the defect
structure is the same as in the prior case, and only the resonant frequencies are scaled with
the altered lattice constant and shifted due to a different r/a. The mode patterns are given
in Fig. 3.3 for the sake of completeness.
3.1.2 L3-type PhCC
The next case follows the data from [3]. Here, a L3cavity with a lattice constant of a= 420 nm,
air hole radius-to-lattice constant ratio r/a =0.29, a slab thickness-to-lattice constant ratio
of t/a =0.6is systematically investigated. For the dielectric permittivity, again !r= 11.56 is
chosen. The case of highest Qis chosen to be verified with Maexle, therefore the shift of the
outer air holes are shifted outward by s/a =0.15. The photonic band structure is calculated
with MPB, see Fig. 3.5 (top left). Normalized frequencies are plotted over the wave vector,
where high-symmetry points in the reciprocal space are marked. Note the photonic band gap
between the first (green) and second (blue) band (the light cone is shown in red). Experimental
data from [3] are shown in the top right corner. Spectral responses over (vacuum-) wavelength
is shown for different outward shifts sof the outer air holes. The calculated spectral responses
of the x-component (y-component) of the electric field are depicted in the middle (bottom).
Each spectrum shows 6 peaks within the band gap (grey-shaded). The mode patterns and
corresponding spatial FTs for the x-component and y-component of the electric field are
depicted in Fig. 3.6 and Fig. 3.7, respectively. In comparison, the frequency of the desired
resonance agrees nicely with the fundamental mode (mode A). The fundamental mode A will
be focus of investigation in Chaps. 5 and 6.
In [4], also a L3-type defect is investigated for the use in strong-coupling experiments with
single QDs. The set of parameters describing the PhCC are a= 300 nm, of r=0.27 ×aand
t=0.9×awith !r= 11.56. An outward shift s/a =0.2is used for investigations [3]. Like in the
previous cases, a photonic band structure is calculated with MPB. Note the band gap between
the first two bands in the top left of Fig. 3.8, where normalized frequencies are plotted over the
wave vector. High symmetry points in the irreducible Brillouin zone are marked. In the top left,
experimental data from [4] are depicted. The spectrum of the active layer in the samples over
the (vacuum-) wavelength is shown on the left, while on the right side spectra of the emission of
three different samples are shown. Note the deviations, occurring due to slight differences in the
samples, e.g. deviating sizes of the air holes, distributed over the whole sample. Calculations
of the spectral response with Maexle are shown in the middle (x-component) and bottom (y-
component) part. Also, the agreement is good. The intensity (logarithmic scale) vs. frequency
in THz (bottom axes; the top axes provides normalized frequencies for better comparison
with the band structure) is plotted. The five occurring resonances are marked from A to E.
65 3.1. VALIDATION OF THE FDTD IMPLEMENTATION
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
normalized frequency [a/λ]
wave vector k
ΓMK Γ
Figure 3.5: Spectral response for the L3PhCC, after [2]. Top left: Band structure of the PhC,
containing the first 5 photonic bands and the light cone (red line). Normalized frequencies are plotted
vs. the wave vector. High-symmetry points of the reciprocal space are marked. Note the 1st band
gap between the first two bands. Calculated with MPB [82]. Top right: Experimental data from [3].
The (vacuum-) wavelength of fundamental mode is shown for different shifts sof the outer air holes.
For comparison, the case for s/a =0.15 is chosen. Middle and bottom: Spectral response (middle:
x-component, bottom: y-component), calculated with Maexle. Intensity on a logarithmic scale over
frequency in THz is shown (the top horizontal axes also provides normalized frequencies for better
comparison with the band structure). The 1st band gap is marked with a grey shade. Six resonances
are marked from A to F.
3.1. VALIDATION OF THE FDTD IMPLEMENTATION 66
Figure 3.6: Spatiospectral response, calculated with Maexle. From top to bottom: mode patterns
belonging to the peaks A-F in Fig. 3.5, middle. From left to right: electric field pattern Ex(rip,ω),
absolute square |Ex(rip,ω)|2and spatial Fourier transform |Ex(kip,ω)|2in-plane of the slab.
67 3.1. VALIDATION OF THE FDTD IMPLEMENTATION
Figure 3.7: Spatiospectral response, calculated with Maexle. From top to bottom: mode patterns
belonging to the peaks A-F in Fig. 3.5, bottom. From left to right: electric field pattern Ey(rip,ω),
absolute square |Ey(rip,ω)|2and spatial Fourier transform |Ey(kip,ω)|2in-plane of the slab.
3.1. VALIDATION OF THE FDTD IMPLEMENTATION 68
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
normalized frequency [a/λ]
wave vector k
ΓMK Γ
Figure 3.8: Spectral response for the L3PhCC, after [4]. Top left: Band structure of the PhC,
containing the first 5 photonic bands and the light cone (red line). Normalized frequencies are plotted
vs. the wave vector. High-symmetry points of the reciprocal space are marked. Note the 1st band
gap between the first two bands. Calculated with MPB [82]. Top right: Experimental data from [4].
Spectrum of the active layer in the sample (left). Photoluminescence of the samples. A modification
(shift of the outer air holes s/a =0.2) is applied. The (vacuum-) wavelength for three different samples
are shown, including the corresponding Q-factors. Note the different resonances due to fabrication
inaccuracies. Middle and bottom: Spectral response (middle: x-component, bottom: y-component),
calculated with Maexle. Intensity on a logarithmic scale over frequency in THz is shown (the top
horizontal axes also provides normalized frequencies for better comparison with the band structure).
The 1st band gap is marked with a grey shade. Due to a slightly different r/a, only five resonances
occur (compared to Fig. 3.5), which are marked from A to E.
69 3.1. VALIDATION OF THE FDTD IMPLEMENTATION
Figure 3.9: Spatiospectral response, calculated with Maexle. From top to bottom: mode patterns
belonging to the peaks A-E in Fig. 3.8, middle. From left to right: electric field pattern Ex(rip,ω),
absolute square |Ex(rip,ω)|2and spatial Fourier transform |Ex(kip,ω)|2in-plane of the slab.
3.1. VALIDATION OF THE FDTD IMPLEMENTATION 70
Figure 3.10: Spatiospectral response, calculated with Maexle. From top to bottom: mode patterns
belonging to the peaks A-E in Fig. 3.8, bottom. From left to right: electric field pattern Ey(rip,ω),
absolute square |Ey(rip,ω)|2and spatial Fourier transform |Ey(kip,ω)|2in-plane of the slab.
Chapter 4
Microdisk Resonator
4.1. SUB-µMICRODISK 72
Figure 4.1: Sub-µMicrodisk: Geometrical Setup. The main axes of the MD is parallel to the z-axes,
the thickness is denoted by t, the radius of the device is denoted by Rand the microdisk has an
isotropic dielectric permittivity. For the environment, the dielectric permittivity is !. Left: side view.
Right: top view. aand bare the big and small halfaxis, respectively.
Microdisk resonators in general provide ultra-high Q-factors [107], corresponding photon life
times up to the millisecond regime. In fact, the higher the azimuthal mode order, the higher
is the Q-factor. The reason is the reduction of bending loss of the confined electromagnetic
wave in the rim of the microdisk. Consequences, which one can take advantage from, are
for example extremely high energy densities and intensities and ultra-narrow line widths,
which can be used e.g. for ultra-fine sensing. Experimentally interesting, and also interesting
for (commercial) applications, is the usability for accurate displacement measurements with
ultra-fine environmental sensing and high-resolution spectroscopy. Also, they can be used for
quantum non-demolishing measurements, optical switches and in integrated optical devices
[26, 108–110].
In this chapter, the focus lies on two different microdisk devices. First, a sub-µmicrodisk
resonator is investigated (after [83]). The effects of a non-perfect circular shape and of a
uniaxial anisotropic environment on the spectral response is numerically analyzed. Second, a
microdisk device with a radius of 3µm, embedded in a LC environment (modeled with the
uniaxial anisotropic material), is calculated and compared with the experimental realization
[26, 111, 112]. The microdisk device itself is modeled with a pure dielectric material, thus no
absorption is taken into account. Substrates and microposts are also neglected. Due to the fact
that the field distribution for a resonance is concentrated in the rim of the device, a post does
not affect the spectral response. The general geometrical setup is depicted in Fig. 4.1. The
main axis of the cylindrically shaped resonator is perpendicular to the x-y-plane. Parameters
used in the simulations are given in the corresponding sections.
4.1 Sub-µMicrodisk Resonator
The general parameters for the numerical investigation of the sub-µmicrodisk resonator are
taken from [83], hence the radius is R= 361nm, thickness of the resonator is t= 265 nm and
the value for the dielectric permittivity (only real part) is taken for GaAs, hence !D= 11.56.
A spatial resolution of ∆x=7.3529 nm is chosen. Since no absorption is taken into account,
the fact that GaAs is absorbing below 800 nm is ignored. Note that this is a model study.
Also, materials like Gallium Nitride with a larger band gap than GaAs, having a comparable
dielectric permittivity in this spectral region, can be used.
In Sec. 2.4.1, the analytic solution for a microdisk resonator surrounded with a PEC mate-
rial, as well as for a dielectric material is shown. Numerically, using the FDTD method, one
can get the spectral response from a broadband excitation with a point source located near
73 4.1. SUB-µMICRODISK
Figure 4.2: Broadband spectral response from the microdisk device, simulation parameters after [83].
Intensity (logarithmic scale) vs. frequency in THz is plotted. Equidistant peaks belong to the same
radial mode order.
the interface of the microdisk with its isotropic environment, shown in Fig. 4.2 from a two-
dimensional calculation volume, where open boundaries (CPML, see Sec. 2.2.4) in-plane and
periodic boundary condition in z-direction are used. The vertical axis shows the intensity of the
TE response ( |Hz|2) on a logarithmic scale over frequencies in THz. Equidistant peaks with
different spacings occur over the whole spectral range. Starting from the low-frequency side,
the azimuthal mode order increases with each peak for a radial mode order of 1(fundamental
modes in-plane). For higher radial modes, this applies, too. Figure 4.3 illustrates this exem-
plarily in the spectral range from 500 THz to 600 THz, covering 5modes (see spectral response
in the top part on a logarithmic scale, plotted over the frequency in THz). The high-intensity
peaks belong to the modes with a radial mode order of 1, while the low-intensity modes have
a radial mode order of 2. Starting at the low-frequency edge, the azimuthal mode order is
incremented from 9(therefore, the mode label is TE1,9) to 11 (TE1,11) for the high-intensity
and high-Qmodes, and from TE2,6to TE2,7for the low-intensity and also low-Qmodes. The
effect of additional lobes in radial direction is visible in the mode patterns for the modes with
radial mode order 2. Due to increased bending loss, the mode pattern also expands strongly
into the environment. Therefore, the Q-factor is reduced. The spatial field distribution in the
mirror plane z=0(quasi-TE) for these modes are depicted in the bottom part of Fig. 4.3,
where the field intensity is color-coded (negative: red, positive: blue, normalized to 1) and the
grey circle shows the interface between the microdisk and the environment. The normalized
absolute value of the spatial FT (color-coded: black: zero, white: 1) of the mode patterns in
the kx-ky-plane are directly below the real-space patterns. Here, one directly sees the reason
for the lower Q-factor of the modes with higher radial order and higher Q-factor for modes
with high azimuthal mode order. The consequence of the second order in radial direction is
a second lobe in the spatial FT as well, yielding large contributions inside the light cone,
|k|⩽klc =2π
λ0(λ0is the vacuum wavelength of the resonance). For waves with contributions
4.1. SUB-µMICRODISK 74
Figure 4.3: Spatial field distribution and intensity in reciprocal space from a two-dimensional calcu-
lation. Top: Spectrum, intensity in a logarithmic scale over frequency in THz. The labels denote the
corresponding mode order TEN,M , where Nis the radial mode order and Mthe azimuthal mode order.
Note, that peaks belonging to higher modes in z-direction do not occur due to the periodic boundary
condition. Middle: mode patterns (normalized, red: +1, blue: −1). Bottom: intensity in reciprocal
space (dark: low, bright: high) of the TEN,M modes in the spectrum. The spatiospectral responses in
reciprocal space have the same number of lobes as the field patterns in real space.
inside the light cone, the conservation law for the parallel wave vector component is fulfilled.
Thus, they are not confined due to TIR, but are diffracted. On the other side, modes with
high radial mode orders have reduced components inside the light cone. Thus, the condition
for TIR is fulfilled for mostly all the waves and the Q-factor increases.
4.1.1 Geometrical Variations
The effect of slight deviations of the parameters from a perfectly circular microdisk resonator
is significant and cannot be underestimated, when one wants to utilize the spectral properties
of the microdisk eigenmodes. In the following sections, the influence of the variation of the
radius of a perfectly circular device, of a numerical excentricity, of an elliptically shaped device
as well as an edge profile is investigated numerically.
75 4.1. SUB-µMICRODISK
Regarding this from a positive point of view, proper shaping of a microdisk resonator leads to
the possibility to tune resonances to the interval of interest. Also, switching the mode order
can be achieved with relatively small changes of the geometry. Contrary, this has a negative
side effect. Slight deviations of the perfectly cylindrical structure result in shifts of resonances,
detuning afore properly tuned QDs for example. Numerical calculations are helpful to properly
predict the effect of geometrical changes. An analytic solution for WGMs with a perturbated
circular shape is not available, because the variables cannot be separated as in [113] for the
perfectly circular shape.
Radius
The modification of the resonance via a radius variation is used in the later Sec. 6.1 to tune
coupled MD resonators into resonance. Figure 4.4 shows the shift of the spectral response.
Every line corresponds to another radius of the considered MD, denoted on the right side.
The absolute square of the TE mode is plotted over the frequency in THz. From the bottom
to the top, the radius of the microdisk device is increased in steps of a spatial discretization
from R−10∆x= 287.471 nm up to R+10∆x= 434.529 nm. First, one notices the strong shift
of the modes with even a slight radius variation. This is due to the fact, that the size of the
device is small compared to the wavelength of the modes, as well as due to the chosen spectral
range, where the azimuthal mode orders are small. For increasing radius, the frequencies are
collectively red-shift. A larger radius includes a larger effective radius of the WGM due to
the continuity condition at the interface between microdisk and environment (see Sec. 2.4.1).
Therefore, the wave vector decreases and the frequency is red-shifted.
The fairly large amount of eigenmodes, available in the shown tuning range, ranging from
TE1,7for R−10∆xto TE1,13 for R+ 10∆xfor modes with a radial order of 1and from
TE2,5for R−10∆xto TE2,10 for R+ 10∆xfor modes with a radial order of 2, can be used
to investigate the coupling between modes with different mode orders. In [114], this coupling
behavior is investigated and explained with the radiation patterns of the eigenmodes. For a
more detailed explanation of the coupling between eigenmodes with equivalent mode order,
see Sec. 6.1.
In large microdisks, for example like the MD in Sec. 4.2, the spectral density of modes is fairly
high, since all kinds of azimuthal and radial mode orders are present. This makes it difficult to
experimentally distinguish between the different peaks emerging in the spectrum and identify
the mode order. A coarse estimation is presented to identify first order radial modes. However,
if other disturbing factors like ovality or an edge profile are present, the estimation is not valid.
4.1. SUB-µMICRODISK 76
500 510 520 530 540 550 560 570 580 590 600
frequency [THz]
R-10dx
R-5dx
R
R+5dx
R+10dx
Figure 4.4: Shift of the resonances with varying radius of the MD. Each line shows the intensity in
arbitrary units versus the frequency in THz. From bottom to top: Radius increases from r<=R−10∆x
to r>=R+ 10∆x. A slight variation of the radius about dR =∆xresults in a tremendous frequency
shift of more than 10 THz. Note the spectral separation of the different modes.
77 4.1. SUB-µMICRODISK
500 510 520 530 540 550 560 570 580 590 600
frequency [THz]
0
0.2770
0.3808
0.4538
0.5105
0.5566
0.5952
Figure 4.5: Spectral response of a MD with different halfaxis, calculated in a three-dimensional
simulation. Every line corresponds to the labeled numerical excentricity !=(1 + (b/a)2(a: large
halfaxis, b: small halfaxis). Intensity on a logarithmic scale is plotted over the frequency in THz. A
collective red-shift with increasing !is observed.
4.1. SUB-µMICRODISK 78
Figure 4.6: Spatiospectral response of TE1,10,1in a MD with numerical excentricity !=0(left) and
!=0.5952 (right). The amplitude is color-coded (red: +1, blue: -1) and the MD is shown in transparent
grey. Stretching of the mode pattern concentrates the field in the left and right corner. Bending loss
is increased there, while contrary bending loss is decreased in the front and back side.
Ellipticity
Figure 4.5 shows the numerical results from the variations of the numerical excentricity, defined
as !=(1−(b/a)2, where aand bare the big and small halfaxis, respectively (refer to Fig.
4.1 for a sketch of the geometry). Due to the radial symmetry with respect to the x- and
y-axis, only the radius in x-direction is altered without loss of generality. Different numerical
excentricities are plotted. From bottom to top, !=0,!=0.2770,!=0.3808,!=0.4538,!=
0.5105,!=0.5566 and !=0.5952, corresponding to a fixed small halfaxis b= 361 nm (along
y) and a big halfaxis a= 361 nm,375.7nm,390.4nm,405.1nm,419.8nm,434.5nm and
a= 449.2nm, respectively. Intensity on a logarithmic scale is plotted over the frequency
in THz. It is apparent, that first all modes are red-shifted and second that different modes
experience a different shift. To be more precisely, modes with radial mode order N=1shift
stronger than higher order modes. Due to the strong spatial confinement, a perturbation of the
radial symmetry also perturbates the confinement and strong localization of the field inside
the rim of the MD. Exemplarily, the spatiospectral response of WGM TE1,10,1is plotted in
Fig. 4.6 (red: +1, blue: -1). The MD is shown in a grey shade. On the left, the case with zero
ellipticity is depicted. One can see the strong spatial confinement of the TE-like mode inside
the resonator structure. On the right side, contrary, the case for !=0.5952 is shown. The
mode pattern is clearly stretched along the x-axis, thus increasing bending loss on the left and
right side and decreasing bending loss at the front and back side. A drop of the Q-factor from
Q0= 28200 down to Q0.5952 = 1050 accompanies the spectral shift of ∆f= 61.1THz.
Edge Profile
As another frequency-altering effect, the edge profile comes into play. Assume the bottom and
top of the MD to have different radii, forming a linear edge profile. Now, the propagating
WGMs in the MD are altered due to an effectively larger device thickness. In Sec. 2.4.1, the
Helmholtz equation for an infinitely extended dielectric slab was derived, eq. (2.48). Standing
waves inside the layer and exponential decaying tails outside are the solutions, when a wave is
assumed to propagate perpendicular to the normal of the layer (guided mode). In the case of a
WGM in a MD resonator, the propagating wave is guided along the rim of the MD (clockwise
and counterclockwise because of degeneracy of the angular equation). With an oblique wall
(formed due to i. e. a larger bottom radius Rbcompared to the top radius Rt), the WGM
snuggles up into the small-angle corner. This increases the effective radius, on which the WGMs
79 4.1. SUB-µMICRODISK
500 510 520 530 540 550 560 570 580 590 600
frequency [THz]
0%
5%
10%
15%
20%
25%
30%
Figure 4.7: Spectral response of a MD with an applied edge profile, calculated in a three-dimensional
simulation. Every line corresponds to the labeled increase of the bottom radius with respect to the top
radius. Intensity on a logarithmic scale is plotted over the frequency in THz. A collective red-shift with
increasing !is observed. An effectively thicker membrane results in higher order modes in z-direction.
are propagating, thus increasing the wavelength. The shift to smaller frequencies is observed
in the numerical data, too. Figure 4.7 shows the variation of the bottom radius with respect to
the top up to Rb=1.3Rt. Every line shows the intensity on a logarithmic scale over frequency
in THz. The black labels denote the corresponding increase of Rb. Collectively, all modes are
red-shifting. Also, due to the increased effective radius for propagation, higher-order modes
4.1. SUB-µMICRODISK 80
Figure 4.8: 3D spatial field distributions of TE1,10,1for different edge profiles. Top: Rb=Rt,Rb=
1.05Rt,Rb=1.1Rt(from left to right). Bottom: Rb=1.15Rt,Rb=1.2Rt,Rb=1.25Rt(from left to
right). With increasing edge profile, the maximum of the lobes moves towards the bottom, thus the
field is concentrated more in the small-angled corner.
Figure 4.9: 3D spatial field distributions of TE1,7,3for different edge profiles. Top: Rb=Rt,Rb=
1.05Rt,Rb=1.1Rt(from left to right). Bottom: Rb=1.15Rt,Rb=1.2Rt,Rb=1.25Rt(from left to
right). With increasing edge profile, the maximum of the lobes moves towards the top, thus the field
is concentrated more in the large-angled corner.
in z-direction emerge, since the oblique side wall includes an effectively thicker MD device.
Spatiospectral responses for the TE1,10,1and TE1,7,3WGM are depicted in Figs. 4.8 and 4.9,
respectively, with frequencies f0,1,10,1 = 568 THz and f0,1,7,3 = 592 THz (without applied edge
profile). In the top lines from left to right, the bottom radius is increased by 0%,5% and 10%,
while in the bottom line from left to right, the bottom radius is further increased to 15% 20%
81 4.1. SUB-µMICRODISK
Figure 4.10: Spatial field distributions of TE1,7,3for different edge profiles in the y-z-plane. Top:
Rb=Rt,Rb=1.05Rt,Rb=1.1Rt(from left to right). Bottom: Rb=1.15Rt,Rb=1.2Rt,Rb=1.25Rt
(from left to right). Concentration of the field in the upper corner is clearly visible.
and 25%. With increasing bottom radius, the WGMs are aligned along the oblique side wall,
while maintaining the mode order.
Additionally to Fig. 4.9, a cut along the y-z-plane shows Fig. 4.10. The grey box denotes
the MD, field amplitudes are color-coded (red: +1, blue: -1) and black lines are iso-intensity
surfaces. One can clearly see, that the intensity of the TE1,7,3is concentrated in the large-
angled corner. Due to edge profile and the comparably weak spatial confinement of the high-
order mode in z-direction, field in the small-angle corner couples to the environment more
easy, resulting in diffraction and hence reducing of the concentration of field. In contrast, the
WGM TE1,10,1concentrates the field in the small-angle corner (note the downward shifted
maxima of the lobes in z-direction).
4.1.2 Mode Tuning with a Uniaxial Anisotropic Liquid Crystal Environ-
ment
Providing a dynamical tuning procedure is of great importance for both, current research
and commercial applications. A microdisk resonator is embedded in an artificial uniaxial
anisotropic environment. The quantity nzis varied from nz=1to nz=3.5, thus it ex-
ceeds the refractive index of the MD, while the other components are fixed to nx=ny=1.
Due to the penetration depth of the evanescent tail of the WGM, leaking into the surrounding
of the MD, the coupling of the electromagnetic field to the environment can be controlled.
Hence, WGMs can be spectrally tuned. Exemplarily, the accessible range for the difference be-
tween the ordinary and extraordinary refractive index of the LC 5CB is about |no−neo|=0.2
4.1. SUB-µMICRODISK 82
Figure 4.11: Mode map of WGMs in a microdisk resonator, embedded in an artificial uniaxial
anisotropic environment. The intensity is color-coded (logarithmic scale, bright: high, dark: low), the
refractive index nzis on the vertical axis and the energy in eV is on the horizontal axis. Several
anticrossing points are observed due to coupling of WGMs via the anisotropic environment.
from ≈1.5to ≈1.7, easily controllable via an external electric field, which provides a large
tuning range. Possible future materials, i.e. meta materials, may provide an even larger range.
Therefore it is interesting to investigate the effect of anisotropy on the spectral properties of
a resonator [115].
Figure 4.11 shows the quasi-TE spectral response of a broad energetic range for different val-
ues of nz(vertical axis) on a mode map. The intensity on a logarithmic scale is color-coded
(dark: low intensity, bright: high intensity). Two global properties are observed, when nzis
increased. First, all modes in the investigated range shift slighty to lower energies. WGMs with
lower azimuthal mode order (low energy side) shift stronger than WGMs with high azimuthal
mode order. Second, strong red-shifts occur, when modes of different mode order approach
each other. Anticrossings emerge at the suspected crossing points and the line widths are
divided equally between both resonances. Thus, the WGMs couple to each other indirectly
via the uniaxial anisotropic environment. All these features are located in an experimentally
accessible range for the birefringence.
An interesting fact occurs, when nzapproaches and exceeds the refractive index of the MD.
As shown in Fig. 4.12, where intensity (normalized) on a logarithmic scale is plotted over the
energy in meV, the usage of a uniaxial anisotropic environment (top) differs substantially from
the usage of an isotropic environment. Four spectra are collimated for each case. The corre-
sponding regime of nzis shown on the right side. For the case of an isotropic environment,
the increase of nzresults in suppression of all WGM before the value for the MD is exceeded.
The condition for TIR cannot be fullfilled for this regime, therefore the confinement is lost.
In contrast, when a uniaxial anisotropic environment is used, confinement is still maintained
83 4.2. 3µMICRODISK
Figure 4.12: Comparison between isotropic and uniaxial anisotropic variation of the environmental
dielectric permittivity. Intensity on a logarithmic scale is plotted over the energy in eV for nz(top) and
n=1,1.8,3.4,3.6(bottom), respectively. In the isotropic case, no confinement is provided for n>1.8,
because the condition for total internal reflection cannot be fulfilled. In contrast, the anisotropic case
still maintains confinement even for exceeding the refractive index of the MD. Dotted lines are a guide
for the eye, showing the spectral position of selected WGMs for nz=1.
and clear peaks are observed in the spectrum. Since the resonances are only TE-like, all com-
ponents of one kind of field drive the other field via the time-dependent Maxwell equations
eq. (2.13). Thus, dielectric permittivities act like a potential for the electromagnetic waves
(see Tab. 2.1) and the confinement is provided by the TE-components, but the coupling be-
tween the WGMs, the anticrossings in other words, are due to the interaction of the non-TE
components with z-component of the environmental permittivity.
4.2 3µMicrodisk Resonator
In the previous section, a microdisk resonator with a large radius-to-thickness ratio is investi-
gated. Now, attention is shifted towards more common circular devices. Thicknesses of some
hundreds of nanometers are used to still have only the fundamental modes occupied perpen-
dicular to the microdisk plane, so only one field maximum concentrated in the quasi-TM plane
at z=0(see also Sec. 2.4.1). The radius is chosen to be comparably large to ensure, that
a post does not affect the propagation of the WGM along the rim of the device. Here, the
radius is R=3.05 µm, thickness t= 240 nm and the dielectric permittivity for the disk is
!= 11.56, corresponding to GaAs at room temperature. Figure 4.13, top, shows a spectrum
of the considered MD resonator. Intensity in arbitrary units is plotted over the energy in eV.
Several equidistant modes of different amplitude are observed. For first order modes, the en-
ergetic distance between modes with neighbored azimuthal mode orders M→M±1can be
estimated via [116]
4.2. 3µMICRODISK 84
Figure 4.13: Top: Spectrum of a MD resonator (R=3.05 µm) in vacuum, intensity in arbitrary units
is plotted over the energy in eV. Several equidistant modes are observed. First order modes in radial
direction exhibit highest intensities and shortest distance, followed by second order modes. Third order
modes are not visible on the linear scale. Bottom: Spatiospectral intensity (normalized) of the WGM
TE3,30,1(blue: low, red: high) in a 3µm microdisk resonator, denoted by the white lines (interface
between MD and environment). The mode pattern indicates, that a centered post has no influence on
the field distribution due to vanishing field components in the center of the microdisk.
85 4.2. 3µMICRODISK
∆εM→M±1=!c
nR,(4.1)
where nis the refractive index of the microdisk device. For the present device, the energy
distance is calculated to be ∆εM→M±1= 19.04 meV. Compared to the simulated spectral
response in Fig. 4.13, top, the difference between the high-intensity peaks is ≈18 meV, which
nicely agrees. Though the values fit nicely, one has to keep in mind, that in a real system
the energetic distance is smaller than the calculated value with eq. (4.1) and only approaches
the calculated value for increasing energy, hence with increasing azimuthal mode order. The
bottom side of Fig. 4.13 shows the normalized intensity pattern (blue: low, red: high) for the
WGM TE3,30,1of a microdisk resonator, obtained from a three-dimensional simulation. For the
calculation, a slightly modified set of parameters is used. A thickness of t= 240 nm, radius of
R=3µm and a dielectric permittivity of !MD = 11.56, similar to GaAs at room temperature in
the desired frequency range, is assumed. White lines show the interface between the microdisk
and the isotropic environment (vacuum, !=1). As can be seen, the intensity is concentrated
along the rim of the device and therefore, neglecting a post in the simulations is justified.
The following section deals with a MD, embedded in a uniaxial anisotropic environment,
mimicking a LC. Numerical results are shown, compared with the experimental data [26] and
discussed.
4.2.1 Mode Tuning with a Uniaxial Anisotropic Liquid Crystal Environ-
ment: Comparison with the Experiment
The geometrical setup in the simulation is equal to Fig. 4.1. Modifications, perturbating the
perfect, round shape, are not applied here. To take the birefringence of a LC environment into
account, the ordinary and extraordinary refractive indices are supposed to be in the range
of 1.5⩽n⩽1.68, respectively [26, 101], to model rod-like, polar LC-molecules. Thus, the
extraordinary refractive index is applied along the molecules, while the ordinary refractive
index applies in the plane perpendicular to the disk. The computational domain has a volume
of Vsim =4×4×1R3(x-, y-, z-extensions) with a spatial cell size of ∆x=R/100 (cubic)
with a simulation time of tsim = 10 ps. A subpixel averaging process is not applied for both,
the microdisk and the environment, since the implemented algorithm supports only subpixel
averaging for isotropic, dielectric media. However, these setting are sufficient to reproduce the
experimental results.
Uniaxial anisotropy means, that the dielectric permittivity tensor has only non-zero diagonal
elements, hence the electric field components are not mixing. As a simple model, it is assumed,
that TE-like modes populate the microdisk device. Hence, Ezand Hx,y vanish in the TE plane
at z=0. Therefore, the displacement field (see Sec. 2.1, eq. (2.9), neglecting the polarization
field P(r,t)) can be written as:
D(r,t)=!0!(r)E(r,t)=!0
n2
o00
0n2
o0
00n2
eo
E(r,t).(4.2)
Magnetic field components are altered by the dielectric tensor via eqs. (2.13) indirectly.
Now, applying an external electric field forces a rotation of the polar LC molecules, to align
the dipole axis along the electric field. In reality, the molecules are not rotating homogeneously
4.2. 3µMICRODISK 86
Figure 4.14: Scanning Electron Microscope image of the microdisk device, used in the experiment
[81, 112]. Left: Side view. The microdisk circumference in the microdisks plane is apart enough from
the post, that the influence of the post on the WGMs can be neglected. Note the perturbated rim and
the edgeprofile, which significantly affects the frequency of a WGM due to non-vanishing fields. Right:
Top view. The shape shows slight ellipticity and a rough interface between the microdisk device and
the environment.
in the whole volume. Especially around the MD, the molecules are anchored to the surface of
the device. Anchoring mechanisms of LC molecules in photonic devices is investigated in detail
in [95] for LC-infiltrated photonic crystal fibers. Therefore, the area near a surface has to be
treated differently. Note that within the numerical evaluation, no special treatment is applied
to take effects like bend or splay from anchored molecules into account. However, the whole
simulation volume is filled homogeneously with the uniaxial anisotropic material, to simulate
the birefringence.
The introduction of an orientation parameter owill act as a map for the progress of orientation
of the LC molecules. Assuming, that parallel orientation with respect to the TE-plane is given
by o=−1, the orientation perpendicular to the TE-plane is equal to o= +1. The intermediate
state, meaning a quasi-isotropic case with disordered orientation, is mapped with o=0. The
suggestion of an orientation parameter is closely related to a simple picture of the director
field nand the ordering parameter S, which describes a state of a LC. Thus, {n,S}→o.
Figure 4.15, top, shows the meaning of the orientation parameter schematically. From left to
right, o=−1→o=0→o= +1. The red rods depict the LC molecules, while blue depicts
the shape of the MD.
A comparison between the experimental data (Fig. 4.15, left) and the numerical data (right)
shows the intensity in arbitrary units, plotted over the energy in eV. For the sake of clarity, the
numerical data show the spectral response only for the three cases of o, shown at the top. On
the left, every line belongs to a higher bias (from top to bottom). Zero bias, 30 V and 100 V are
marked. A strong collective shift of the WGMs is observed, for increasing the bias voltage from
0V to 30 V. The effect of the altered orientation of the LC molecules is affecting all resonances.
Above 30 V, a saturation in the spectral shift occurs, after a spectral repositioning of ∆ε≈5
meV. The numerical data (right) cannot reproduce the saturation effect properly, since the
orientation parameter is only defined between o=−1(parallel) and o= +1 (perpendicular).
However, the trend of the spectral shift is in very good agreement with the experiment. A
global shift of all WGMs is clearly visible. Assuming, that the corresponding WGM, observed
87 4.2. 3µMICRODISK
Figure 4.15: Top: Alignment of the LC molecules according to the orientation parameter o:o=−1,
parallel orientation of the LC molecules. o=0, quasi-isotropic case. o= +1, perpendicular orientation
of the LC molecules. Bottom left: Experimental data. Intensity in arbitrary units is plotted over the
energy in eV. The chosen energy range shows 5 different modes. Two main features are observable.
First, with increasing bias, starting at 0V up to 100 V (labels on the right), all modes collectively shift
towards higher energies. Second, above 30 V, a saturation effect emerges. Bottom right: Numerical
data from the FDTD simulation. Intensity in arbitrary units is plotted over the energy in eV. Special
values of the orientation parameter oare chosen: o=−1, parallel orientation of the LC molecules (red).
o=0, quasi-isotropic case (green). o=−1, perpendicular orientation of the LC molecules (blue). The
trend of the (collective) spectral shift is reproduced nicely.
in the experiment, agrees with the numerically calculated WGM at ε= 967.408 meV, the
reorientation of the LC shifts the resonance to ε= 970.196 meV, resulting in a shift ∆ε=2.788
meV.
A detailed view shows, that first order radial modes shift less than higher order radial modes.
Additionally, the higher the azimuthal mode order, the less is the shift. Due to a stronger
spatial confinement, first order modes in radial direction have less field in the environment
compared the higher order modes. Therefore, the coupling of the electromagnetic field to the
LC environment is weaker. Figure 4.16 shows the shift of 1st-order WGMs with low (left) and
high (right) azimuthal mode order, TE1,35,1and TE1,48,1, respectively. Intensity in arbitrary
units is plotted over the energy in eV. For both cases, the energy range is chosen to be
the same size to show the difference in the shift from the parallel orientation (red) to the
perpendicular orientation (blue), passing the disordered state (green). As can be seen, both
modes shift differently. According to the penetration depth of the evanescent tail into the LC
environment, which is shorter for the WGM with higher azimuthal order, the resonance in the
right graph shifts less than the resonance in the left graph. Black labels denote the shift of
the blue curve with respect to the red curve. For low azimuthal mode order, the shift yields
∆ε=3.21 meV, while the shift for the high azimuthal mode order is ∆ε=2.88 meV. Hence,
the azimuthally low-ordered mode can be tuned 10% more.
Due to the observed shift compared with the experiment, it is concluded that the LC initially
was in a parallel alignment, at least within coherence length around the microdisk. With bias
applied, the LC molecules start to rotate due to their dipole moment to orientate along the
4.2. 3µMICRODISK 88
Figure 4.16: Spectral response of the MD with uniaxial anisotropic environment. Intensity in arbitrary
units is plotted over the energy in eV. Red: o=−1, green: o=0, blue: o= +1.Left: spectral shift of
TE1,35,1(left) and TE1,36,1(right). A shift of ∆ε=3.21 meV occurs. Right: spectral shift of TE1,48,1.
A shift of ∆ε=2.88 meV occurs.
electric field. According to the progress of the orientation, the refractive index background
changes, affecting the evanescent field of the WGMs. With increasing twist, ordinary and ex-
traordinary refractive index exchange due to the perpendicular alignment or the LC molecules.
The simple model using the orientation parameter ogives a satisfying explanation for the spec-
tral shift of the resonance and reproduces the experimental data very well.
However, two main differences emerge. First, the spectral shift of the WGM due to the ori-
entation of the LC molecules in the experiment is approximatively twice as large as in the
theory. Second, the spectral position of the investigated eigenmodes are different. Both dif-
ferences are explained by a non-perfect shape of the microdisk. In Sec. 4.1.1, the effect of
a non-perfect cylindrical shape on the resonance of the microdisks eigenmodes is discussed.
All kinds of introduced perturbations on the perfect shape have significant influence on the
frequency. A view on Fig. 4.14, showing a SEM image of the device, used for the experiment
[26, 81, 112], reveals a rough circumference, as well as a slight elliptically shape. While the
rough surface acts as a scatterer for the electromagnetic field and therefore induces additional
loss and frequency shifts, the latter introduces additional modes with increasing mode orders
in z-direction when the slope of the side wall decreases. Thus, the possibility for comparing
different modes in the experimental and numerical data is at place. The estimation eq. 4.1,
however, is only valid for first order radial modes, but gives a good approximation, for which
modes one has to look for in the experimental data.
Chapter 5
Photonic Crystal Cavities
5.1. PHCC: PREPARATIONS 90
This chapter deals with the tuning possibilities of PhCC modes in two-dimensional PhC
structures. Tuning in this sense means, to change the spectral properties of a resonance, say
the frequency and the cavity decay time. Therefore, different aspects are taken into account.
First, mode tuning is possible via the modification of the PhC slab itself. The effect is a
permanent tuning, mostly effecting all resonances in the system. Permanent tuning can be
realized in different ways. One can locally modify the slab properties, the PhC parameters like
lattice constant or the air hole radius for example. Another way is to evaporate (or with other
epitaxial methods) an additional layer, either only on the top surface of the PhC slab, or on
all surfaces. The latter would have a significantly larger effect on the whole system, since the
conditions for not only the TIR are changed, but also for the Bragg conditions for the in-plane
confinement of the light.
The goal of the tuning mechanism addresses some different points: first, modifying the cavity
decay time. To have high Q-factors is essential to investigate strong coupling between a cavity
mode and a single QD [4, 117] or i. e. a nitrogen vacancy center [118]. Ultra-high Q-factors
can be realized with a double heterostructure, say a waveguide with a locally changed lattice
constant and air hole radii [13]. On the other hand, when coupling cavities among themselves,
the injection of the photons from one to another cavity is important. Therefore, cavities with
a lower Q-factor are already utilized for trapping from a coupled waveguide to a single defect
cavity, and subsequent emission of the photon [119]. There is already a proposed method to
dynamically change the Q-factor for efficient in- and outcoupling of photons into a cavity from
a close waveguide [120]. Additionally, the usage of the optoelectronical device also dictates the
importance of the tuning range of the resonance. Other tuning mechanisms can be realized
via nonlinearities, where local variations of the refractive index occur due to the Kerr effect
or thermo-optical effects [121].
5.1 Preparations
To obtain reasonable results from the simulations, calculations concerning the spatial dis-
cretization, spatial and temporal simulation domain are performed. The discretization is a
crucial point when calculating structures with strong curvatures, like in the case of the PhC,
since the air hole radii in general are smaller than the wavelength of the light in the PhCC.
Hence, when the spatial resolution is not sufficiently good, the waves in the PhC cannot sample
the curvature sufficiently, thus the structure is not interpreted as a circle. This dramatically
changes the spectral properties, because small deviations from the perfect circular structure
mean to simulate a different geometry. The spatial computational domain has, if chosen too
small, influence on the spectral response of the geometry, too. A resonant mode in a cavity
has evanescent field contributions, which penetrate the surroundings in all three dimensions.
When the evanescent parts pierce the CMPL, the fields are leeched out of the simulation
volume, meaning to reduce the energy in the system and hence to e.g. reduce the Q-factor.
For the spectral analysis, the Harmonic Inversion, see Sec. 2.8, is used [105]. Despite this
method provides converged results for short simulation times, this is not independent of the
decay time of a resonance, for example. Figure 5.1 shows the behavior of a specific resonance
and its decay, thus Q-factor, in dependence of the in-plane size of the simulation domain and
the spatial grid resolution. A L3-type PhCC in a hexagonal lattice with a lattice constant of
a= 260 nm, air hole radius r=0.35 ×a(= 91 nm), slab thickness t=0.692 ×a(= 180 nm)
made from GaAs (!= 11.56) is considered. A size- and resolution-stable frequency and Q-
91 5.2. PERMANENT MODE TUNING OF A L3-TYPE PHCC
314.84
314.88
314.92
314.96
315
315.04
5 10 15 20 25 30 35
frequency [THz]
simulation domain in x-direction [a]
1000
2000
3000
4000
5000
6000
5 10 15 20 25 30 35
Q-factor
simulation domain in x-direction [a]
313
313.5
314
314.5
315
315.5
316
10 15 20 25
frequency [THz]
grid points per a
1000
2000
3000
4000
5000
6000
10 15 20 25
Q-factor
grid points per a
Figure 5.1: Determination of the simulation domain parameters. Top: minimum size determination
along the x-direction. Frequency (left) and Q-factor (right) are plotted versus the length in units of
the lattice constant. A size of 15 lattice constants is found to be necessary to obtain sufficient results.
Bottom: determination of the sampling rate per lattice constant. Frequency (left) and Q-factor (right)
are plotted versus the number of grid points per lattice constant. Most influencing factor here is the
sampling rate of the air holes, to let a propagating wave interpret the interfaces to be perfectly circular
shaped. A sampling rate of 16 grid points per lattice constant is found to be sufficient.
factor is obtained for a sampling rate of 16 grid points per lattice constant (spatial resolution
dx =a/16) and at least 15 lattice constants extension in x-direction. To be sure, 19 lattice
constants extension are chosen for further investigations. The y-direction is supposed to have
the same number of air holes. For the z-direction, a size of 2×ais chosen, which is sufficient
for the numerical calculations (not shown).
5.2 Permanent Mode Tuning of the L3Photonic Crystal Cavity
The method of choice to tune the spectral properties of resonances from a cavity in a two-
dimensional PhC slab is the method of gentle confinement [3]. There, the Q-factor is increased
by almost one order of magnitude, while maintaining the effective modal volume Veff. Both
quantities, the Q-factor and the effective modal volume, enter the Purcell-factor as the fraction
Q/Veff, which determines the enhanced spontaneous emission in a weakly coupled cavity-
quantum dot system [9]. Hence, the cavity decay time directly influences the Purcell factor.
To understand the method of gentle confinement, a k-space analysis is performed. Assume a
one-dimensional dielectric slab with a finite extension of length Lin x-direction and thickness
tin z-direction, surrounded by perfect mirrors for |x|⩽L/2and vacuum for |z|⩽L/2. Two
5.2. PERMANENT MODE TUNING OF A L3-TYPE PHCC 92
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
electric field amplitude [arb. u.]
real space coord. [L]
-15 -12 -9 -6 -3 0 3 6 9 12 15
k-space amplitude [arb. u.]
k-space coord. [2π/λ]
Figure 5.2: Principle of the method of gentle confinement, after [3]. Left: two different kinds of
(hypothetical) wave functions, confined in a one-dimensional cavity of length L, cladded by vacuum:
a step-like envelope function (red) and a Gaussian envelope function (green), modulated with a cosine
with wavelength λ=L/2.5. Amplitude in arbitrary units is plotted over the spatial coordinate in
cavity lengths L. Right: spatial FT of wave functions on the left. Absolute value in arbitrary units is
plotted versus the wave vector in units of 2π/λ. The black arrows indicate the left and right border of
the light cone. For the FT of the step-like envelope, large contributions inside the light cone appear,
while for the Gaussian envelope function, almost no contributions are located inside the light cone.
cases concerning the wave function of the confined field, shown in Fig. 5.2, are now considered.
The left side shows the electric field distribution in x-direction. A step-like envelope function
(red curve) and a Gaussian envelope function (full width at 1/e, green curve), modulated with
a cosine function with wavelength λ=L/2.5are used. Note that the step-like envelope has a
discontinuous transition at the interface ±L, while the Gaussian envelope has a smooth profile
and is continuous at the interface. The right side depicts the spatial FT (absolute value) of
the field profiles in arbitrary units. Black arrows indicate the light cone barriers for λ=L/2.5
at k=±1(in units of 2π/λ). The FT of the step-like function has non-zero contributions for
all wave vectors, while the Gaussian profile has only contributions around the resonance.
Regarding the conservation law for the wave vector k, a wave can leak out of the cavity, when
the tangential component of k(here |kx|) is 0⩽|kx|⩽k0[3]. The condition for TIR at the
cavity-vacuum interface is not fulfilled. For |kx|>k
0, the conservation law is violated, hence
the wave cannot leak out of the cavity and is highly confined. Thus, generating a more gentle
envelope for a field distribution of a confined mode yields a higher Q-factor.
Utilizing the method of gentle confinement to the L3-type PhCC (the PhC slab is laying in the
x-y-plane and the cavity is aligned along the x-axes), the geometrical setup results in Figure
5.3. The two-dimensional dielectric slab (grey block) with a hexagonal array air holes in a
three-dimensional computation volume (white box) is shown. A line defect is introduced by
leaving away three air holes in a line (center), a L3-type cavity. On the left side, the unmodified
geometry can be seen, while on the right, the modified geometry is depicted. The outer holes
on the left and right side of the line defect are shifted away from the cavity center by a shift-to-
lattice constant ratio s/a =0.3to make the modification clearly visible. The lattice constant
of the hexagonal air hole array is a= 300 nm (hole-to-hole distance along x-direction), the slab
thickness-to-lattice constant ratio is t/a =0.6and the hole radius-to-lattice constant ratio is
r/a =0.3. As a dielectric permittivity for the slab, a value of != 10.595 is chosen, which is
93 5.2. PERMANENT MODE TUNING OF A L3-TYPE PHCC
Figure 5.3: Modification process of the Q-optimization of the L3-type PhCC. Left: Unmodified geo-
metrical setup of the simulation volume (white box). The dielectric slab (light grey block), permittivity
!= 10.595, is perforated with air holes in a hexagonal lattice with a lattice constant a= 300 nm, slab
thickness t= 180 nm (=0.6a) and air hole radius r= 90 nm (=0.3a). Three missing air holes in a
line in the center introduce the L3-type cavity (along x-direction). Right: Air hole shift of the nearest
holes located left and right at the ends of the cavity (x-direction). For clarity, a shift of s= 90 nm
(=0.3a) is chosen.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
normalized frequency [a/λ]
wave vector k
ΓMK Γ
260 280 300 320 340 360 380
0.26 0.28 0.3 0.32 0.34 0.36 0.38
intensity [arb. u.]
frequency [THz]
normalized frequency [a/λ]
Figure 5.4: Left: Band structure for the investigated PhC (normalized frequencies vs. the wave
vector). High-symmetry points in the irreducible Brillouin zone are marked. The first 5 bands (1:
green, 2: blue, 3: magenta, 4: turquoise, 5: yellow), including the light cone (red) are plotted. Note
the band gap between the first two bands. Calculations are performed with MPB [82]. Right: Spectral
response of the PhCC. Intensity in arbitrary units is plotted vs. frequency in THz (lower axes; the upper
axes shows normalized frequencies in units of a/λ0,λ0is the vacuum wavelength). Used parameters
are a= 300 nm, slab thickness t= 180 nm (=0.6a) and air hole radius r= 90 nm (=0.3a). For the
band structure calculations, supercell of 4 unit lengths in z-direction, 16 points per lattice constant
for the real space sampling and 64 k-points between high-symmetry points are used.
similar to a GaAs slab at T=4◦K [122]. The corresponding photonic band structure is shown
in Fig. 5.4, where normalized frequencies are plotted over the wave vector. High-symmetry
points in the irreducible Brillouin zone are marked. Note the band gap between the first two
bands.
5.2. PERMANENT MODE TUNING OF A L3-TYPE PHCC 94
Figure 5.5: Field distribution in the TE-plane of the y-component for the fundamental mode for the
unmodified L3-type PhCC. Cutting along the y-axes at x=0(left) and along the x-axis at y=0(top)
shows discontinuities at the interfaces between the air holes and the defect region in the dielectric slab
(black circles). The dotted lines are a guide for the eye.
Figure 5.6: Intensity in reciprocal space of the fundamental mode for the unmodified PhCC. k-space
intensity (vertical axis, arbitrary units) is plotted over the x-component of the wave vector in reciprocal
nm. The grey-shaded area denotes the light cone. Note the contributions of the k-space intensity within
the light cone.
In this chapter, the focus lies on the fundamental mode, since this mode provides already
the highest Q-factor even without modification. Spectrally, this mode is located at the low
frequency side of the band gap at f= 281.3THz (see Fig. 5.4) with a decay rate of Γ= 219.77
GHz, resulting in a Q-factor of slightly above Q= 4000. The spatial field distribution of the
y-component of the electric field in the TE-plane of the slab (at z=0) is shown in Fig.
5.5. The two-dimensional, color-coded plot shows the field distribution normalized to 1 (red:
1, blue: -1) with a superimposed PhCC lattice (grey circles). Note that this is only a part
95 5.2. PERMANENT MODE TUNING OF A L3-TYPE PHCC
278
279
280
281
282
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
frequency [THz]
shift [a]
0
5
10
15
20
25
30
35
40
45
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Q-factor [103]
shift [a]
278
279
280
281
0.1 0.15 0.2 0.25 0.3 0.35 0.4
frequency [THz]
r’ [a]
0
5
10
15
20
25
30
35
40
45
0.1 0.15 0.2 0.25 0.3 0.35 0.4
Q-factor [103]
r’ [a]
Figure 5.7: Modification of the PhCC of L3-type for Q-optimization: shift (s) and radius variation
(r#) of the first outer air hole. Top line: resonance (left) and Q(right) alteration due to the first air
hole shift-to-lattice constant ratio 0⩽s/a ⩽0.35,∆s/a =0.01. The resonance is shifted successively
towards the upper band edge of the first band gap, where it reaches its minimum at s/a =0.23. The
maximum in the Q-factor, Q≈3.3×104, is reached at for s/a =0.185, indicated by black arrows.
Bottom line: resonance (left) and Q(right) alteration du to the first air hole radius-to-lattice constant
ratio 0.1⩽r#/a ⩽0.38,∆r#/a =0.01. A continuous increase in the resonance is observed. The Q-factor
reaches his maximum at r#/a =0.2,Q≈41500, marked with a black arrow.
of the simulation volume. Following the dotted line to the left side, a cut along the y-axes
(x=0) is shown, while when following the dotted lines upwards, a cut along the x-axes
(y=0) is depicted. The spatial extent is given in nm. The field is spatially concentrated in
the defect area, as it is expected, and the envelope of the electric field distribution has already
a Gaussian-like shape. When the field reaches a discontinuous change in the permittivity (the
dotted line are a guide for the eye), the slope changes drastically, too. Therefore, the Gaussian
behavior is perturbated. Figure 5.6 depicts the absolute value of the spatial FT of the x-cut
from Fig. 5.5 in arbitrary units, plotted over the wave vector in x-direction in inverse nm. The
grey-shaded area denotes the light cone for the fundamental mode. Non-zero contributions are
locate within the light cone.
Following the method of gentle confinement, the defect area will be modified. Like in [3], the
focus lies on the first-neighbored air holes. The outer air holes are successively shifted away
from the cavity center in steps of ∆s/a =0.01 from 0⩽s/a ⩽0.35. This modifies also the
effective length of the cavity (along x), hence an additional effect is obviously a decrease in
the resonance frequency. Thus, the hole shift pushes the resonance closer towards the upper
edge of the 1st band. Figure 5.7 shows the effect of the modification of the first air hole on
the spectral properties. In the top line, the resonance and Q-factor change is plotted versus
5.2. PERMANENT MODE TUNING OF A L3-TYPE PHCC 96
277.8
278
278.2
278.4
278.6
278.8
279
279.2
-0.2 -0.1 0 0.1 0.2 0.3 0.4
frequency [THz]
shift [a]
0
5
10
15
20
25
30
35
40
45
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Q-factor [103]
shift [a]
277.8
278
278.2
278.4
278.6
0.1 0.2 0.3 0.4
frequency [THz]
r’’ [a]
25
30
35
40
45
50
0.1 0.15 0.2 0.25 0.3 0.35 0.4
Q-factor [103]
r’’ [a]
Figure 5.8: Modification of the PhCC of L3-type for Q-optimization: shift (s) and radius variation
(r#) of the second outer air hole. Top line: resonance (left) and Q(right) alteration due to the second air
hole shift-to-lattice constant ratio 0⩽s/a ⩽0.35,∆s/a =0.01. The resonance is shifted successively
towards the upper band edge of the first band gap, where it undergoes saturation for s/a =0.4. The
maximum in the Q-factor appears slightly above s/a =0with strong deviations, altering around the
maximum reached with the first air hole modification (a more fine sampling of the region around
s/a =0did not show any clear maximum). Bottom line: resonance (left) and Q(right) alteration due
to the second air hole radius-to-lattice constant ratio 0.1⩽r## /a ⩽0.38,∆r## /a =0.01. A continuous
increase in the resonance is observed. The Q-factor starts spreading for r#/a ⩾0.15, hence no clear
maximum is found. Numerical uncertainties are assumed to be the reason.
the shift of the air hole in units of the lattice constant, starting from s/a =0to s/a =0.35.
A decrease of the resonance is observed, about 0.7%, accompanied by a drastic change of the
cavity decay time. The Q-factor increases from approximatively 4000 to 33000 for a shift-to-
lattice constant ratio of s/a=0.185. Furthermore, a decrease of the radius-to-lattice constant
ratio r$decreases the frequency by an amount of approximatively 1THz, while increasing the
Q-factor to 41500. Hence, the modification of the first air hole results in an overall increase
of the photon life time about one order of magnitude. In [123], an expanded method of the
gentle confinement is proposed, regarding the next-nearest neighbors of the L3defect. Second
and third air hole modification yield an improvement of 166% compared with [3]. However,
numerical evaluations of further modifications, the shift and radius variation of the second
air hole show no further improvement for the chosen set of parameters. Figure 5.8 plots the
frequency change and Q-factor variation versus the shift of the second air hole in units of the
lattice constant in the top line, like in Fig. 5.7.
97 5.2. PERMANENT MODE TUNING OF A L3-TYPE PHCC
Figure 5.9: Field distribution in the TE-plane of the y-component for the fundamental mode for the
modified L3-type PhCC. A cut along the y-axes at x=0shows no difference (left). However, cutting
along x-axis at y=0(top) shows a continuous field distribution (black circles) due to the shifted air
holes. The dotted lines are a guide for the eye.
Figure 5.10: Intensity in reciprocal space of the fundamental mode for the modified PhCC. k-space
intensity (vertical axis, arbitrary units) is plotted over the x-component of the wave vector in reciprocal
nm. The grey-shaded area denotes the light cone. Though the fundamental mode is spectrally shifted,
this slight deviations are not visible on this scale. Note the clear reduction of the k-space intensity
within the light cone, compared with Fig. 5.6.
Figure 5.9 and 5.10 show the field distribution in the x-y-plane in real space and k-space,
respectively, for a modified defect region, similar to Figs. 5.5 and 5.6 for the unmodified case.
For modification, the parameters for the largest Q-factor are chosen, therefore s=0.185 ×a
and r$=0.2×a. The cut along the x-axes shows clear reduction of the perturbation of the
envelope function from a perfect Gaussian shape (black circles), resulting in a more gentle
confinement. In contrast to Fig. 5.10, the intensity of the Fourier spectrum for the modified
5.2. PERMANENT MODE TUNING OF A L3-TYPE PHCC 98
case (black) shows a substantial difference around the light cone: the intensity profile itself
remains approximatively the same, but it is pushed away from the light cone. Thus, less
contributions are inside the light cone, meaning, that less modes couple to vacuum modes.
Therefore, the Q-factor is increased. For better comparison, the cuts in real and k-space along
the x-axes are plotted together in Fig. 5.11, top. Note the strong differences in the black circles
while the rest of the field distribution is mostly unchanged. Since the frequency shifts due to
the modification, the light cone also changes. The differences between the light cone for the
fundamental mode is not visible on this scale.
The reciprocal space intensity for the fundamental mode in the kx-ky-plane is shown in Fig.
5.11. The normalized intensity is color-coded (red:1, blue:0), leaky modes are located inside the
light cone (white circle). On the left side, the k-space profile for the unmodified case is shown.
Inside the light cone, non-zero contributions limit a high Q-factor (white shade, stretching
along ky). On the right side, the k-space intensity is shown for the modified structure. Signif-
icantly less contributions are located within the light cone.
In Fig. 5.12, the band structure of the given PhC is shown again, including the tuning range
in a dark-grey-shade. The photonic band gap is shown in a light grey shade. Spectrally, the
fundamental mode can be tuned above half the band gap size with only shifting the air holes
outward of the defect area. The left part shows a more detailed band structure around the
photonic band gap. Figure 5.13 compares the applied modification steps, normalized to 1for
better comparison. Red shows the resonance of the fundamental mode without modification,
in green the resonance for an outward shift of the 1st air hole is depicted and blue is the
resonance with outward shift with additional radius decrease. The decrease of the line width
of more than one order of magnitude can be seen, resulting in an increase of the Q-factor from
4000 to 41500.
99 5.2. PERMANENT MODE TUNING OF A L3-TYPE PHCC
Figure 5.11: Electric field Distribution for the unmodified (red) and modified (blue) case (top), real
part of Ey(ω, x)versus the x-coordinate in nanometer. The black circles show the effect of the shift
of the first outer air hole. Middle: Cut through the spatial FT of the y-component of the electric field
along x-direction. Intensity versus wave vector kxin inverse nanometers, for the unmodified (green)
and modified (blue) cavity. The grey shaded area denotes the light cone (since the frequencies of the
fundamental mode with and without modification differ only slightly, a difference in the width of the
light cone is not visible on this scale). Clearly visible is the reduction of contributions inside the light
cone. Bottom: spatial FT of the y-component (intensity) of the electric field in the x-y-plane. The
white circle indicates the light cone. Left: unmodified cavity, right: modified cavity. A reduction of the
contributions inside the whole light cone are apparent (along the y-direction), pushed outwards along
the x-direction.
5.2. PERMANENT MODE TUNING OF A L3-TYPE PHCC 100
Figure 5.12: Band structure for the investigated PhC with lattice constant a= 300 nm, slab thickness
t= 180 nm (=0.6a) and air hole radius r= 90 nm (=0.3a). In z-direction, a supercell of 4 unit lengths
is used. Left: First 5 bands (1: green, 2: blue, 3: magenta, 4: turquoise, 5: yellow), including light cone
(red), plotted is a normalized frequency [c/a] vs. |k|. Right: Zoom around the 1st band gap. High-
symmetry points of the first Brillouin zone are marked on the horizontal axes. The light grey-shaded
denotes the band gap, the dark grey shaded-area shows the modification range of the resonance of
the fundamental mode with hole shifting and air hole radius modification of the first air hole. The
black dashed line shows the optimized fundamental mode frequency with Q≈41500. Calculations are
performed with MPB [82].
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
278 278.5 279 279.5 280 280.5 281 281.5 282
intensity [arb. u.]
frequency [THz]
219.77 GHz24.89 GHz21.03 GHz
Figure 5.13: Spectral response of the unmodified cavity (red), with shift of the first air hole (green)
and with radius shrinking (blue). The frequency is shifted towards the band edge of the first band,
while the line width is reduces by more than one order of magnitude, resulting in an increase of the
Q-factor from ≈4000 (without) over ≈33000 (shifted first outer air hole) to ≈41500 (shift+ shrunken
first outer air hole).
101 5.3. COMPARISON WITH THE EXPERIMENT: L7-TYPE PHCC
5.3 Permanent Mode Tuning of the L7Photonic Crystal Cavity
Generalizing the permanent tuning possibilities of resonances in a cavity, one maybe wants to
retune a optical resonance to match an embedded quantum mechanical oscillator, for example
a QD or a nitrogen vacancy [118]. Difficulties or inaccuracies in the fabrication process may
have implicated defects in the dielectric resonant structure. For example, holes, to be expected
perfectly circular in the whole sample to maintain periodicity, can get a slightly elliptical
shape. The effect on the spectral properties of the sample is then a lifting of the degeneracy
for double-degenerate modes, hence a splitting of the resonance. Also, the surface of a wall of
a hole can be irregularly rough, resulting in disturbance of the perfectly assumed periodicity.
Thus, a resonance shifts. Figure 5.14 shows the SEM image of a PhC slab in the top left corner
[112], where clear deviations from a perfect circular shape and rough walls can be seen. The
small black bar in the middle of the image has a length of about 20 nm. Both effects now
affect e.g. the coupling to a single QD, which is located inside the resonator in a high-field
position to maximize the coupling and whose parameters are chosen such, that the resonance
matches the cavity resonance. The unwanted detuning due to the fabrication process can be
corrected with a subsequent tuning process, which is explained in the following:
To get rid of such unwanted detuning, one can spectrally tune the single QD using its temper-
ature dependence of the energy gap (which also affects the dielectric medium) or apply electric
or magnetic fields to make use of the quantum confined Stark effect [66] (which not affects the
optical properties of the cavity). Sometimes, this retuning possibilities are not enough to make
up the fabrication processing contingented mismatches of the resonances. Another possibility
to permanently retune the sample into resonance is to change the optical properties of the
cavity by evaporation of another material onto the surface. Controlled to only evaporate on
one side of a dielectric slab and not the walls of the air holes, for example, or uncontrolled
by evaporating the whole surface of the sample. These changes will affect only the spectral
properties of the confined field, but not of the QD.
An unmodified L7-type PhCC in a hexagonal lattice of air holes in a dielectric slab (GaAs)
with a permittivity != 11.6281(= 3.412)with lattice constant a= 300 nm, thickness-to-
lattice constant ratio t/a =0.8, air hole radius-to-lattice constant ratio of r/a =0.26667 (80
nm) is used for the numerical investigation. The slab lies in the x-y-plane. Numerical data
obtained from the simulation are then compared with experimental results [112]. To ensure
numerical precision, the lower limit for the spatial sampling rate (including a subpixel averag-
ing [54] with 15 points per pixel edge length) and the extension of the simulation domain are
determined. In the top right part of Fig. 5.14, the strongest resonance within the first band
gap is traced, showing the intensity of the electric field plotted over frequency in THz while
increasing the spatial sampling rate from bottom to top, starting at 10 and incrementing by
1(left). For a sampling rate of 12 points per lattice constant aand above, the simulation
provides usable data. The bottom left side of Fig. 5.14 shows the dependency of the reso-
nance of the simulation domain in x-direction (the y-direction is assumed to be constant for
all calculations), starting at 11 ×aup to 29 ×a. Strong deviations for small x-extensions
for are observed, induced by the long size of the cavity itself(8×a), while the frequency is
converged for extensions of 23 ×aand above. The spatial sampling rate and the size of the
simulation domain in x-direction for the numerical analysis are determined to be ∆x=a/16,
∆y=√3a/32 and lx= 25 ×a, respectively, to obtain reliable results. The z-direction has to
be treated differently. There, a silicon dioxide (SiO2) layer is used on the top side of the PhCC
(the air holes are not affected inside), covering uniformly the whole sample. The evaporation
5.3. COMPARISON WITH THE EXPERIMENT: L7-TYPE PHCC 102
235 240 245 250 255 260 265 270
Intensity [arb. u.]
frequency [THz]
10
15
20
25
30
242.64 242.65 242.66 242.67 242.68 242.69 242.7
extension in x-direction [a]
frequency [THz]
235 240 245 250 255 260 265 270
Intensity [arb. u.]
frequency [THz]
Figure 5.14: L7-type PhCC for permanent tuning with evaporated SiO2layers of different thickness.
Top left: SEM image of the GaAs slab with e-beam written and wet-etched air holes in a hexagonal
array with r/a =0.4167 (125 nm). The image shows distinct deviations of the air holes froma
perfectly circular shape, as well as rough walls. Top right: Determination of the minimum in-plane
spatial sampling rate of a lattice constant for sufficient results. The intensity versus frequency in THz
is plotted for different sampling rates, going from 10 (lower red) to 19 (upper red). A sampling of 12
points per lattice constant is found to be sufficient enough, 16 are chosen for the calculations to ensure
numerical accuracy. Bottom left: Determination of the needed extension of the simulation domain in
x-direction (extension in units of the lattice constant versus the resonance of the strongest mode). For
the numerical calculation, a length of 25 ×ais chosen. Bottom right: Intensity (DFT) of the electric
field (y)-component versus frequency in THz for different cell sizes inside the GaAs slab and additional
SiO2layer. Red: uniform grid with ∆z=∆x. Green: Non-uniform grid with a minimum ∆z=5nm.
Blue: Non-uniform grid with a minimum ∆z=2.5nm. Magenta: Non-uniform grid with a minimum
∆z=2nm. Due to the marginal difference between the different versions of the non-uniform grid, the
minimum spatial cell size is chosen to be ∆z=5nm.
process of the SiO2allows thin layers (thicknesses in the range of some nanometers), hence
a detailed resolution in the computational region perpendicular to the slab is necessary to
take the layers of different thicknesses properly into account. Therefore, a non-uniform grid in
z-direction is used, reducing the grid edge length in the slab, including the SiO2layer, down
to 5nm (2.5nm and 2nm). Figure 5.14, bottom right, depicts the differences in the spectral
response of the plain PhCC on the bottom right for the three cases of a nonuniform grid
( green: 5nm, blue: 2.5nm, magenta: 2nm), together with a uniform grid (red), spatially
sampled with ∆z=∆x. For the nonuniform grid, a discrepancy compared to the uniform grid
103 5.3. COMPARISON WITH THE EXPERIMENT: L7-TYPE PHCC
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
normalized frequency [a/λ]
wave vector k
ΓMK Γ
Figure 5.15: Left: Band structure for the PhC base structure (normalized frequencies vs. the wave
vector). High-symmetry points in the irreducible Brillouin zone are marked. The first 5 bands (1:
green, 2: blue, 3: magenta, 4: turquoise, 5: yellow), including light cone (red) are plotted. Note the
band gap between the first two bands. For the band structure calculations, supercell of 4 unit lengths in
z-direction is used. Calculations are performed with MPB [82]. Right: Geometrical setup. A L7-type
defect is introduced in a PhC with a hexagonal lattice of air holes. Used parameters are a= 300 nm,
slab thickness t= 240 nm (=0.8a) and air hole radius r= 80 nm (=0.26667a). The cuts show the
different permittivities of the dielectric material: blue: !=1, red: !GaAs = 11.6281(= 3.412), yellow:
!SiO2=2.25. The thickness of the SiO2-layer is fixed for 50 nm for better visibility.
is visible. Since all versions of the nonuniform grid show the same behavior, the nonuniform
grid with 5nm spatial discretization is assumed to be converged and used for the numerical
calculations for the PhCC with evaporated SiO2of different layer thickness. For the dielectric
permittivity for the SiO2layer, !=2.25 is used in the numerical evaluation.
The photonic band structure for the PhC is depicted in Fig. 5.15, left. Normalized frequen-
cies are plotted over the wave vector, where high-symmetry points in the Brillouin zone are
marked. Between the first two photonic bands. a gap emerges. The right side shows the geo-
metrical setup. Note that only a section from the simulation volume is shown here for better
visualization. The dielectric slab is shown in light-grey (see the defect area with the missing
air holes). Cuts along the x- and y-direction show the permittivity distribution in more de-
tail. Blue color has !=1, red is the dielectric slab, and yellow marks the SiO2layer, whose
thickness is varied to tune the resonance. Note that the SiO2-layer is only on the top side, the
holes are not filled.
It is expected, that the samples with evaporated SiO2layers will be red-shifted due to the
extension of non-vacuum material in the z-direction (asymmetrically just at the top side),
hence changing the conditions for the wave vector component kzto be still confined. Also, the
additional layer makes leakage of the field out of the cavity perpendicular to the slab easier,
which will result in a drop of the Q-factor, as well as a drop in intensity. Above a certain
thickness of the additional layer, a saturation effect is expected. Thus, the tuning range will
be limited.
5.3. COMPARISON WITH THE EXPERIMENT: L7-TYPE PHCC 104
0
1
2
3
4
5
6
220 230 240 250 260 270 280 290 300 310
Intensity [arb. u.]
frequency [THz]
1st band gap
A
B
C
DEF
Figure 5.16: Spectrum (top) and spatiospectral response (bottom, normalized) of the geometrical
setup in Fig. 5.15, right, without the additional SiO2-layer. The labels A-C in the spectrum correspond
with the left row, the labels D-F correspond with the right row. The color-coding is shown below. Grey
circles show the air hole-slab interface.
105 5.3. COMPARISON WITH THE EXPERIMENT: L7-TYPE PHCC
Identification of the peaks in the spectral response with the corresponding spatiospectral re-
sponse is given in Fig. 5.16. Intensity in arbitrary units is plotted over the frequency in THz
in the top figure. Black vertical lines mark the edges of the 1st band gap. Capital letters label
the six modes A-F of interest. The bottom part of Fig. 5.16 shows the spatiospectral response
of the y-component of the electric field from the six modes in the TE-plane (z=0). The
patterns for the modes A-C are shown in the left row from top to bottom, while the right row
shows the patterns for the modes D-F from top to bottom. The amplitude is color-coded (red:
positive, blue: negative). Grey circles show the air holes. The modes A and C dominate the
spectral response due to their high intensity.
In Figs. 5.17-5.21, numerical results with the modification of the slab are shown. Figure 5.17,
top, shows the accumulated energy in the x-y- and x-z-plane without modification. The nor-
malized intensity is color-coded (black: 0, white: 1.). Spatial coordinates are given in nm for
both plots. As in the case of the L3-type PhCC, the energy is highly concentrated in the
defect area (left). The logarithmic color scale chosen, reveals the evanescent parts, leaking out
of the defect region mainly in a 30◦angle. Note, that the accumulated energy distribution
is clearly dominated by the fundamental mode, labeled with A in Fig. 5.16. Using different
thicknesses of the SiO2-layer, the normalized energy distribution at different cuts are depicted
in the lower part of Fig. 5.17. The thickness of the additional layer is increased in steps of
10 nm (red: 0nm, green: 10 nm, blue: 20 nm, magenta: 30 nm, turquoise: 40 nm, yellow: 50
nm, black: 60 nm, orange: 70 nm). In the first line, the left and right plot differ in linear and
logarithmic scale of the z-axis, respectively, while in the lower line both plots have a linear
z-axis. Cutting along y=0nm, z= 55 nm, the results are shown in the middle line. The
effect of the additional SiO2-layer is a drop in the total accumulated energy of 23% (peak
intensity). The logarithmic scale shows no significant changes along the x-axis. More drastic
changes are expected along the z-direction. The lower line shows a cut along x=0nm, y=0
nm (left) and x= 2260 nm, y=0nm (right). The normalized accumulated energy is plot-
ted over the z-coordinate (note the difference on the vertical axes). Black vertical lines mark
the interfaces between the different dielectric media. At positive z-coordinates, the additional
SiO2-layer is located. Again, the different colors denote the different thicknesses of the layer.
On the left, the accumulated energy is observed inside the defect region, where three facts
attract attention. First, the accumulated energy drops with increasing SiO2-layer thickness.
Second, the distribution of the energy shows a slight asymmetry with more contribution at the
interface between the PhC slab and the additional layer. Third, no gradient of the maximum
with the layer thickness is observed. The first fact is clear, since the fields in the cavity are
also located inside the additional layer. From there, due to the lower refractive index contrast,
the fields can leak out of the GaAs defect region into the SiO2defect region due to a modifi-
cation of the condition for total internal reflection. Thus, energy gets lost. The reason for the
slight asymmetry is, that the additional layer increases the thickness of the basis slab, thus
generating an effectively more thick slab. Including already the explanation for the latter fact,
thus the strong spatial confinement in the defect region, the field and therefore the energy is
still concentrated in the center of the GaAs slab, but the altered continuity condition at the
interface to the additional layer allows more field in the SiO2layer.
Observing the energy more far away from the defect region, x= 2260 nm (right), the situa-
tion is substantially different. First, the intensity is five orders of magnitude smaller than in
the defect region. Second, the asymmetry is more obvious with clearly visible discontinuous
changes at the interfaces and third, a clear gradient with the layer thickness occurs, pointing
towards the additional layer. The evanescent tail on the upper side of the device penetrates
5.3. COMPARISON WITH THE EXPERIMENT: L7-TYPE PHCC 106
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
-4000 -3000 -2000 -1000 0 1000 2000 3000 4000
accumulated energy (normalized)
x-coordinate [nm]
z=55nm
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
-4000 -3000 -2000 -1000 0 1000 2000 3000 4000
accumulated energy (normalized)
x-coordinate [nm]
z=55nm
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
-300 -200 -100 0 100 200 300
accumulated energy (normalized)
z-coordinate [nm]
GaAs
SiO2
x=0nm
0e+00
1e-05
2e-05
3e-05
4e-05
5e-05
6e-05
-300 -200 -100 0 100 200 300
accumulated energy (normalized)
z-coordinate [nm]
GaAs
SiO2
x=2260nm
Figure 5.17: Accumulated energy distribution for the modified L7-type PhCC. Top: linear (left)
and logarithmic scale (right) for the accumulated energy in the x-y- (upper) and x-z-plane for the
unmodified sample. The strongest mode at 242.677 THz dominates the energy pattern. The logarithmic
scale shows the evanescent parts, leaking out of the cavity. Middle: Cut along the x-axes at y=0
nm and z= 55 nm in a linear scale (left) and a logarithmic scale (right). Bottom: Cut along the
z-axes at x=0,y=0(left) and x= 2260 nm, y=0(right, corresponding to ≈7.5333 ×a) for the
different thicknesses of the additional SiO2layer (from 0-70 nm). Vertical lines: GaAs slab thickness
and the maximum SiO2thickness. Vertical (diagonal) arrows: displacement of the maximum of the
acculumated energy with incrementing layer size.
deeper into the additional layer with increasing thickness due to weaker spatial confinement.
Therefore, the maximum of the field is not located in the center of the basis slab consisting
of GaAs, but also shifts to towards the SiO2-layer.
The effect on the spectral shift of the fundamental mode is depicted in Fig. 5.18. In the upper
graph, the frequency shift with increasing layer thickness is shown. On the left and right side,
107 5.3. COMPARISON WITH THE EXPERIMENT: L7-TYPE PHCC
the spectra for 0nm and 70 nm SiO2-layer thickness are plotted, respectively. The frequencies
in THz are on the vertical axes, while the horizontal axis shows the intensity in arbitrary
units. Capital letters mark the different modes within the first band gap (same labels mark
the same modes as in Fig. 5.16, top), dotted lines and dashed arrows connect corresponding
modes and are a guide for the eye. Both spectra are separated by the frequency shift (red
crosses) with increasing layer thickness (top axis). With increasing layer thickness, the modes
are collectively red-shifted. Due to the thicker slab, the wavelengths in z-direction increase.
Also, saturation of the frequencies of the modes is observed for larger additional layers. In Fig.
5.17, bottom right, one can see the effect in more detail for mode A (mode B is also shown
in the spectral region, but due to the small amplitude it is not clearly visible). The different
colors correspond to different layer thicknesses of the additional SiO2layer (red: 0nm, green:
10 nm, blue: 20 nm, magenta: 30 nm, turquoise: 40 nm, yellow: 50 nm, black: 60 nm, orange:
70 nm). For clarity, the left plot shows the spectral position of the peak maxima of modes
A and B in THz (horizontal axis) with increasing layer thickness in nm (vertical axis). The
saturation effect is explained with the evanescent tail of the confined modes in z-direction. As
long as the coupling of the field to the vacuum is sufficiently large, the modes are red-shifted
due to the leakage out of the basic GAs slab into the SiO2-layer and further into the vacuum
environment. As soon as the additional layer thickness exceeds a critical length lC, the effect
of a more thick layer vanished. Assuming the picture of a planar GaAs waveguide of finite
thickness and infinitively extended in the x-y-plane, the field decays exponentially outside of
the slab. With λ= 337.07 nm being the material wavelength of the fundamental mode for the
unmodified PhCC system, shown in Fig.5.15, the field decays to 1/e at 62 nm away from the
interface. Therefore, when the thickness of the additional layer exceeds this coherence length,
the saturation takes place.
A clear shift for both modes with increasing layer thickness is observed. For larger additional
layers, the peak position starts to saturate. However, the amplitudes and Q-factors drop
further with saturated spectral peak position. Due to the presence of the field in the additional
layer, the condition of total internal reflection is generally weakened because of the decreased
refractive index contrast, first at the interface between the basic GaAs slab and the SiO2-layer,
and second at the interface the SiO2-layer and the vacuum environment. This automatically
leads to an increase in the cavity decay rate, since the fields leak more out of the basic slab.
Hence, the Q-factor drops. Figure 5.19, left, shows the drop of the Q-factor on a logarithmic
scale for the six modes, labeled with capital letters A-F, in dependence of the SiO2-layer
thickness (vertical axis).
The comparison with experimental data [112] is shown in the Figs. 5.20 and 5.21. The first Fig.
compares the spectral response of the unmodified (left) and modified (54 nm additional layer
of SiO2, right), the latter compares the numerically obtained relative frequency shift (left)
and drop in the Q-factor (right). In Fig. 5.20, the intensity of the y-component of the electric
field in arbitrary units is plotted over the frequency in THz. The 1st band gap is marked
with black lines. Red shows the experimental data, green shows numerically obtained data.
One observes a clear difference in the spectral position of the cavity modes for both cases.
Inaccuracy in the fabrication process, as shown in Fig. 5.14, top left, leads to perturbation of
the periodicity in the sample. The periodicity is the fundamental aspect of the PhC slab to
provide best confinement. Also, deviations in the air holes around the defect affect the cavity
resonance directly (see for example Fig. 5.7, top left). Nevertheless, a deviation of up to 10
THz is unusually large. From the numerical point of view, accuracy can mostly be increased
with a more fine spatial grid. However, Fig. 5.14, top right and lower line, show that the
5.3. COMPARISON WITH THE EXPERIMENT: L7-TYPE PHCC 108
0
1
2
3
4
5
6
242 243 244 245 246 247 248
Intensity [arb. u.]
frequency [THz]
0
10
20
30
40
50
60
70
242 243 244 245 246 247 248
Thickness of SiO2 layer [nm]
frequency [THz]
Figure 5.18: Spectral properties of the resonances with increasing additional SiO2layer thickness.
Top: frequency shift of the strongest modes (intensity in arbitrary units, DFT) within the first band
gap (for clarity and better visualization of the small red-shift, a smaller range is shown). The left side
shows the spectral response for 0nm thickness, the right side for a thickness of 70 nm. In between, the
spectral displacement of the resonance with increasing layer thickness in 10 nm steps is depicted. The
arrows serve for clarity, as well as the dotted lines, to visualize the red shift of the resonances. Bottom:
Tracing the strongest mode. DFT intensity (left) and peak position (right) for different thicknesses.
For large thicknesses, a saturation starts to take place.
computational volume, the spatial discretization as well as the nonuniform grid in z-direction,
are checked to be converged. The frequencies of the cavity relative to each other show good
agreement between the experiment and the FDTD simulation. For the frequency shift with
increasing SiO2-layer thickness, Fig. 5.21, left, shows the experimental data in red and the
numerical data in green. Above a layer thickness of 40 nm, the experimental data shows a
saturation of the shift. In the numerics, saturation starts above a thickness of 70 nm (not
109 5.3. COMPARISON WITH THE EXPERIMENT: L7-TYPE PHCC
0
10
20
30
40
50
60
70
1e3 1e4
Thickness of SiO2 layer [nm]
Q-factor
ACEDBF
0
10
20
30
40
50
60
70
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Thickness of SiO2 layer [nm]
intensity (Ey) [arb. u.]
ACE
B
D
F
Figure 5.19: Left: drop of the Q-factor (horizontal axis, logarithmic scale) depending on the SiO2
layer thickness in nm (vertical axis). Right: drop of the intensity of the cavity modes (horizontal axis,
arbitrary units) in dependence of the SiO2-layer thickness in nm (vertical axis). The capital letters
label the modes corresponding to Fig. 5.18, top.
shown). Moreover, the experimental data exhibit a frequency shift of more than a factor 2.
Again, disorder-induced perturbations in the sample due to inaccuracies in the fabrication
process cause the difference. In general, the trend of the frequency shift agrees nicely. On
the right side of Fig. 5.21, the Q-factors are compared. A difference is expected due to the
differences in the frequencies of the modes, since Q∝f. However, a factor of 2.6separates the
numerical data (green) from the experimental data (red). Since the simulation assumes perfect
periodicity and a perfectly smooth surface on all interfaces, the numerically obtained Q-factors
show the upper limit. Experimental data are naturally interfered by the already mentioned
fabrication inaccuracies or dirtiness. Though the drop of the Q-factor is nice reproduced within
the simulation.
5.3. COMPARISON WITH THE EXPERIMENT: L7-TYPE PHCC 110
220 230 240 250 260 270 280 290 300 310
Intensity [arb. u.]
frequency [THz]
1st band gap
0nm SiO2
0nm SiO2
220 230 240 250 260 270 280 290 300 310
Intensity [arb. u.]
frequency [THz]
1st band gap
54nm SiO2
50nm SiO2
Figure 5.20: Comparison between the experimental (red) and the numerical (green) data. Left:
unmodified cavity, SiO2-layer thickness 0nm. Right: modified cavity, SiO2-layer thickness 54 nm (50
nm was used in the FDTD simulation). Intensity in arbitrary units over frequency in THz is shown.
The black arrows assign the peaks in the experimental data with peaks in the numerical data. Vertical
lines mark the upper and lower edge of the 1st band gap. Note that the band gap stays unchanged
under modification.
-2.5
-2
-1.5
-1
-0.5
0
0 10 20 30 40 50 60 70
frequency shift [THz]
Thickness of SiO2 layer [nm]
2
4
6
8
10
0 10 20 30 40 50 60 70
Q-factor [103]
Thickness of SiO2 layer [nm]
Figure 5.21: Comparison between the experimental (red) and the numerical (green) data. Left: the
spectral shift in THz is plotted over SiO2-layer thickness in nm. Right: drop of the Q-factor (labeled
with D in Fig. 5.18) is plotted over SiO2-layer thickness in nm. The trend of the spectral behavior is
reproduced nicely.
Chapter 6
Coupled Systems
6.1. COUPLED SUB-µMDS 112
The investigation of coupled resonator systems is the content of this chapter. The focus lies
on resonators in photonic band gap materials (PhCC) and resonators based only on TIR (mi-
crodisks). The first provide small effective mode volumes accompanied with high Q-factors, but
the spatial extensions (several lattice constants) need to be large to ensure proper confinement
in the plane of the PhC slab. Therefore, single devices in the regime of several micrometers
are standard. The latter provide ultra-high Q-factors, accompanied with mode volumes of one
order of magnitude larger and more, compared to PhCCs. Nevertheless, the size of a single
device and therefore the capability for dense integration on chips makes it attractive. Appli-
cations for coupled resonator systems are fast semiconductor lasers [124], signal transmission
[125], e.g. via coupled optical resonator waveguides (CROWs) and quantum information pro-
cessing devices.
The embedding of QDs in a microresonator affects the optical properties of the microcavity
only slightly due to its low crossection. The application range of such coupled systems is broad,
covering single photon sources, lasing systems and quantum computing, just to mention some.
Also, such systems are useful in current research to investigate the light-matter interaction in
more detail.
6.1 Coupled Sub-µMicrodisk Resonators
The interaction between resonators based only on TIR is discussed here. For the sake of
less computational effort, two-dimensional calculations of the sub-µmicrodisk resonators are
performed (see Sec. 4.1). Hence, the parameters for the simulations are a microdisk radius
R= 361 nm and a dielectric permittivity of != 11.56. Note that this model for the dielectric
material does not include absorption. The regime of strong cavity-cavity interaction, thus the
line splitting of the coupled system exceeds the line width of the isolated, uncoupled cavity,
is investigated numerically. The computational domain has an extension of 745 cells in each,
x- and y-direction, plus 20 cells of CPML (see Sec. 2.2.4). Square shaped cells with an edge
length of ∆x=∆y=7.35294 nm and a simulation time of tsim =2.5ps are sufficient to gain
reliable results.
6.1.1 Strong Interaction Between Two Sub-µMicrodisks
The geometrical setup of a system of two coupled microdisk resonators is shown in the insets
of Fig. 6.2. Different distances between the microdisks are taken into account. The detuning
in frequency between the two microdisk resonators is performed via a radius variation of one
microdisk, while the other microdisk has a constant radius, therefore the WGMs of the latter
have constant frequency. Section 4.1.1 shows the frequency dependency of WGMs with radial
mode order M=1and M=2and different azimuthal orders. Slight variations of the radius
strongly affect the resonance of the WGMs. The WGM TE1,10 will be in the focus for the
investigation of the strong resonator-resonator interaction, since this mode provides a high
Q-factor of Q=3.52 ×106at a frequency f= 555.742 THz. Loss in perpendicular direction
is not taken into account, since the simulation domain is two-dimensional (TE-plane) and
periodic boundary conditions are applied in z-direction.
In Fig. 6.1, left, the spectral response of the coupled system (green) is shown, compared
with the isolated, single MD (red). The intensity on a logarithmic scale is plotted over the
frequency in THz. A reduced frequency range around the TE1,10 WGM is shown. Both MDs
are separated by the distance d= 14.7nm. One observes a strongly asymmetric splitting
113 6.1. COUPLED SUB-µMDS
Figure 6.1: Beating between the odd and even eigenmode of two resonant MDs. Left: line splitting
(green) of the coupled system. The isolated, single MD resonance of the TE1,10 WGM is shown in red
for comparison. Accompanied by the formation of an ultra-low-Qand ultra-high-Qmode, the single
resonance splits asymmetrically into two new eigenmodes with even (low-frequency mode) and odd
(high-frequency mode) symmetry, respectively. Right: electromagnetic energy transfer between the two
MDs. Red: energy recorded in the right-placed MD. Green: energy recorded in the left-placed MD (see
Fig. 6.3 for the mode patterns of the even and odd mode). The inset shows the beating on a shorter
time scale to show the time needed to transfer the energy from one disk to the other (and vice versa).
of the MD resonance. Both peaks are separated by ∆=2 .0679 THz. This asymmetric line
splitting is accompanied by an asymmetric loss spitting as well. Thinking about the coupling
between a single QD and an optical mode (refer to Sec. 6.2 for the investigation of a single
QD to a high-QWGM in a MD), one expects a symmetric line and loss splitting due to the
strong coupling between the oscillators. Here, the coupling between the closely spaced MDs
induces the asymmetry in the line splitting, as will be explained later. The loss splitting finds
its explanation in the radiation patterns of the even and odd eigenmode later, too.
On the right side of Fig. 6.1, the transfer of the electromagnetic energy between the two MDs
is shown. The electromagnetic energy in arbitrary units is plotted over the time in ps. The red
curve belongs to the right MD, green to the left MD (see Fig. 6.3 for the positioning of the
disks and the corresponding mode patterns for the even and odd mode). The slight difference
in the total amplitude of the energy results from the deformation of the mode pattern for the
even mode (the field is more concentrated in the right-placed MD) due to the close spacing,
hence the strong interaction, while the pattern for the odd mode has equal amplitude in both
MDs. Therefore, the recorded energy in the left MD (green in Fig. 6.1, right) is lower. A
clear beating between the two MDs is observed. The time needed to transfer the energy from
one MD to the other equals the difference between two maxima (as denoted with a black
arrow). For the transfer, the time is estimated to be ttransfer = 483.455 fs (averaged), which
corresponds to a transfer frequency of ftransfer =2.0684 THz and is in perfect agreement with
the line splitting ∆=2 .0679 THz.
Figure 6.2 shows the line splitting (left) and the Q-splitting (right) of the coupled system for
different distances. On the vertical axis, the detuning of one microdisk with respect to the
second, fixed microdisk, is given as 1−r/R in percent, where Ris the radius of the fixed
microdisk. This emphasizes the strong impact of a slight radius variation. Frequencies in THz
are given on the horizontal axis. As a guide for the eye, the grey dashed line shows the isolated,
single microdisk resonance of the WGM TE1,10 at f= 555.742 THz. The different distances
6.1. COUPLED SUB-µMDS 114
Figure 6.2: Coupling between two MDs. One MD is detuned with respect to the other, fixed microdisk
via a radius variation, see inset. Left: Anticrossing of the coupled system for different spacings between
the MDs, the detuning in 1−r/R in percent is shown on the vertical axis, the frequency in THz on the
horizontal axis. The color-coding for different gap sizes is red: d= 14.7nm, green: d= 29.4nm, blue:
d= 44.1nm, magenta: d= 58.8nm, turquoise: d= 73.5nm. For small spacings, a strong asymmetry
in the line splitting is observed. With increasing gap size, the asymmetry vanishes. Right: Q-splitting
of the coupled system. On the vertical axis, the detuning is shown, while the horizontal axis shows the
Q-factor on a logarithmic scale. The color-coding for the different gap sizes is the same like on the left
side. For small spacings, the modes split asymmetrically into a ultra-low- and ultra-high-Qmode. For
increasing spacing, the loss splitting decreases with decreasing asymmetry.
dare color-coded (red: d= 14.7nm, green: d= 29.4nm, blue: d= 44.1nm, magenta:
d= 58.8nm, turquoise: d= 73.5nm). At first, one notices the strong asymmetry of the
eigenstates of the coupled system, which gradually decreases with increasing distance. For
d= 58.8nm and d= 73.5nm, the line splitting is almost symmetric. This is explained with
a tight-binding (TB) approach, as it is well-known for electronic systems in semiconductor
physics [63, 64]. For CROWs, a TB approach is successfully applied to calculate the dispersion
relation [79, 80]. The basis of the TB approach is the wave equation for one localized mode
EΩ(r,t)of an isolated, single cavity with eigenfrequency Ω. The mode is assumed to be real
and orthonormal according to
;dr!r,0(r)EΩ(r)·EΩ!(r)=δΩΩ!,(6.1)
where !r,0(r)denotes the dielectric profile of the single resonator. For a time-harmonic mode,
the wave equation yields
∇×[∇×EΩ(r)] = !r,0(r)Ω2
c2EΩ(r).(6.2)
Describing the system of two coupled cavities, the eigenmodes with frequency ωof the coupled
system are a superposition of the modes of the single defects, weighted with Aand B:
Eω(r)=AEΩ(r)+BEΩ(r+r$).(6.3)
Here, r$is the displacement of the second resonator with respect to the first resonator. Equation
(6.3) also fullfills eq. (6.1), but with !r,0(r)=!r,0(r+r$)for the dielectric profile. Now, first one
uses eq. (6.3) in eq. (6.2). After multiplication of EΩ(r), followed by multiplication of EΩ(r+r$)
115 6.1. COUPLED SUB-µMDS
from the left side, one spatially integrates over the whole space. Subsequent use of eq. (6.1)
yields the new eigenfrequencies of the coupled system, depending on three parameters. The
TB-frequencies are
ω2
1,2 =Ω21±β1
1±α1+∆α.(6.4)
and the TB-parameters α1,β1and ∆αare given by the integrals
α1=;dr!(r)EΩ(r)·EΩ(r+r$)(6.5)
β1=;dr!r,0(r+r$)EΩ(r)·EΩ(r+r$)(6.6)
∆α=;dr(!r(r)−!r,0(r)) |EΩ(r)|2.(6.7)
The overlap-type integral in α1induces the asymmetry in the mode splitting, enhanced by
∆αat very short distances. ∆αdecays asymptotically to zero with the exponentially decaying
field amplitudes of the resonant modes outside of the resonator. Hence, this quantity can
often be neglected. Figure 6.3, left, depicts the accumulated energy of the coupled system on
a logarithmic scale, while the middle and right show the mode patterns of the even and odd
eigenmode, respectively, for a separation distance of d= 14.7nm. As a guide for the eye, the
black line depicts the MD-to-environment interface. As the accumulated energy in the coupled
system shows, the TE1,10 is the dominant mode, since the energy distribution counts 20 lobes,
corresponding to the 10 minima and 10 maxima in the field distribution. The occurrence of
an even and odd mode confirms the fact of strong resonator-resonator interaction. Note the
strong deformation of the field for the even mode, especially in the left-placed MD, compared
with the odd mode. This is due to the strong interaction between the two resonators. For
larger separation distances, this deformation vanishes.
Beside the line splitting, a loss splitting occurs for the coupled system. Asymmetric line and
loss splitting is numerically investigated in [126] and experimentally proves in [114] for two
coupled microdisk resonators whose isolated, single resonances are of the same radial mode
order. In Fig. 6.2, right, the Q-factors of the eigenstates of the coupled system are shown
on a logarithmic scale (horizontal axis). The color-coding for the distances is the same like
in the left plot. As can be seen, for close positioned microdisk cavities, the Qsplitting is
largest, resulting in an ultra-low-Qmode (low-frequency mode, even field distribution) and
an ultra-high-Qmode (high-frequency mode, odd field distribution). See Fig. 6.3 for the field
distribution of the even (left) and the odd mode (right). For the ultra-low-Qeigenmode, the
cavity decay time yields TC,↓=1.19 ps (corresponding to a Q-factor of Q=2071), while the
cavity decay time ultra-high-Qeigenmode is TC,↑= 94.94 ps (corresponding to a Q-factor
of Q=165900). This is connected with an enhanced and suppressed radiation perpendicular
to the coupling direction (the coupling direction is along the x-axis), respectively. Figure 6.4
shows the absolute value of the radiation pattern of the even (left) and the odd mode (middle)
on a logarithmic scale for a resonator spacing of d= 14.7nm. The color-coding is blue for
low and red for high intensity. Substantial differences are observed between the even and odd
case. The radiation of the even mode is only suppressed along the dark blue lines due to
destructive interference. The selective mode patterns have a slightly asymmetric form. For the
very short spacing of the resonators, the coupling between the microdisks is so strong, that
6.1. COUPLED SUB-µMDS 116
Figure 6.3: Spatiospectral response of the coupled MD system in the TE-plane (x-y-plane). The
microdisk radius is r= 361 nm and the spacing between the MDs is d= 14.7nm. Black lines show the
interface of the MD to the vacuum environment. Left: Accumulated energy (normalized, logarithmic
scale, red: high, blue: low). One can see, that the TE1,10 WGM dominates the radiation pattern. Middle
and right: mode patterns (normalized, linear scale) of the even (middle) and odd (right) eigenmode of
the coupled system. The even mode has a deformed pattern, especially in the left-placed MD. Also the
amplitude in the left MD is lower than in the right MD. For the odd mode, the amplitude is divided
equally. Color-coding: red: +1, blue: -1.
Figure 6.4: Intensity patterns (normalized, logarithmic scale, red: high, blue: low) for the even (left)
and odd (middle) eigenmode of the coupled system. The odd mode has decreased radiation along the
y-axis due to destructive interference of the patterns of the single MDs. Right: difference between even
and odd intensity pattern. The enhanced radiation along the y-axis is clearly visible. Black circles
denote the interface between the MDs and the vacuum environment.
the mode pattern is substantially altered. Due to constructive interference of the radiation
patterns of the left and right microdisk, the intensity is high, especially along the y-direction.
Therefore, the resonator decay time is reduced, hence the Q-factor is reduced. The intensity
in y-direction is decreased for the odd mode compared to the intensity of the even mode.
Destructive interference along the y-direction suppresses radiation and the resonator decay
time increases. Thus, the Q-factor increases. For comparison, the left shows difference of the
even and odd radiation pattern (logarithmic scale). A clear radiation perpendicular to the
coupling direction (x-direction) is observed, which proves the assumption of suppression and
enhancing of radiation due to destructive and constructive interference, respectively.
117 6.2. SINGLE QD STRONGLY COUPLED TO A WGM OF A SUB-µMD
Figure 6.5: A single QD strongly coupled to the high-Qmicrodisk WGM TE1,10. Left: Spectral
response of the coupled system, the intensity on a logarithmic scale is plotted over the frequency in
THz. Inset: geometrical setup (black circle: MD, blue triangle: QD). As a guide for the eye, the black
dashed line denotes the isolated, single MD resonance TE1,10. Red and blue: the QD is detuned to
the MD resonance. Single peaks in the spectrum left and right show the QD resonances, while the
MD resonance stays unchanged. Green: the QD on resonance with the MD resonance of TE1,10.A
clear line splitting around the isolated, single MD resonance is observed, and the VRS is ∆= 2g= 46
GHz. Right: Anticrossing of the upper and lower polariton branch. Frequency in THz is plotted over
the detuning of the QD in THz with respect to the MD resonance TE1,10. Grey dashed lines are a
guide for the eye, horizontal line: isolated, single MD resonance, diagonal line: QD resonance. A clear
repelling of the upper and lower polariton branch can be seen, hence the QD is strongly coupled to
the MD mode TE1,10.
6.2 A Single Quantum Dot Strongly Coupled to a WGM of a
Sub-µMicrodisk Resonator
For the numerical investigation, two-dimensional calculations are performed. As the electro-
magnetic resonator, the sub-µmicrodisk resonator is used. For the coupling, the high-QWGM
TE1,10 is chosen. Its resonance is at f= 555.742 with a Q-factor of Q=3.52×106. The QD is
placed at a high-field position near the rim inside the microdisk. Spectrally, the QD frequency
is variable to tune the QD into resonance with the WGM. Parameters for the QD are the
dipole matrix element µy=3eÅ and the phenomenological damping γQD =0. For the carrier
wave functions, a Gaussian ellipsoid with a radius of r= 30 nm in x- and y-direction is used:
Ψ=C·e−(x2+y2)/r2. Initially, the QD is in the ground state, therefore nQD =0. For the
simulation, the amplitude of the exciting source is chosen small, so that no nonlinear effects
occur.
Figure 6.5 shows the interaction between the high-Qresonator mode TE1,10 and the QD. A
scheme of the system can be found in the upper left corner, where the black circle shows the
interface of the microdisk with the vacuum environment and the light-blue triangle shows the
QD. On the left, the spectral response on a logarithmic scale is depicted, plotted over the
frequency in THz. Black dashed line denotes the resonance of the isolated, single microdisk
resonator. For a (red- and blue-) detuned QD, the WGM and the QD are not interacting.
Therefore, the two peaks show the isolated resonances of the QD and the WGM. When the
QD is on resonance with the microdisk WGM (green), a line splitting occurs, the VRS. The en-
ergy is periodically transferred between the quantum-mechanical and the optical resonator due
to the strong coupling. In the quasi-particle picture, this is described with cavity-polaritons,
6.3. COUPLED PHCCS 118
the upper and lower polariton branch. Detuning the QD further into the blue, the strong
coupling vanishes and the two oscillators are not coupled any more (blue curve). In Fig. 6.5,
the two polariton branches are shown. The emission frequency of the coupled system in THz
is plotted over the detuning of the QD in THz with respect to the isolated, single WGM
resonance. In the top left corner, the geometrical setup is sketched. In a system, which is
not strongly interacting, the emission frequency would behave like the light-grey dashed lines.
The vertical line stands for the WGM mode of the microdisk, while the diagonal line shows
the variable QD frequency. At zero detuning, both lines cross. In the strong coupling regime,
where the energy of the WGM is absorbed by the QD before it leaks out of the resonator
and is re-emitted before it decays non-radiatively, the lines split up, depending on the Rabi
frequency. Therefore, changing the frequency of a detuned QD into resonance, leads to an
anticrossing between the WGM and the QD, the polariton branches. On resonance, the line
splitting yields ∆= 46 GHz. The splitting is quite small for such a high-Qmode, but the
high Q-factor is compensated by a fairly large mode volume, compared to a L3-type PhCC,
where the mode volume is below (λ0/n)3(λ0is the vacuum wavelength of the resonance and
nis the refractive index of the PhC slab [3, 4]). For microdisk devices, this value exceeds the
effective mode volume of a PhCC [127]. Therefore, the interplay between the Q-factor and the
effective mode volume restricts the line splitting to a value of several tens of GHz.
6.3 Coupled Photonic Crystal Cavities
In contrast to Sec. 6.1, photonic band gap materials are utilized here. The L3-type cavity,
as introduced in Sec.5.2 is used for the numerical investigation. Therefore, three-dimensional
calculations are performed to take into account all kinds of radiation loss. The PhCC is
characterized by a hexagonal lattice of air holes in a purely dielectric slab. The lattice constant
is a= 300 nm, the air hole radius r=0.3×a, the slab thickness t=0.6×aand the dielectric
permittivity of the slab is != 10.595. The parameters are similar to [3] to obtain a TE-
like photonic band gap in the desired frequency range. For the computational domain, the
parameters are set to be Vsim = 29 ×14√3×2a3,tsim =3ps, ∆x=∆z=a/16 and
∆y=(3/4∆x, if not mentioned otherwise. For more accurate calculations, an !-averaging
subroutine [54] with 10 points per cell length is used (see Sec. 2.2.3).
6.3.1 Strong Cavity-Cavity Interaction between Two Photonic Crystal Cav-
ities
The investigated geometrical setup is depicted in Fig. 6.6, left, where the PhC slab is shown
in grey. Two L3-type defects are aligned linearly next to each other. The next-neighbored air
holes along the cavity axis are modified according to the investigations in Sec. 5.2. Therefore,
the hole radius is shrunken to r=0.2×aand the center of the air hole is shifted outwards
of the cavity centers by s=0.18 ×a. The Q-factor of the fundamental mode for the isolated
cavity is Q= 41500. Figure 6.6, right, shows the spectral response of the coupled cavity-system
(green) around the fundamental mode frequency, compared with the spectrum of the isolated
cavity (red). A slightly asymmetric line splitting around the fundamental mode frequency
occurs. The eigenmodes of the coupled system, labeled with A and B, are spectrally separated
by g= 378 GHz. For the isolated cavity with the mentioned modifications, the line width is
γC= 21 GHz. Therefore, since the line splitting of the coupled system obviously exceeds the
line width of the isolated system, strong cavity-cavity interaction is present. In the top right
119 6.3. COUPLED PHCCS
Figure 6.6: Geometry (left) and spectrum (right) for linearly aligned PhCCs. The spectrum shows
intensity in arbitrary units over frequency in THz. Red: spectrum of an isolated single cavity. Green:
coupled cavity response. The even and odd modes are labeled with A and B, respectively. The inset
shows the spectrum on a logarithmic scale. In the top right corner, the Q-factors for the modes A and
B are given.
Figure 6.7: Mode patterns of the coupled cavity system with linear alignment. Left: even mode.
Right: odd mode. The labels A and B refer to the frequencies in Fig. 6.6.
corner, the Q-factors for the even mode (QA) and the odd mode (QB) is given. Besides the
line splitting in frequency, a Q-splitting occurs [128]. The splitting in loss is explained by the
spatial FT of the mode patterns just above the PhCC slab. Larger components inside the light
cone allow stronger leakage perpendicular to the slab. As shown in [128] for linearly aligned
cavities, the Q-splitting is asymmetrically around the value of the isolated, single cavity Q-
factor. Additionally, the spatiospectral response approves the strong interaction between the
two cavities. In Fig. 6.7, the mode patterns are shown in the TE-plane (z=0) with color-coded
amplitude (red: positive, blue: negative). On the left, the even mode is depicted, while the right
side shows the odd mode with color-coded amplitude (red: positive, blue: negative). Hence,
the symmetric and asymmetric field distribution approve the strong cavity-cavity interaction.
From the field distribution for the fundamental mode of the isolated cavity, Fig. 5.9, one sees,
that the evanescent parts of the mode pattern penetrate the air hole lattice in a 30◦angle with
respect to the cavity main axis. For efficient field injection from one cavity into the other cavity,
6.3. COUPLED PHCCS 120
Figure 6.8: Geometry (left) and spectrum (right) for angle-aligned PhCCs. The spectrum shows
intensity in arbitrary units over frequency in THz. Red: spectrum of an isolated single cavity. Green:
coupled cavity response. The even and odd modes are labeled with A and B, respectively. The inset
shows the spectrum on a logarithmic scale. In the top right corner, the Q-factors for the modes A and
B are given.
Figure 6.9: Mode patterns of the coupled cavity system with angled alignment. Left: even mode.
Right: odd mode. The labels A and B refer to the frequencies in Fig. 6.8.
it is obvious to have maximum overlap between the single cavity mode patterns. Therefore,
two more alignments of the coupled cavity system are investigated. First, the second cavity is
aligned in a 60◦angle with respect to the first cavity main axis to match the air hole lattice.
Then, the second cavity is shifted away from the first cavity such, that the two centers of
the cavities are connected with a virtual straight line, having a 30◦angle with respect to the
cavity axes. A scheme of the alignment is shown in Fig. 6.8, left. The second alignment scheme
consists of two laterally aligned cavities, where the second cavity is shifted with respect to the
first cavity such, that the centers of both cavities are again connected with a virtual straight
line, having a 30◦angle with respect to the cavity axes. See Fig. 6.10, left, for a scheme. Note,
that the lateral alignment gives rise to the nearest alignment of the two cavities, regarding the
distance between the centers, compared with the linear and angled alignment.
Figure 6.8, right, shows the spectrum of the coupled system for the angled alignment (green)
compared with the spectral response of the isolated, single cavity. Intensity in arbitrary units
121 6.3. COUPLED PHCCS
Figure 6.10: Geometry (left) and spectrum (right) for lateral aligned PhCCs. The spectrum shows
intensity in arbitrary units over frequency in THz. Red: spectrum of an isolated single cavity. Green:
coupled cavity response. The even and odd modes are labeled with A and B, respectively. The inset
shows the spectrum on a logarithmic scale. In the top right corner, the Q-factors for the modes A and
B are given.
Figure 6.11: Mode patterns of the coupled cavity system with lateral alignment. Left: even mode.
Right: odd mode. The labels A and B refer to the frequencies in Fig. 6.10.
is plotted over the frequency in THz. Like in the case of the linearly aligned cavities, a line
spitting occurs, but the asymmetry is more pronounced. The even and odd eigenmodes are
split by g=2.233 THz, which is a factor of ≈6more than in the case of linearly aligned
cavities. Therefore, strong cavity-cavity interaction is at place. The mode patterns (color-
coded, red: positive, blue: negative) of the eigenmodes of the coupled system are depicted in
Fig. 6.9, where the even (left) and odd (right) field distribution according to the resonances
A and B in Fig. 6.8 are shown. Despite the mode pattern is not developed so clearly, one can
see, that the field injection is enhanced, because the line splitting is larger than in the linearly
coupled case, thus the coupling is increased. The Q-factors here are slightly split, but for both
eigenmodes of the coupled system, the values are below the Q-factor of the isolated, single
cavity.
The case for laterally aligned cavities is shown in Fig. 6.10 schematically on the left. Regarding
the center-to-center distance between the cavities, both cavities are closer to each other than
6.3. COUPLED PHCCS 122
Figure 6.12: Left: Time signals of the coupled system for broadband excitation (blue, center frequency
is the fundamental mode resonance) and narrowband excitation for the low-Q(red) and high-Q(green)
eigenmode. Right: energy transfer between the two coupled PhCCs. Electromagnetic energy in arbi-
trary units is plotted over the time in ps. The red and green (multiplied by 0.3) curves are recorded
in the centers of the two cavities. Inset: beating on short time scale.
in the two previous cases. Therefore, an increased coupling is expected, resulting in a larger
line splitting. Figure 6.10, right, confirms this fact. The intensity of the coupled system (green)
and, for comparison, of the isolated, single cavity (red), are plotted over frequencies in THz. A
comparably large line splitting of g=7.099 THz emerges. Due to the (spatially) exponentially
decaying evanescent field, penetrating the neighboring cavity, the coupling strength between
the cavities is enhanced due to increased spatial overlap of the single uncoupled mode patterns.
This also explains the strong asymmetry in the line splitting due to eq. (6.4), which is induced
by the overlap-type integral α1, see eq. (6.7). The Q-factors for the even and odd mode are
given in Fig. 6.10, too. Loss splitting is observed like in the case of the coupled MDs in Sec.
6.1.1. In [128], the splitting of the Q-factors is explained to be due to the radiation pattern
perpendicular to the PhC slab plane, where the confinement is dominated by TIR, for the case
of 2 linearly aligned L3-type defects in a PhC. Thus, the same argument as for the coupled
MDs applies here. Constructive (destructive) interference of the radiation perpendicular to
the slab cause the even and odd mode (compare Fig. 6.11 for the field distribution) to be less
(stronger) confined. Figure 6.12 shows the time evolution of the field envelope (normalized)
over ps, resulting from broadband excitation (blue, center frequency is the fundamental mode
resonance) and narrowband excitation (red and green), resonant with the even and odd mode
(low-Qand high-Qmode), respectively. The different Q-factors are clearly visible, since the
even mode decays faster than the odd mode.
To see the transfer time of the electromagnetic energy from one cavity to the other, Fig. 6.12
shows the electromagnetic energy in arbitrary units over the time in ps. Each curve is recorded
in the center of one of the cavities. The green curve is scaled by 0.3since the amplitudes of
the confined modes are slightly different. On the whole time scale, fast oscillations are visible,
perturbated by the energy of the other cavity modes (see Sec. 3.1.2). Since the fundamental
mode of the L3-type defect is the dominant mode, the beating between the two cavities can
be seen even when the other modes are present. The inset shows the time from 6ps to 6.5
ps. The distance between two maxima, indicated by the black arrow, is the transfer time,
which is needed, to transfer the electromagnetic energy from one cavity to the other and back.
123 6.3. COUPLED PHCCS
Figure 6.13: Avoided crossing of two strongly interacting PhCC cavities with a L3-type defect.
Detuning in units of the outward shift of the outer air holes in s/a is plotted over the frequency
in THz. Red: frequency-dependency of the isolated, single cavity. Green: frequency-dependency of
the coupled system. The low-frequency branch corresponds to the fixed cavity. When zero detuning
is approached, the fixed resonance is repelled by the approaching resonance of the cavity with the
variable air hole position.
This time is estimated to be ttransfer = 139.922 fs, corresponding to a transfer frequency of
ftransfer =7.146 THz, equal to the line splitting.
For the purpose of investigating an avoided crossing between PhCC modes, two cavities are
aligned laterally for enhanced field injection from one into the other cavity (see Fig. 6.10,
left). One cavity has a fixed resonance. Here, the common parameters for the outer air hole
shift is used, but the radius is unmodified (refer to Sec. 5.2). Therefore, the frequency of the
fundamental mode is fixed to fC= 279.055 THz with a Q-factor of Q≈33000. The second
cavity is modified via the shift of the outer air holes, to spectrally tune the resonance through
the resonance of the first, fixed cavity. Due to the air hole shift-dependent frequency shift of the
cavity resonance (see Fig. 5.7), the variable resonance is expected to push the fixed resonance
apart when the detuning between the cavities decreases. After the resonant case, the fixed
mode is expected to return to its isolated frequency. Hence, no common avoided crossing,
say anti-crossing, is expected here. Figure 6.13 shows the behavior of the coupled, detuned
cavities. The detuning (vertical axis) is given in units of the air hole modification s/a, where 0
means being on resonance with the fixed cavity. Therefore, the vertical axis maps the resonance
only indirectly. For better comparison, the single, isolated cavity resonance is shown in red.
Frequencies in THz are given on the horizontal axis. As can be seen, the modes of the coupled
system split around the isolated, single cavity resonance. When the detuning approaches 0,
the branch of the fixed cavity is clearly pushed away. This clear avoided crossing emphasizes
the strong cavity-cavity interaction.
Distance dependency of the coupled PhCCs is crucial, when one wants to integrate them
in a dense manner on a chip. Therefore, the distant-dependent line splitting of two coupled
6.3. COUPLED PHCCS 124
Figure 6.14: Left: Overview of the investigated alignments of the PhCCs. Right: distant-dependent
line splittings of the alignments from the left. Same colors mean same alignment. The frequency in
THz is plotted over the center-to-center distance between the cavities. A light-grey dashed line is a
guide for the eye to show the resonance of the fundamental mode for the isolated, single cavity.
cavities, using the three investigated alignments, is depicted in Fig. 6.14. The left side shows
the three alignment schemes. On the right side, frequency of the eigenmodes of the coupled
system is plotted over the center-to-center distance. A grey-dashed line shows the isolated,
single PhCC resonance of the fundamental mode. Red shows the frequencies of the even and
odd eigenmodes (left and right branch, respectively) of the laterally aligned PhCCs, blue shows
the angle-coupled PhCC eigenmodes and green shows the inline coupled cavities. As can be
seen in the Figs. 6.6, 6.8 and 6.10, the line splitting is largest for the lateral alignment, with
direct field injection due to the proper alignment. With increasing distance, the asymmetry in
the line splitting vanishes, since the asymmetry-inducing term in eq. 6.4 gets less important
due to less spatial overlap. The lateral alignment has the largest line splitting over the whole
investigated distance range, therefore the coupling is largest in this case.
6.3.2 Strong Cavity-Cavity Interaction between Three Photonic Crystal
Cavities: Pioneering more effective CROW s
In the previous Sec. 6.3.1, the focus lied on the coupling between two PhCCs, properly aligned
for enhanced field injection into the neighboring cavity. Now, this scheme is expanded by
introducing a third cavity. An interesting application is the CROW. Waveguides in PhCs are
already investigated well, e.g. the W1waveguide [129, 130], which is basically just a straight
missing line in the air hole lattice. W1waveguides exhibit, of course depending on a proper
choice of the parameters (especially the ratio of the air hole radius and the lattice constant
r/a), large bandwidths below the light line and also low loss [131]. However, an alternative
to a W1waveguide is a CROW, where for example the transmission is based on the coupling
efficiency between the neighboring cavities.
Utilizing the laterally aligned cavities, since they exhibit the largest line splitting of the in-
vestigated alignments in the previous section, the addition of the third cavity can be done
in different ways. First, the three cavities can be V-aligned. Second, the third cavity can be
aligned such, that they are <-shaped. Third, the third cavity is aligned laterally with the same
shift like the second cavity with respect to the first cavity, hence the three cavity-centers are
125 6.3. COUPLED PHCCS
Figure 6.15: Geometries of the three cavities, making advantage of the lateral alignment for enhanced
coupling. Left: V-aligned cavities. Middle: <-aligned cavities. Right: /-aligned cavities.
Figure 6.16: Spectral response from three coupled cavities in different alignments. Intensity on a
logarithmic scale is plotted over the frequency in THz. The insets denote the coupling geometries (the
dark-blue element contains the exciting light source). Red: isolated, single resonance of the fundamental
mode. Green: response of the coupled system. Grey dashed lines are a guide for the eye, showing the
eigenmodes at the most far (most close) frequency compared to the single cavity resonance.
connected via a virtual line, having a 30◦angle with the cavity main axes. The three cavi-
ties are /-aligned. Figure 6.15 shows the different alignments (from left to right: V-, <-and
/-aligned). Now, the spectral response is investigated, depending on the arrangement of the
three cavities. In Fig. 6.16, the intensity on a logarithmic scale is shown over the frequency
in THz. Red shows the isolated, single PhCC resonance of the fundamental mode for com-
parison, while green shows the spectral response of the coupled systems. In the insets, the
corresponding alignments of the three cavities is shown. One observes, that the /-coupled
geometry provides the largest line splitting, therefore the coupling is the largest. Thus, this
geometrical alignment is preferable for CROWs, consisting of several coupled L3-type defect
cavities in a PhC. For the case of V-coupled system, the line splitting clearly exceeds the one
from the inline-coupled system, containing only two cavities, see Fig. 6.6. Note, that here the
6.4. A SINGLE QD COUPLED TO A PHCC 126
distance between the top left and right cavities is larger. Equivalent spacing in the case of two
linearly aligned cavities result in less efficient coupling. Therefore, the alignment proposed
here is more efficient to transmit light along the cavity axis. The interaction between the
corresponding cavities is larger than in the linearly aligned case, thus the coupling between
the top left and right cavity is enhanced using the third cavity. The same applies for the
case of the <-coupled cavities for light transmission perpendicular to the cavity axis. This is
interesting, because the hexagonal character of the lattice dictates the alignment of a single
defect. Light can be transmitted efficiently along different symmetry directions only by proper
arrangement. A 90◦-bend, for example, is problematic, because the hexagonal air hole lattice
provides no useful alignment scheme for L3-type defects. Guiding light perpendicular to the
cavity axis can be done efficiently with a <-aligned array.
The field patterns of the eigenmodes of the three-fold coupled system for the lateral alignment
are depicted in Fig. 6.17, showing the y-component in the TE-plane (x-y-plane). From left
to right, the plot shows the field amplitude Ey(x, y, 0,ω)(normalized, red: +1, blue: −1)
in the TE-plane, the real-space intensity |Ey(x, y, 0,ω)|2and the reciprocal space intensity
|Ey(kx,k
y,0,ω)|2(blue: low, red: high). From top to bottom, each pattern corresponds to the
eigenmode of the upper spectrum in Fig. 6.16, thus the even mode at the frequency f= 271.5
THz with a Q-factor of Q≈5500, the center peak with vanishing amplitude in the center of
the middle cavity at the frequency f= 277.7THz with a Q-factor of Q≈8800 and the odd
mode at the frequency f= 283.6THz with a Q-factor of Q≈17000. The different Q-factors
are explained with radiation patterns perpendicular to the slab. For the even eigenmode,
constructive interference from all three cavities leads to increased energy loss, therefore to a
low Q-factor. One also observes a long evanescent tails in the TE-plane, leaking into the air hole
lattice of the PhC in a 30◦angle with respect to the cavity main axis. The other eigenmodes
have odd symmetry, hence leading to destructive interference in the perpendicular direction,
which yields a larger Q-factor. Also, the spatial confinement is stronger due to a stronger
decay of the evanescent tails.
6.4 A Single Quantum Dot Coupled to a Photonic Crystal Cav-
ity
Till now, light-mediated coupling between optical resonators was investigated. Now, the iso-
lated, single L3-type PhCC is extended with a quantum-mechanical oscillator, a QD. For this
purpose, microscopic, dynamic equations of motion for the coherence and occupancy of the QD
(see Sec. 2.5 for details) are evaluated numerically in every time step, using a RK-integrator
(see Sec. A.1). Calculating the optical polarization field P(r,t)of the QD, P(r,t)is coupled
back self-consistently to drive Maxwells equations (Sec. 2.5.1). Strong coupling between a sin-
gle QD and an eigenmode of a microcavity is well-investigated [4, 117, 132]. The focus of this
section, however, is the effect of a high-intensity excitation of the QD. It is well-known, that,
when the energy of an exciting pulse is large enough, so-called Mollow triplets occur.
6.4.1 QDs under strong excitation: Mollow Triplets
Excitation with strong light fields yield additional peaks e.g. in a fluorescence or transmission
spectrum [133, 134]. Regarding the picture of bare and dressed states (see Sec. 2.6) brings light
into the situation. Assume an atom-like oscillator with one transition ωL=!gap/!between the
127 6.4. A SINGLE QD COUPLED TO A PHCC
Figure 6.17: Spatiospectral response in the TE-plane from three laterally coupled cavities. Only
the y-component of the coupled fundamental mode is shown. From left to right: Field amplitude
(normalized, red: +1, blue: −1). Real-space intensity (blue: low, red: high). Reciprocal space intensity
(blue: low, red: high). From top to bottom: even mode at f= 271.5THz (Q≈5500). Center peak at
f= 277.7THz (Q≈8800). Odd mode at f= 283.6THz (Q≈17000).
ground |0*and excited state |1*. Bare and dressed states are described with a second quantum
number, the photon number N, in addition to the quantum number of the atomic state. Thus,
the full description of a state yields |0,N*, exemplarily for the ground state with Nphotons.
Figure 6.18 shows the bare and dressed states schematically. Energy increases from bottom to
top. Horizontal lines denote the states, bare states on the left and dressed states on the right.
The transition frequency between the bared states is ωL. However, the two bared states may
have a detuning δand are therefore degenerate for zero detuning. Illuminating the atom with
an intense light field lifts the degeneracy. The bared states split by the Rabi frequency ΩR,
which is dependent of the intensity of the light field. Four transitions are possible:
•a transition connecting the dressed state with the atom in its ground state |0*while
6.4. A SINGLE QD COUPLED TO A PHCC 128
Figure 6.18: Left: energy levels from the quantum optics point of view. Bare states are separated by
the gap energy εQD =!ωLof the QD. Excitation with an intense laser pulse lifts the degeneracy of
the bare states. Dressed states occur, split by the Rabi energy !ΩR. Now, four transitions are possible.
Right: Spectrum of a atom-like TLS under excitation with an intense laser pulse. The main peak
belongs to the main transition (2x) in the scheme on the left. The upper and lower side band peak
belong to the transition from |1,N −1*to |0,N +1*and from |0,N*to |1,N*, respectively. Colors
belong to the scheme on the left.
increasing the photon number Nby 1 (red),
•a transition connecting the dressed state with the atom in its excited state |1*while
increasing the photon number Nby 1 (red),
•a transition connecting the atomic ground state to the atomic excited state without
changing the photon number N(blue) and
•a transition connecting the atomic excited state and N−1photons with the atomic
ground state and N+1photons (green).
These splittings yield three peaks in the spectrum. Two additional side peaks next to the
central peak at ωL, at ω=ωL±ΩRoccur. These peaks have half the intensity of the central
peak since they are each fed by one transition, while the central peak is fed by two transitions.
A spectrum is shown in Fig. 6.18, right, to illustrate the Mollow triplet. The colors are equal
to the colors of the transitions in the scheme on the left side. Lorentzian curves are used for
the illustration. Since the Mollow triplet is intensity-dependent, placing a QD into a high-
field position of a high-QPhCC will lead to an increased ΩRdue to the strong spatial light
confinement in the defect area and the small cavity decay rate. Therefore, one can control the
splitting of the Mollow triplet with the Q-factor.
Since the dressed states are a picture from quantum optics, one is still able to see the splitting in
a semiclassical approach, like the FDTD, where the electromagnetic field is treated classically
and the QD is treated quantum-mechanically.
129 6.4. A SINGLE QD COUPLED TO A PHCC
Figure 6.19: Left: air hole shift dependency (vertical axis, s/a) over frequency (horizontal axis, THz).
Right: air hole shift dependency (vertical axis, s/a) over the Q-factor. Calculations are performed with
a coarse grid with ∆x=∆z=a/10,∆y=(3/4∆x. Similar to Fig. 5.7, top.
6.4.2 Mollow Triplets in a Coupled Cavity-QD System
For the investigations, a PhCC with a L3-type defect is used (similar to Sec. 5.2). The para-
metric set is a lattice constant of a= 300 nm, slab thickness-to-lattice constant ratio t/a =0.6,
air hole radius-to-lattice constant ratio t/a =0.3and a dielectric permittivity of != 10.595,
corresponding to GaAs at T≈4◦K [122]. For the purpose of a Q-depending Mollow triplet,
the outer air hole modification is variable in the shift, but fixed in the radius (r/a =0.3).
The fundamental mode is in the focus of this section. A spatiospectral response of the y-
component of the electric field is shown in Fig. 5.5 for the unmodified defect area and in
Fig. 5.9 for the modified defect area. A comparison shows, that the mode pattern is almost
not changed. Especially the field maximum in the center of the defect area is not altered.
Therefore, the QD is placed there. To describe the QD, the transition frequency is always
chosen to be resonant with the cavity resonance, the dipole matrix element is µy=3eÅ
(since only the y-component of the fundamental mode has a field maximum in the center, the
other dipole matrix elements are not taken into account) and the phenomenological dephasing
is γphen =1ns−1. For the spatial extension of the quantum dot, a Gaussian ellipsoid in all
three dimensions is used. The radius is chosen to be fairly large with rQD = 50 nm. In [85]
it is shown, that the effect of the size of a QD in a square lattice on the radiative dephasing
is negligible for extensions larger than 30 nm. Therefore, the size of the QD is not expected
to have an noticeable influence. As initial conditions, the QD is assumed to be in the ground
state. Otherwise, the situation would be non-physical due to a lack in surrounding fields.
The computational domain is set to Vsim = 19 ×9.5√3×2a3and tsim = 10 ps. For the sake
of computational time and effort, the spatial discretization is set to be more coarse than in
the previous cases, ∆x=∆z=a/10 and ∆y=(3/4∆x. Note that this slightly changes
the spectral properties of the fundamental mode. Varying the position of the outer air hole
results in Fig. 6.19, where the change in the resonance is shown on the left, and the change
in the Q-factor is shown on the right side. The shift of the outer air holes is given in units
of the lattice constant a. Frequencies on the left are given in THz. The general behavior of
the frequency and the Q-factor remains with the coarse spatial grid. However, the resonance
of the fundamental mode is at fC= 277.4THz with a cavity decay time of TC≈0.01 ns,
6.4. A SINGLE QD COUPLED TO A PHCC 130
Figure 6.20: Left: Excitation amplitude (vertical axis) in arbitrary units over frequency in THZ
(horizontal axis). The emission intensity of the QD is color-coded (logarithmic scale). Right: Excitation
amplitude (vertical axis) in arbitrary units over time in ps (horizontal axis). The population of the
conduction band of the QD is color-coded. From top to bottom: Shift of the outer air holes: s/a =0.02
(top), s/a =0.2(middle) and s/a =0.38 (bottom). The modification maps the Q-factor. Refer to Fig.
6.19 for the shift-dependent Q-factor.
corresponding to Q= 8919 for the optimal case with an air hole shift-to-lattice constant ratio
s/a =0.2.
Figure 6.20 is built up in the following manner: from top to bottom, the Q-factor of the
cavity is gradually altered. De facto, the shift of the outer air holes maps the cavity Q-factor
according to Fig. 6.19, right, as well as the frequency of the fundamental mode, left. The outer
air holes of the cavity are shifted outwards by s/a =0.02,s/a =0.2and s/a =0.38 (from
131 6.4. A SINGLE QD COUPLED TO A PHCC
Figure 6.21: Left: Shift of the outer air holes from s/a =0.02 to s/a =0.38 (vertical axis) over
frequency in THZ (horizontal axis). The emission intensity of the QD is color-coded (logarithmic
scale). Right: Shift of the outer air holes from s/a =0.02 to s/a =0.38 (vertical axis) over time in ps
(horizontal axis). The population of the conduction band of the QD is color-coded. The modification
maps the Q-factor. Refer to Fig. 6.19 for the shift-dependent Q-factor. From top to bottom: Increasing
intensity of the exciting laser pulse. The amplitudes refer to 1,15 and 30 in Fig. 6.20.
top to bottom). On the left side, Fig. 6.20 shows the emission intensity of the QD (color-
coded) on a logarithmic scale, depending on the exciting laser pulse intensity on the vertical
axis in arbitrary units over the frequency in THz. The intensity is increased linearly from 1
to 30. On the right side, the temporal evolution of the conduction band population of the
QD (color-coded) is shown for the corresponding cases. The exciting laser pulse amplitude is
plotted on the vertical axis (in arbitrary units), while the time in ps is on the horizontal axis.
6.4. A SINGLE QD COUPLED TO A PHCC 132
Starting from the first plot for the conduction band population for a outer air hole shift of
s/a =0.02, the Q-factor is fairly low (Q= 755). For lowest intensity, the exciting laser field
is not able to invert the QD. Above, Rabi oscillations are observed. For increasing intensity,
the Rabi frequency ΩRincreases linearly due to the linear dependence on the exciting field.
The Q-factor is mapped in the population, too. Since the field decays exponentially in time in
the cavity, the Rabi oscillations are dilated. In the spectrum for the coupled system, one can
see the isolated cavity resonance at f= 279.41 THz for the lowest investigated amplitude.
Increasing the amplitude of the exciting laser pulse lets the single resonance split. Population
is built up in the QD and multiple Rabi flops are observed. Now, increasing the Q-factor of
the cavity to Q= 8919 results in a clear Mollow triplet with narrow side band peaks, similar
to Fig. 6.18 for the dressed state picture (note the resonance of the isolated, single cavity
shift f= 277.47 THz according to Fig. 6.19 due to the modification). The population of the
QD oscillates with an increased Rabi frequency ΩR. With increasing Q-factor, the side bands
of the Mollow triplets in the emission intensity of the QD become more narrow. Due to a
more narrow line width of the cavity mode, the excitation of the QD is spectrally smaller,
too, therefore the dressed states are excited more resonantly, yielding sharp side band peaks.
This also is reflected in the conduction band population of the QD, which flops with a Rabi
frequency ΩR≈3THz, equal to the spectral shift of the side band peak frequency with respect
to the main transition. Following the decrease of the Q-factor with larger shift of the outer
air holes, one finds a decreased splitting of the Mollow triplet due to the lower Q-factor of the
cavity. Also, the cavity field cannot fully invert the QD.
The situation in Fig. 6.21 is similar to Fig. 6.20. Here, the amplitude of the exciting pulse in
arbitrary units is gradually increased from top to bottom (1,15 and 30). The alignment of the
subfigures is the same like in Fig. 6.20. On the left, the emission intensity of the QD is shown
with frequencies in THz on the horizontal axis. The right side depicts the conduction band
population of the single QD (color-coded), coupled to the cavity and the horizontal axis shows
the time in ps. On the vertical axis, the shift sof the outer air holes is shown in units of the
lattice constant a. Thus, this axis indirectly maps the cavity Q-factor, according to Fig. 6.19.
Small and large shift means low Q-factors, while the optimal shift of the outer air holes is
about s/a =0.2. Generally, one sees the main transition line tracing the isolated, single PhCC
resonance in Fig. 6.19. The QD transition is resonant with the cavity mode for all cases. For an
exciting laser pulse amplitude of 10 and above, the Mollow triplet is clearly visible. Side band
peaks reduce their width with increasing Q-factor of the cavity. For increasing amplitude, the
side bands shift away linearly from the main peak, as it is expected.
Chapter 7
Conclusion and Outlook
133
134
Conclusion
Two different nano-scale sized structures are investigated within this thesis, using the Finite-
Difference Time-Domain method for numerical evaluation of the three-dimensional Maxwells
equations. Due to their different method to confine light in small spatial regions, both kinds of
resonators have certain advantages where the other has disadvantages and vice versa. However,
strong resonator-resonator interaction and also strong coupling between a single quantum dot,
which is described with dynamic equations to calculate optical polarization field, is shown.
The effects of pertubating the perfectly circular shape of a microdisk resonator is investigated
numerically, together with radius variations. Ellipticity and non-vertical side walls affect the
resonances dramatically. Modifications of the mode patterns arise, resulting in stronger cou-
pling to the environment, hence less confinement. The Q-factors are therefore affected strongly,
too. One can also make advantage from these facts. An embedded quantum dot, for example,
can be slightly detuned. Thus, by applying proper methods, the resonance can be adapted.
Also, the spatial mode profile can be altered, to let the quantum dot be placed in a posi-
tion with high field amplitude, when a proper modification is performed. Also, the coupling
between different mode orders in a microdisk resonator can be analyzed well, because the
resonance tuning using the mentioned parameters of the microdisk is comparably large. The
effect of a uniaxial anisotropic environment is calculated, too. Even when the permittivity
tensor elements exceed the permittivity of the resonator, confinement is still maintained. Ad-
ditionally, anticrossing features between different mode orders are observed, resulting from
the coupling between the different modes via the anisotropic environment. Experimental data
of a microdisk resonator with radius R=3µm in a liquid crystal environment show a clear
red-shift for all resonances in the investigated frequency range. A comparison with numerical
data, using the uniaxial anisotropy to model the LC environment, is in good agreement.
For the purpose of strong interaction between photonic crystal cavities and strong coupling
between single quantum dots and photonic crystal cavities, the optimization with the method
of gentle confinement is discussed in detail. Effects of the modification of the first two neigh-
bored air holes of a L3-type defect are investigated. The Q-factor is increased by one order of
magnitude. Experimental data of a permanent tuning mechanism of a L7-type defect by evap-
orating a thin additional layer homogeneously only on the top surface of a photonic crystal
cavity (without covering the air holes inside), are compared with numerical data. A red-shift
in frequency and drop in the Q-factor and amplitude of the resonances are nicely reproduced
by the simulations.
The coupling between microdisk resonators, photonic crystal cavities and quantum dots is in-
vestigated. For the microdisk-microdisk coupling, strongly asymmetric line and loss splitting
emerges for small gap spacings between the resonators. A tight-binding approach, equivalent
to the approach known from semiconductor physics, shows that the asymmetric line splitting
results from an overlap integral-type tight-binding parameter, which decreases drastically with
increasing gap size. The line splitting is explained by radiation patterns of the new eigenmodes
of the systems, where constructive and destructive interference yield the low and high Qmode,
respectively. The energy transfer between the resonators agrees perfectly with the line split-
ting. Strong coupling is observed for a single quantum dot coupled to a high-Qwhispering
gallery mode in a microdisk resonator. Placing the quantum dot in a high-amplitude posi-
tion in the microdisk, a symmetric line splitting of 46 GHz occurs. When the quantum dot
frequency is tuned through the microdisk resonance, repelling between the upper and lower
polariton branch is observed. A more intense focus lies on the coupling using photonic crys-
135
tal cavities with L3-type defects. Analyzing the field pattern of the fundamental mode leads
to alignment schemes for more efficient field injection between coupled cavities. The beating
frequency between the two laterally aligned cavities agrees perfectly with the line splitting.
Distant-dependent calculations are performed. It is shown, that the proposed alignment scheme
also applies for three coupled cavities. Possible construction of waveguides, based upon pho-
tonic crystal cavities is proposed to efficiently guide light along different symmetry directions
of the air hole lattice. Coupling a single quantum dot to a cavity mode in a photonic crystal
under intense excitation is investigated at the last. Q-factor dependent simulations show the
occurrence of Mollow triplets with increasing intensity of the exciting laser field.
Outlook
The recent progress in spectral mode tuning in optical resonators like microdisks and photonic
crystal cavities is already far. Permanent tuning by direct modification and dynamic mode
tuning by embedding a resonator in a liquid crystal environment have been shown. Also, the
usage of a liquid crystal resonator yields a tuning range two orders of magnitude larger than
using isotropic resonators. Finding a way for providing an anisotropic, dynamically tunable
photonic crystal cavity is a big challenge. By Combining an anisotropic resonator with an
anisotropic environment, the tuning range may be further increased. Also it may be possible
to dynamically tune the cavity decay times, due to full control of the penetration depth of the
mode patterns into the environment.
For the coupling between photonic crystal cavities and embedded quantum dots, it is also worth
to investigate the coupling between multiple quantum dots in a single photonic crystal cavity.
The knowledge of the energy transfer between the quantum dots can lead to new devices, new
technologies in nanooptics and photonics. Also, for multiple coupled photonic crystal cavities
and quantum dots in different cavities, which are spatially uncoupled but radiatively coupled
via the confined electromagnetic modes in the coupled cavity system, can reveal interesting
facts about the light-matter interaction and can give deeper insight into it, e.g. formation
of new quasi-particles. Under strong excitation, side bands of Mollow triplets can be pushed
towards resonances of other eigenmodes of the coupled system. The possibilities are merely
unlimited due to the variety of possible combinations of these resonator systems. With this, a
step towards quantum information processing with devices on a single chip can be done.
136
Appendix A
Numerical Methods
A.1 Runge-Kutta Algorithm
For the integration of the system of inhomogeneous, ordinary differential equations (ODE) of
first order for the coherence and density of the nonlinear treated quantum dot, the Runge-
Kutta algorithm of 4th is used, since it is well established and of great stability. The integration
steps will only be depicted shortly, a more detailed description can be found in [84].
Assume an inhomogeneous ODE in the form
˙y(x)=f(y(x),x).(A.1)
Basically, the algorithm transforms the eq. (A.1) into an integral equation:
y(x+∆x)=
x+∆x
;
x
f(y(ξ),ξ)dξ≈y(x) + ∆x
j
+
i=1
γif(yi(xi),x
i),(A.2)
where ∆xis the discretization. In an explicit Runge-Kutta algorithm, f(yi(xi),x
i)≈ki(y, x)
is set. In 4th those functions are given by the weightings γi:
γ1k1(y, x)=1
6f(y, x)
γ2k2(y, x)=1
3f(y+k1
2,x+∆x
2)
γ3k3(y, x)=1
3f(y+k2
2,x+∆x
2)
γ4k4(y, x)=1
6f(y+k3
2,x+∆x).(A.3)
137
A.2. LAGRANGE INTERPOLATION 138
A.2 Lagrange Interpolation
Since the Runge-Kutta integration algorithm, Sec. A.1, for the time-depending coherence and
density equations of the QD, eqs. 2.58-2.59, needs the future time steps at t+0.5∆tand
t+∆t, which are not provided by the FDTD algorithm, one needs to extrapolate these values
from earlier time steps. One method for the extrapolation, actually used for interpolation, but
also applicable for this purpose, is the Lagrange interpolation scheme, which is a polynomial
interpolation scheme. In general, one creates a polynom p(x)of order (n−1), whose ith element
equals the function, which wants to be interpolated, at the ith element.
p(xi)=
n
+
j
ljn(xi)f(xj).(A.4)
With ljn(xi)=δij, this condition is fulfilled. Hence, the coefficients ljn(xi)take the form
ljn(x)=
n
=
i=1,i*=j
x−xi
xj−xi
.(A.5)
For the extrapolation of the needed future time steps, it is enough to use a 3rd order scheme,
resulting in (for reasons of readability, the spatial indices are omitted)
En+1
2=3
8En−2−5
4En−1+15
8En
En+1 =En−2−3En−1+3En.(A.6)
Appendix B
Parameters and Constants for the
Numerical Simulations
B.1 Constants of Nature
List of the constants of nature in SI units.
Constant Symbol [Dimension] Value
Planck constant h[Js] 6.626176 ×10−34
reduced Planck constant ![Js] 1.054589 ×10−34
vacuum permeability µ0[Js2C−2m] 4π10−7
vacuum permittivity !0[C2J−1m] 8.854188 ×10−12
speed of light in vacuum c0[ms−1]2.997925 ×108
Table B.1: List of Constants of Nature.
B.2 Parameter List for the Quantum Dot
Parameter Quantum Dot Sec. 6.2 Sec. 6.4
transition frequency ωL[Hz] varying varying
dipole matrix element dµ[eÅ] 3 3
phenomenological dephasing rate γ[Hz] 0 1 ×109
background !11.56 10.595
Gaussian ellipsoidal wave function radius x[m] 30 ×10−950 ×10−9
Gaussian ellipsoidal wave function radius y[m] 30 ×10−950 ×10−9
Gaussian ellipsoidal wave function radius z[m] 30 ×10−950 ×10−9
Table B.2: List of Parameters for the Quantum Dot.
139
B.3. PARAMETER LIST FOR THE FDTD SIMULATIONS 140
B.3 Parameter List for the FDTD Simulations
Parameter MD, Chap. 4+6 PhCC, Chap. 5+6
radius [m] 361 ×10−9(3050 ×10−9) - - -
lattice constant [m] - - - a/16 (a/10)
thickness [m] 265 ×10−9(240 ×10−9)0.6×a
air hole radius ∆x[m] - - - 0.3×a
spatial grid cell size ∆x[m] 7.3529 ×10−9a/16 (a/10)
spatial grid cell size ∆y[m] 7.3529 ×10−9a/16 (a/10)
spatial grid cell size ∆z[m] 7.3529 ×10−9a/16 (a/10)
dielectric permittivity slab !slab 11.56 11.56
dielectric permittivity environment !env 1 1
simulation time tsim [s] 5×10−12 5×10−12 (1×10−11)
CPML cells 20 20
Table B.3: List of Parameters for the PhC Cavities and Microdisks.
CPML Parameter Value
αCPML, max 0.05
σCPML, max 0.8
κCPML, max 5
mCPML 3
mαCPML 1
Table B.4: List of CPML Parameters. For more information, refer to [60].
Bibliography
[1] M. Shirane, S. Kono, J. Ushida, S. Ohkouchi, N. Ikeda, Y. Sugimoto, and A. Tomita.
Mode identification of high-quality-factor single-defect nanocavities in quantum dot-
embedded photonic crystals. J. Appl. Phys., 101(073107), 2007.
[2] Y. Tang, A. M. Minatairov, J. L. Merz, V. Tokranov, and S. Oktyabrsky. Characteriza-
tion of 2d-photonic crystal nanocavities by polarisation-dependent photoluminescence.
Proceedings of 2005 5th IEEE Conference on Nanotechnology, 1:35 – 38 vol. 1, 2005.
[3] Y. Akahane, T. Asano, B.-S. Song, and S. Noda. High-q nanocavity in a two-dimensional
photonic crystal. Letters to Nature, 425:944–947, 2003.
[4] T. Yoshie, A. Scherer, J. Hendrickson, G. Kithrova, H. M. Gibbs, G. Rupper, C. Eli,
O. B. Shchekin, and D. G. Deppe. Vacuum rabi splitting with a single quantum dot in
a photonic crystal nanocavity. Letters to Nature, 432:200–204, 2004.
[5] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brian.
Quantum computers. Nature, 464(7285), 2010. Reviews.
[6] A. Grodecka, C. Weber, A. Knorr, and P. Machnikowski. Interplay and optimization of
decoherence mechanisms in the optical control of spin quantum bits implemented on a
semiconductor quantum dot. Phys. Rev. B, 73(12):125306, 2006.
[7] A. Grodecka and P. Machnikowski. Partly noiseless encoding of quantum information
in quantum dot arrays against phonon-induced pure dephasing. Phys. Rev. B, 76(12):
205305, 2007.
[8] A. Grodecka, P. Machnikowski, and J. Lacak. Reducing pure dephasing of quantum bits
by collective encoding in quantum dot arrays. J. Phys.: Conf. Ser., 30(1):41, 2006.
[9] E. M. Purcell. Spontaneous emission probabilities at radio frequencies. Physical Review,
69, 1946.
[10] M. Pelton, J. Vučković, A. Scherer, and Y. Yamamoto. Three-dimensionally confined
modes in micropost microcavities: Quality factors and pucrell factors. IEEE Journal of
Quantum Electronics, 38(2):170–177, 2002.
[11] V. S. C. Manga Rao and S. Hughes. Single quantum-dot purcell factor and βfactor in
a photonic crystal waveguide. Physical Review B, 75(205437), 2007.
[12] L. Sanchis, M. J. Cryan, J. Pozo, I. J. Craddock, and J. G. Rarity. Ultrahigh purcell
factor in photonic crystal slab microcavities. Physical Review B, 76(045118), 2007.
141
BIBLIOGRAPHY 142
[13] B.-S. Song, S. Noda, T. Asano, and Y. Akahane. Ultra-high-qphotonic double-
heterostructure nanocavity. Letters to Nature, 4:207–210, 2005.
[14] D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala. Ultra-high-qmicro-
toroid on a chip. Letters to Nature, 421:925–928, 2003.
[15] G. Kithrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer. Vacuum rabi splitting
in semiconductors. Nature Physics, 2:81–90, 2006. Review article.
[16] A. A. Abdumalikov, O. Astafiev, and Y. Nakamura. Vacuum rabi splitting due to
strong coupling of a flux qubit and a coplanar waveguide resonator. Physical Review B,
78(189502(R)), 2008.
[17] S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer. Observation of strong
coupling between a micromechanical resonator in an optical cavity field. Nature, 460,
2009.
[18] S. Gröblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab,
and M. Aspelmeyer. Demonstration of an ultracold micro-optomechanical oscillator in
a cryogenic cavity. Nature Physics, 5, 2009.
[19] J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris. Strong and
tunable nonlinear optomechanical coupling in a low-loss system. Nature Physics, 6, 2010.
[20] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D.
Whittaker, K. W. Lehnert, and R. W. Simmonds. Sideband cooling micromechanical
motion to the quantum ground state. arXiv:1103.2144v1 [quantum physics], 2011.
[21] Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong. Design of dispersive
optomechanical coupling and cooling in ultrahigh-q/v slot-type photonic crystal cavities.
Opt. Express, 18(23), 2010.
[22] Y.-G. Roh, T. Tanabe, A. Shinya, H. Taniyama, E. Kuramochi, S. Matsuo, T. Sato, and
M. Notomi. Strong optomechanical interaction in a bilayer photonic crystal. Physical
Review B, 81(121101(R)), 2010.
[23] E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and
I. Robert-Philip. Optomechanical coupling in a two-dimensional photonic crystal defect
cavity. Phys. Rev. Lett., 106(20):203902, 2011.
[24] F. R. K. Reinitzer. Beiträge zur kenntniss des cholesterins. Monatshefte für Chemie,
1888.
[25] O. Lehmann. über fliessende krystalle. Zeitschrift für Physikalische Chemie, 1889.
[26] K. A. Piegdon, S. Declair, J. Förstner, T. Meier, H. Matthias, M. Urbanski, H.-S. Kitze-
row, D. Reuter, A. D. Wieck, A. Lorke, and C. Meier. Tuning quantum-dot based
photonic devices with liquid crystals. Opt. Express, 18(8):7946–7954, 2010.
[27] M. Humar, M. Ravnik, S. Pajk, and I. Muševič. Electrically tunable liquid crystal optical
microresonators. Nature Photonics, 3(10):595 – 600, 2009.
143 BIBLIOGRAPHY
[28] Sir J. Pendry, D. Schurig, and D. R. Smith. Controlling electromagnetic fields. Science,
312(5514):1780–1782, 2006.
[29] U. Leonhardt. Optical conformal mapping. Science, 312(5781):1777–1780, 2006.
[30] A. Taflove and S. C. Hagness. Computational Electrodynamics - The Finite-Difference
Time-Domain Method, 3rd edition. Artech House, Boston, 2005. 3rd Edition.
[31] A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. John-
son. Meep: A flexible free-software package for electromagnetic simulations by the fdtd
method. Computer Physics Communications, 181:687–702, 2010.
[32] J. B. Schneider. Understanding the finite-difference time-domain method, 2010.
www.eecs.wsu.edu/~schneidj/ufdtd.
[33] L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt. Advanced finite element method
for nano-resonators. Proceeding of SPIE, 6115, 2006.
[34] K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba. Full-vectorial finite element method
in a cylindrical coordinate system for loss analysis of photonic wire bends. Opt. Express,
14(23):11128–11141, 2006.
[35] T. Weiland. A discretization method for the solution of maxwell’s equations for six-
component fields. Electronics and Communications AEUE, 31(3):116–120, 1977.
[36] T. Weiland. Time domain electromagnetic field computation with finite difference meth-
ods. International Journal of Numerical Modelling: Electronic Networks, Devices and
Fields, 9:259–319, 1996.
[37] R. Schuhmann and T. Weiland. Stability of the fdtd algorithm on nonorthogonal grids
related to the spatial interpolation scheme. IEEE Transactions on Magnetics, 34(5):
2751–2754, 1998.
[38] F. Abelès. Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans
les milieux stratifiés. Ann. Phys., 3(596):706, 1950.
[39] M. Born and E. Wolf. Principles of Optics. Cambridge, 2006. 7th Edition, 4th Printing.
[40] S. Teitler and B. W. Henvis. Refraction in stratified, anisotropic media. J. Opt. Soc.
Am., 60(6):830–834, 1970.
[41] L. Wang and S.I. Rokhlin. Stable reformulation of transfer matrix method for wave
propagation in layered anisotropic media. Ultrasonics, 39:413–424, 2001.
[42] J.S. Hesthaven and T. Warburton. Nodal high-order methods on unstructured grids: 1.
time-domain solution of maxwells equations. J. Comput. Phys., 181:186–221, 2002.
[43] J. S. Hesthaven and T. Warburton. Nodal Discontinuous Galerkin Methods: Algorithms,
Analysis, and Applications. Springer, 2007. 1st Edition.
[44] K. Stannigel, M. König, J. Niegemann, and K. Busch. Discontinuous galerkin time-
domain computations of metallic nanostructures. Opt. Express, 17:14934–14947, 2008.
BIBLIOGRAPHY 144
[45] X. Ji, T. Lu, W. Cai, and P. Zhang. Discontinuous galerkin time domain (dgtd) methods
for the study of 2-d waveguide-coupled microring resonators. J. Lightwave Technol., 23:
3864, 2010.
[46] G. Tang, R. L. Panetta, and P. Yang. Application of a discontinuous galerkin time
domain method to simulation of optical properties of dielectric particles. Appl. Opt., 49:
2827–2840, 2010.
[47] J. C. Maxwell. A dynamical theory of the electromagnetic field. Philosophical Transac-
tions of the Royal Society of London, 155:459–512, 1865.
[48] J. D. Jackson. Klassische Elektrodynamik. Gruyter Berlin, 2006. 4th Edition.
[49] D. J. Griffiths. Introduction to Electrodynamics. Pearson Benjamin Cummings, 2008.
3rd Edition.
[50] K. S. Yee. Numerical solution of initial boundary value problems involving maxwell’s
equations in isotropic media. IEEE Trans. Antennas and Propagation, 14:302–307, 1966.
[51] C. Wiebeler, 2010-2011. private communication.
[52] Y.-P. Li S.-L. Qui. Q-factor instability and its explanation in the staircased fdtd simu-
lation of high-q circular cavity. Science, 313, 2006.
[53] N. Okada and J. B. Cole. Simulation of whispering gallery modes in the mie regime
using the nonstandard finite-difference time domain algorithm. J. Opt. Soc. Am. B, 27
(4):631–639, 2010.
[54] A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos,
S. G. Johnson, and G. Burr. Improving accuracy by subpixel smoothing in fdtd. Optics
Letters, 31:2972–2974, 2006.
[55] A. F. Oskooi, C. Kottke, and S. G. Johnson. Accurate finite-difference time-domain
simulation of anisotropic media by subpixel smoothing. Optics Letters, 34:2778–2780,
2009.
[56] G. Mur. Absorbing boundary conditions for the fintite-difference approximation of the
time-domain method electromagnetic field equations. IEEE Transactions on Electro-
magnetic Compatibility, EMC-23(4):377–382, 1981.
[57] J. P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves.
Journal of Computational Physics, 114(2):195–200, 1994.
[58] Z.S. Sacks, D.M. Kingsland, R. Lee, and J.F. Lee. A perfectly matched anisotropic
absorber for use as an absorbing boundary condition. IEEE Trans. Antennas and Prop-
agation, 43:1460Ð1463, 1995.
[59] M. Kuzuoglu and R. Mittra. Frequency dependence of the constitutive parameters of
causal perfectly matched anisotropic absorbers. IEEE Microwave and Guided Wave
Letters, 6:447–449, 1996.
145 BIBLIOGRAPHY
[60] J. A. Roden and S. D. Gedney. Convolutional pml (cpml): An efficient fdtd implemen-
tation of the cfs-pml for arbitrary media. Microwave Optical Tech. Lett., 27(6):334Ð339,
2000.
[61] M. M. Nielson and I. L. Chuang. Quantum Computation and Quantum Information.
Cambridge, 2004.
[62] J. D. Joannopoulus, S. G. Johnson, J. N. Winn, and R. D. Meade. Photonic Crystals:
Molding the Flow of Light. Princeton University Press, 2008. 2nd Edition.
[63] H. Ibach and H. Lüth. Festkörperphysik. Einführung in die Grundlagen. Springer, 2002.
6th Edition (03/2002).
[64] N. W. Ashcroft and N. D. Mermin. Festkörperphysik. Oldenbourg, 2005. 2nd Edition
(03/2005).
[65] O. Madelung. Introduction to Solid-State Theory. Springer, 1996.
[66] H. Haug and S. W. Koch. Quantum Theory of the Optical and Electronic Properties of
Semiconductors. World Scientific Publishing, 2001.
[67] W. W. Chow and S. W. Koch. Semiconductor - Laser Fundamentals. Springer, 1999.
[68] T. Kuhn. Density Matrix Theory of Coherent Ultrafast Optics and Kinetics of Semicon-
ductors, pages 173–187. Chapmann Hall, 1998. edited by E. Schöll.
[69] H. Haug and A.-P. Jauho. Quantum Kinetics in Transport and Optics of Semiconductors.
Springer, 1996.
[70] C. Ell. Microscopic Theory of Semiconductors, chapter II. World Scientific, 1995.
[71] P. Y. Yu and M. Cardona. Fundamentals of Semiconductors. Springer, 1998.
[72] J. Sajeev. Strong localization of photons in certain disordered dielectric superlattices.
Phys. Rev. Lett., 58(23):2486–2489, 1987.
[73] E. Yablonovitch. Inhibited spontaneous emission in solid-state physics and electronics.
Phys. Rev. Lett., 58:2059, 1987.
[74] J. D. Joannopoulos, P. R. Villeneuve, and S. Fan. Photonic crystals. Solid State Com-
munications, 102(2-3):165 – 173, 1997. Highlights in Condensed Matter Physics and
Materials Science.
[75] J.-M. Lourtioz, H. B., V. Berger, J.-M. Gerard, D. Maystre, and A. Tchelnokov. Photonic
Crystals: Towards Nanoscale Photonic Devices. Springer, 2008. 2nd Edition, Translated
by M. P. de Fornel.
[76] J. Heebner, R. Grover, and T. A. Ibrahim. Optical Microresonators: Theory, Fabrication
and Applications. Springer, 2008.
[77] S. Brandt and H. D. Dahmen. Elektrodynamik: Eine Einführung in Experiment und
Theorie. Springer, 2005. 4th Edition.
BIBLIOGRAPHY 146
[78] S. Blume. Theorie Elektromagnetischer Felder. Hüthig, 1991. 7th Edition, edited by
Karl-Heinrich Wittich.
[79] M. Bayindir, B Temelkuran, and E. Özbay. Tight-binding description of the coupled
defect modes in three-dimensional photonic crystals. Phys. Rev. Lett., 84(10):2140–2143,
2000.
[80] D. Leuenberger, R. Ferrini, and R. Houdré. Ab initio tight-binding approach to photonic-
crystal based coupled cavity waveguides. J. Appl. Phys., 95(3):806–809, 2004.
[81] C. Meier, 2009-2011. private communication.
[82] S. G. Johnson and J. D. Joannopoulos. Block-iterative frequency-domain methods for
maxwell’s equations in a planewave basis. Opt. Express, 8(3):173–190, 2001.
[83] Q. Song, H. Cao, S. T. Ho, and G. S. Solomon. Near-ir subwavelength microdisk lasers.
Appl. Phys. Lett., 94(061109), 2009.
[84] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. O. Flannery. Numerical Recipes
- The Art of Scientific Computing. Cambridge, 2007. 3rd Edition.
[85] K. Zimmermann. Zur theorie der elektromagnetischen kopplung in periodischen
quantenpunkt-strukturen. Diploma thesis, Technische Universität Berlin, 2003.
[86] R. Eichmann. Lineare optische Eigenschaften dielektrisch strukturierter Halbleiter. PhD
thesis, Phillipps-Universität Marburg, 2002. PhD thesis.
[87] M. O. Scully and M. S. Zubairy. Quantum Optics. Cambridge, 2006. 5th Printing.
[88] C. C. Gerry and P. L. Knight. Introductory Quantum Optics. Cambridge, 2006.
[89] J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and
A. Blais. Ultrastrong coupling regime of cavity qed with phase-biased flux qubits. Phys-
ical Review A, 80(032109), 2009.
[90] T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-
Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross. Circuit quantum
electrodynamics in the ultrastrong coupling regime. Nature Physics, 6:772–776, 2010.
[91] G. Günter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato,
C. Cuiti, A. Tredicucci, A. Leitenstorfer, and R. Huber. Sub-cycle switch-on of ultra-
strong light-matter interaction. Nature, 458(12):178–181, 2009.
[92] D. Ehmer. Mode calculations for one-dimensional photonic resonators. Bachelor thesis,
Universität Paderborn, 2010.
[93] Q. Chen and D. W. E. Allsopp. One-dimensional coupled cavities photonic crystal filters
with tapered bragg mirrors. Optics Communications, 281(23):5771–5774, 2008.
[94] D. Ehmer, 2010-2011. private communication.
[95] A. Lorenz. Switchable Waveguiding in Photonic Liquid Crystal Structures. PhD thesis,
Universität Paderborn, 2010. PhD thesis.
147 BIBLIOGRAPHY
[96] T. Ergin, N. Stenger, P. Brenner, Sir J. Pendry, and M. Wegener. Three-dimensional
invisibility cloak at optical wavelengths. Science, 328(5976):337–339, 2010.
[97] I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev. Anisotropic
metamaterials emulated by tapered waveguides: Application to optical cloaking. Phys.
Rev. Lett., 102(21):213901, 2009.
[98] G. W. Grey, K. J. Harrison, and J. A. Nash. New family of nematic liquid crystals for
displays. Electronic Letters, 9(6), 1973.
[99] S. Inui, N. Ilmura, T. Suzuki, H. Iwane, K. Miyachi, Y. Takanishi, and A. Fukuda.
Thresholdless antiferroelectricity in liquid crystals and its application to displays. J.
Mater. Chem., 6(4), 1996.
[100] J. Li and S.-T. Wu. Extended cauchy equations for the refractive indices of liquid
crystals. J. Appl. Phys., 95(3):896–901, 2004.
[101] I.-C. Khoo and S.-T. Wu. Optics and nonlinear optics of liquid crystals, page 68. World
Scientific Publishing Co. Pte. Ltd., 1993.
[102] R. Chen and H. Guo. Efficient calculation of matrix elements in low storage filter
diagonalization. J. Chem. Phys., 111(2):464–471, 1999.
[103] M. R. Wall and D. Neuhauser. Extraction, through filter-diagonalization, of general
quantum eigenvalues or classical normal mode frequencies from a small number of
residues or a short-time segment of a signal. i. theory and application to a quantum-
dynamics model. J. Chem. Phys., 102(20):8011–8022, 1995.
[104] V. A. Mandelshtam. On harmonic inversion of cross-correlation functions by the filter
diagonalization method. J. Theoretical and Computational Chemistry, 2(4):497–505,
2003.
[105] V. A. Mandelshtam and H. S. Taylor. Harmonic inversion of time signals. J. Chem.
Phys., 107(17):6756–6769, 2003. See also erratum, ibid, vol. 109, no. 10, p. 4128 (1998).
[106] S. G. Johnson and M. Frigo. Implementing FFTs in practice. In C. Sidney Burrus,
editor, Fast Fourier Transforms, chapter 11. Connexions, 2008.
[107] A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki. Review
of applications of whispering-gallery mode resonators in photonics and nonlinear optics.
IPN Progress Report 42-162, 2005.
[108] B. R. Johnson, M. D. Reed, A. A. Houck, D. I. Schuster, Lev S. Bishop, E. Ginossar,
J. M. Gambetta, L. DiCarlo, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf. Quantum
non-demolition detection of single microwave photons in a circuit. Nature, 6(9):663–667,
2010.
[109] Y.-P. Huang and P. Kumar. Interaction-free all-optical switching in ξ(2) microdisks for
quantum applications. Optics Letters, 35(14):2376–2378, 2010.
[110] M. Soltani, S. Yegnanarayanan, and A. Adibi. Ultra-high q planar silicon microdisk
resonators for chip-scale silicon photonics. Optics Express, 15(8):4694–4704, 2007.
BIBLIOGRAPHY 148
[111] K. A. Piegdon, M. Offer, A. Lorke, M. Urbanski, A. Hoischen, H.-S. Kitzerow, S. Declair,
J. Förstner, T. Meier, D. Reuter, A. D. Wieck, and C. Meier. Self-assembled quantum
dots in a liquid-crystal-tunable microdisk resonator. Physica E: Low-dimensional Sys-
tems and Nanostructures, 42(10):2552–2555, 2010.
[112] K. A. Piegdon, 2009-2011. private communication.
[113] Lord Rayleigh. The problem of the whispering gallery. Cambridge University, Cambridge
UK, 5:617, 1912.
[114] M. Benyoucef, J. B. Shim, J. Wiersig, and O. G. Schmidt. Quality-factor enhancement
of supermodes in coupled microdisks. Optics Letters, 36(8):1317–1319, 2011.
[115] S. Declair, C. Meier, T. Meier, and J. Förstner. Anticrossing of whispering gallery modes
in microdisk resonators embedded in an anisotropic environment. Phot. Nano. Fund.
Appl., 8(4):273 – 277, 2010.
[116] C. Meier. Novel Photonic Materials. Habilitation, Fakultät für Physik, Universität
Duisburg-Essen, 2007.
[117] J. P. Reithmaier, G. S¸ek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V.
Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel. Strong coupling in a single
quantum dot-semiconductor microcavity. Nature, 432(141108):200–204, 2004.
[118] J. Wolters, A. W. Schell, G. Kewes, N. Nüsse, M. Schoengen, H. Döscher, T. Hannappel,
B. Löche, M. Barth, and O. Benson. Enhancement of the zero phonon line emission from
a single nitrogen vacancy center in a nanodiamond via coupling to a photonic crystal
cavity. Applied Physics Letters, 97(141108), 2010.
[119] S. Noda, A. Chutinan, and M. Imada. Trapping and emission of photons by a single
defect in a photonic bandgap structure. Letters to Nature, 407:608–610, 2008.
[120] Y. Tanaka, J. Upham, T. Nagashima, T. Sugiya, T. Asano, and S. Noda. Dynamic
control of the q-factor in a photonic crystal nanocavity. Letters to Nature, 6:862–865,
2007.
[121] T. Uesugi, B.-S. Song, T. Asano, and S. Noda. Investigation of optical nonlinearities in
an ultra-high-qsi nanocavityin a two-dimensional photonic crystal slab. Opt. Express,
14(1):377–386, 2006.
[122] NSM. Nsm archive - optical properties of semiconductors. URL
http://www.ioffe.rssi.ru/SVA/NSM/Semicond/GaAs/optic.html.
[123] Y. Akahane, T. Asano, B.-S. Song, and S. Noda. Fine-tuned high-q photonic crystal
nanocavity based on a gentle confinement of light. pages 882 –883, 2004.
[124] H. Altug, D. Englund, and J. Vučković. Ultrafast photonic crystal nanocavity laser.
Nature, 2(7):484–488, 2006.
[125] D. O’Brien, M. D Settle, T. Karle amd A. Michaeli, M. Salib, and T. F. Krauss. Coupled
photonic crystal heterostructure nanocavities. Opt. Express, 15(3):1228–1233, 2007.
149 BIBLIOGRAPHY
[126] S. Declair, T. Meier, and J. Förstner. Numerical investigation of the coupling between
microdisk modes and quantum dots. Physica Status Solidi C, 8(4):1254–1257, 2011.
[127] K. Srinivasan, M. Borselli, and O. Painter. Cavity q, mode volume, and lasing threshold
in small diameter algaas microdisks with embedded quantum dots. Opt. Express, 14(3):
1094–1105, 2006.
[128] K. A. Atlasov, K. F. Karlsson, A. R., B. D., and E. Kapon. Wavelength and loss
splitting in directly coupled photonic-crystal defect microcavities. Opt. Express, 16(20):
16255–16264, 2008.
[129] P. Strasser, G. Stark, F. Robin, D. Erni, K. Rauscher, R. Wüest, and H. Jäckel. Op-
timization of a 60◦waveguide bend in a inp-based 2d planar photonic crystal. J. Opt.
Soc. Am. B, 25(1):67–73, 2008.
[130] D. Pergande and R. B. Wehrspohn. Polarization-dependent transmission properties of
iosoi w1 photonic crystal waveguide. Phot. Nano. Fund. Appl., In Press, Corrected
Proof:–, 2011. doi: DOI:10.1016/j.photonics.2011.02.001.
[131] E. Dulkeith, S. J. McNab, and Y. A. Vlasov. Mapping the optical properties of slab-type
two-dimensional photonic crystal waveguides. Phys. Rev. B, 72(3):115102, 2005.
[132] J. Kasprzak, S. Reitzenstein, E. A. Muljarov, C. Kistner, C. Schneider, M. Strauss,
S. Höfling, A. Forchel, and W. Langbein. Up on the janes-cummings ladder of a quantum-
dot/microcavity system. Nature Materials, 9:304–308, 2010.
[133] R. E. Grove, F. Y. Wu, and S. Ezekiel. Measurement of the spectrum of resonance
fluorescence from a two-level atom in an intense monochromatic field. Phys. Rev. A, 15
(1):227–233, 1977.
[134] M. Florescu and S. John. Single-atom switching in photonic crystals. Phys. Rev. A, 64
(3):033801, 2001.
Danksagung
Jeder Frau und jedem Mann sei gedankt, die mich in den letzten drei Jahren auf dem Weg zu
dieser Arbeit begleitet und geführt haben.
Besonderen Dank möchte ich meinem Doktorvater Dr. Jens Förstner aussprechen. Als sein
erster Promotionsstudent war ich sicherlich kein leichter Schüler, der oft mit kleinen und auch
großen Fragen und Problemen zu ihm kam. Weiterhin möchte ich Prof. Dr. Torsten Meier
danken, der mich während meiner Arbeit immer unterstützt hat und als Zweitgutachter in
kurzer Zeit viel zu lesen hatte.
Meinen Arbeitskollegen und der gesammten Arbeitsgruppe sowie der anderen Arbeitsgruppen
der theoretischen und experimentellen Physik an der Universität Paderborn gilt ebenfalls
besonderer Dank für die früchtetragenden Diskussionen und Unterhaltungen, Kaffeepausen
und lustige Stammtischabende. Hervorzuheben sind mein Büronachbar Herr Dr. Matthias
Reichelt und der mittlerweile in der Unendlichkeit der freien Marktwirtschaft entschwundene,
ehemalige Kollegen Herr Dr. Christian Thierfelder, der mit seinem überragenden Wissen immer
einen Rat wußte.
Desweiteren möchte ich meiner Familie danken, die mich in allen Lebenslagen unterstützt
hat und immer die richtigen Ratschläge zur richtigen Zeit hatten. Insbesondere möchte ich
meiner Mutter danken, die Dank mir in den letzten drei Jahren eine Menge grauer Haare mehr
bekommen hat.
Besonderer Dank gebührt meiner Freundin Ania, die es in den letzten Monaten nicht leicht
mit mir hatte, mir aber trotz alledem immer hilfreich zur Seite stand.
Nicht vergessen möchte ich meine Freunde, die ich in der ganzen Zeit nur selten besuchen
konnte und dadurch teilweise der Draht zueinander verloren ging. Besonders hervorzuheben
ist Herr Hagen Schäfer, der mich trotz der großen Entfernung häufig in Paderborn besucht
und immer ein offenes Ohr hat.
Weiterer Dank gebührt der Deutschen Forschungsgemeinschaft DFG, die mir im Rahmen des
Graduiertenkollegs GRK 1464 Micro- and Nanostructures in Optoelectronics and Photonics
die Anfertigung dieser Arbeit erst ermöglicht hat.
