Three dimensional finite difference time domain simulations of photonic crystals [Elektronische Ressource] / von Christian Hermann

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• Physik

Informations

• Publié par: universitat_stuttgart
• Publié le: 01 janvier 2004
• Nombre de lectures: 98
• Langue: English
• Poids de l'ouvrage: 4 Mo

Extrait

Three Dimensional Finite-Di erence
Time-Domain-Simulations of Photonic Crystals
Von der Fakult at fur Mathematik und Physik der Universit at Stuttgart
zur Erlangung der Wurde eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigte Abhandlung
von
Christian Hermann
aus Mullheim
Hauptberichter: Prof. Dr. O. Hess
Mitberichter: Prof. Dr. H. Herrmann
Tag der mundlic hen Prufung: 18. Juni 2004
Theoretische Quantenelektronik
Institut fur Technische Physik, DLR
Pfa en waldring 38-40
D-70569 Stuttgart
2004Acknowledgements
First of all I want to thank Prof. Dr. Ortwin Hess for his support and the possibility
to work in his group. Many thanks also for providing me with excellent facilities and
infrastructure.
I also would like to thank Prof. Dr. Hans Herrmann and Prof. Dr. Tilman Pfau for
agreeing to be an examiner.
Many thanks go to the research groups in Stuttgart and Guildford:
Dr. Stefan Scholz for introducing me into the eld and laying the foundation of
the FDTD-codes.
Achim Hamm, Andreas Klaedtke and Christian Simmendinger for their patience
while solving my computer problems, in particular Andreas Klaedtke for imple-
menting the PML-boundaries.
Dr. Edeltraud Gehrig, Dr. Dietmar Preisser and Klaus B ohringer for many helpful
discussions.
Heiko Pittner, Hui Bian, Kosmas Tsakmakidis and Durga Aryal for bearing my
supervisions.
... and all of them for the excellent working atmosphere.
Special thanks I would like to express to Cecile Jamois from Max-Planck-Institute Halle
for an excellent cooperation and many inspiring suggestions and discussions.
Thanks to my parents for making it possible for me to achieve my aims.
And, last but not least, many thanks to my family, Joschua and Monika, for all your
love and support.
34Contents
1 Summary 7
2 Introduction and Overview 11
3 Principles of Photonic Crystals 15
3.1 Electromagnetism of Periodic Dielectrics . . . . . . . . . . . . . . . . . . 15
3.1.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.2 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.3 The Concept of Brillouin-Zone and Bandstructure . . . . . . . . . 19
3.1.4 The Bandgap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.5 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.6 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 One-Dimensional Photonic Crystals: The Bragg Mirror . . . . . . . . . . 25
3.3 Two-Dimensional Crystals . . . . . . . . . . . . . . . . . . . . . 28
3.3.1 Basic Structures and Properties . . . . . . . . . . . . . . . . . . . 29
3.4 Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.1 Point Defects in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.2 Point- and Line-Defects in 2D . . . . . . . . . . . . . . . . . . . . 44
3.5 Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Photonic Crystal Slab Structures 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Principles of Slab Waveguides . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Slab Modes and Lightcone . . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 Backfolding of Slab Modes . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Interaction of In-Plane Periodicity and Vertical Structure. . . . . . . . . 59
4.3.1 Air-Bridge Structure . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.2 In nite Low-Index Cladding . . . . . . . . . . . . . . . . . . . . . 70
4.3.3 Finite Lo . . . . . . . . . . . . . . . . . . . . . . 73
4.3.4 In uence of Substrate . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5Contents
5 Spontaneous Emission in Inverted Opals 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Weak and Strong Coupling Regime . . . . . . . . . . . . . . . . . . . . . 93
5.3 Classical and Quantum Description Within the Weak Coupling Regime . 94
5.3.1 Quantum Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3.2 Classicalh . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 Single Dielectric Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5 Inverted Opals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.5.1 Geometrical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.5.2 Bandstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.6 Spontaneous Emission Rate . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.6.1 Space and Polarisation Dependance . . . . . . . . . . . . . . . . . 104
5.6.2 In uence of Finite Size . . . . . . . . . . . . . . . . . . . . . . . . 108
5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 The FDTD-Method and its Application in Photonic Crystal Analysis 113
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Basic Components and Methods . . . . . . . . . . . . . . . . . . . . . . . 113
6.2.1 The Time-Stepping Algorithm and the Yee-Cell . . . . . . . . . . 114
6.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.4 Data Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3 Application to Photonic Crystals . . . . . . . . . . . . . . . . . . . . . . 132
6.3.1 Bandstructure and Eigenmode Calculations . . . . . . . . . . . . 132
6.3.2 Transmission and Re ection of Partially Periodic Systems . . . . 144
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7 Conclusions and Outlook 151
Bibliography 155
61 Summary
In this work the optical properties of three dimensional photonic crystal structures are
studied theoretically by means of a full vectorial three dimensional nite-di erence time-
domain (FDTD) numerical simulation. First, the principles of lower dimensional pho-
tonic crystal structures are comprehensively discussed to assess the underlying physical
principles of eigenmode and bandgap formation. For consistency, results are obtained
on the basis of the FDTD-method. Then these principles are generalised to two di eren t
classes of three dimensional photonic crystal systems. First, we address the technolog-
ically important case of photonic crystal slab structures, i.e. two dimensional crystal
patterns in vertically nite layer structures, where the localisation of the electromag-
netic elds perpendicular to the crystal plane is achieved by total internal re ection.
Starting with simple air-bridge structures the systems studied are extended to more com-
plex vertical structures, like patterned insulator-on-silicon-on-insulator (IOSOI) cladding
structures mounted on a substrate. The main focus thereby is laid on the quantitative
description of e ects that are related to the coupling of the index-guided localised pho-
tonic crystal modes to the environment. Next to the technologically extremely important
calculation of radiation losses, it is, in particular, the discovery of cladding modes (i.e.
modes that have no exponential decaying behaviour in the claddings) for frequencies
in the bandgap region that is important and pioneering for the understanding of these
systems and their potential application in integrated optics. Second, we focus on the
fundamental physical properties of three dimensional inverted opal structures. Within
the weak coupling regime, we analyse the strong frequency, polarisation and space de-
pendance of the spontaneous emission rate (which is equivalent to the local density of
states) of a dipole source embedded within our inverted opal in a crystallite of nite size.
Thereby, strong enhancement e ects are demonstrated close to dielectric interfaces for
certain frequencies due to the continuity conditions of the electric eld modes. These
results are essential for the interpretation of luminescence experiments, because in real
systems there is always an inhomogenous distribution of emitters and an approximation
based on a total density of states becomes invalid. Finally, the basic building blocks
of the FDTD-code like core algorithm, boundary conditions and data extraction are
discussed in detail. Moreover, special adaptations of these components with respect to
photonic crystal analysis are discussed and examplary results for the dependance on
internal simulation parameters on the results are shown.
71 Summary
8Zusammenfassung
In der vorliegenden Arbeit werden die optischen Eigenschaften von dreidimensionalen
photonischen Kristallen theoretisch mit Hilfe von voll-vektoriellen nite-di erence time-
domain (FDTD)-Simulationen untersucht. Zun achst werden die Prinzipien von niederdi-
mensionalen photonischen Kristall-Systemen diskutiert, insbesondere um die zugrun-
deliegenden physikalischen Mechanismen zu kl aren, die fur das Zustandekommen von
Eigenmoden und Bandluc ken verantwortlich sind. Die zugeh origen exemplarischen Ergeb-
nisse werden mit Hilfe der FDTD-Methode berechnet. An