Optimization of photonic band structures [Elektronische Ressource] / von Markus Richter

Markus Richter

Sujets

• Mathematics

Informations

• Publié par: karlsruher_institut_fur_technologie
• Publié le: 01 janvier 2010
• Nombre de lectures: 20
• Langue: English
• Poids de l'ouvrage: 9 Mo

Extrait

Optimization of
Photonic Band Structures
Zur Erlangung des akademischen Grades eines
DOKTORS DER NATURWISSENSCHAFTEN
von der Fakultat fur Mathematik des
Karlsruher Instituts fur Technologie (KIT)
genehmigte
DISSERTATION
von
Dipl.{Math. techn. Markus Richter
aus Frankfurt am Main
Tag der mundlichen Prufung: 10. November 2010
Referent: Prof. Dr. Willy Dor er
Koreferent: Prof. Dr. Christian WienersPreface
This dissertation is the result of a research project I conducted at the Institute for
Applied Mathematics and Numerical Analysis of the Karlsruhe Institute of Tech-
nology (KIT) in the years 2006 to 2010. During that time I worked as a research
assistant at the institute under the supervision of Prof. Dr. Willy Dor er. The
project was embedded within the research program of the DFG Research Training
Group 1294 \Analysis, Simulation and Design of Nanotechnological Processes" at
the Department of Mathematics.
At this point I would like to take the opportunity to thank a number of people,
who supported me during my research. First and foremost I would like to thank
my advisors, Prof. Dr. Willy Dor er and Prof. Dr. Christian Wieners, for giving
me the opportunity to work on an interesting and challenging problem, and for
guiding me along my way towards this dissertation. I am particularly thankful for
the many times I was able to discuss my ideas with them, or to seek their advice.
I would also like to mention Prof. Wieners’ personal e orts in helping me with his
numerical software library M++.
Furthermore, I would like to thank my colleagues at the institute and my
fellow members of the Research Training Group 1294 for the good collaboration
and for the many interesting discussions. I am especially indebted to Dr. Alexander
Bulovyatov, who implemented some numerical algorithms, on which I could base
my own ones. I also would like to mention Branimir Anic, Markus Burg, Dr.
Thomas Dohnal, Dr. Christian Engstrom, Markus Feist, Dr. Thomas Gauss, Dr.
Vu Hoang, Florian Keller, and Daniel Maurer. A special thanks goes to Benjamin
Exner, Andre Fuetterer, Dr. Desiree Hilbring, and Andreas Kuhn for proofreading
parts of this dissertation.
I thank my family for their constant encouragement and support. I would
also like to thank my closest friends for their company and for the many hours
spent talking and laughing together. Finally, I thank Melanie for her kind, loving
support.
Markus Richter
Karlsruhe, October 2010
2Contents
1 Introduction 6
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Aims of This Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Preliminaries 13
2.1 Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Local Lipschitz Continuity . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 The Mathematical Model 21
3.1 Modelling Photonic Crystals . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Crystal Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Wave Propagation in Linear Dielectrics . . . . . . . . . . . . . . . . 30
3.4 Bloch Modes in Periodic Media . . . . . . . . . . . . . . . . . . . . 37
3.5 The Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . 40
4 Spectral Theory 45
4.1 The Formal Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Sobolev Spaces of Periodic Functions . . . . . . . . . . . . . . . . . 46
4.3 The Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Riesz{Schauder Theory . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5 Auchmuty’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 Photonic Band Structures and Band Diagrams . . . . . . . . . . . . 77
4.7 The Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . 84
5 Photonic Band Structure Optimization 90
5.1 The Formal Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
34 CONTENTS
5.2 Regularity of the Goal Functionals . . . . . . . . . . . . . . . . . . 92
5.3 Existence of Optima in the TM Setting . . . . . . . . . . . . . . . . 96
5.4 Existence of Optima in Other Settings . . . . . . . . . . . . . . . . 103
5.5 Optimization Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6 Nonsmooth Analysis 110
6.1 Generalized Di erentials . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2 Generalized Di erential Calculus . . . . . . . . . . . . . . . . . . . 116
6.3 Dierentiability of the Gap Width Functionals . . . . . . . . . . . . 120
6.4 The Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . 125
7 A Generalized Gradient Method 127
7.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.3 Incorporating Optimization Constraints . . . . . . . . . . . . . . . . 133
7.4 Preserving Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.5 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.6 Choosing Descent Directions . . . . . . . . . . . . . . . . . . . . . . 140
7.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8 A Level-Set Method 148
8.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.3 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9 Numerical Results 158
9.1 Maximizing TM Band Gaps . . . . . . . . . . . . . . . . . . . . . . 158
9.1.1 Results of the Generalized Gradient Method . . . . . . . . . 159
9.1.2 Results of the Level Set Method . . . . . . . . . . . . . . . . 163
9.2 Maximizing TE Band Gaps . . . . . . . . . . . . . . . . . . . . . . 168
9.2.1 Results of the Generalized Gradient Method . . . . . . . . . 168
9.2.2 Results of the Level Set Method . . . . . . . . . . . . . . . . 171
9.3 Maximizing Band Gaps of a 3D Photonic Crystal . . . . . . . . . . 173
10 Summary, Conclusions and Outlook 176
10.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
10.2 Comparison of the Optimization Algorithms . . . . . . . . . . . . . 178
10.3 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
10.4 Final Remarks and Outlook . . . . . . . . . . . . . . . . . . . . . . 179CONTENTS 5
A A FEM Toolbox for MATLAB 180
A.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
A.2 Data Structures and Algorithms . . . . . . . . . . . . . . . . . . . . 182
A.3 Some Customized Features . . . . . . . . . . . . . . . . . . . . . . . 185
A.4 Implementation Examples . . . . . . . . . . . . . . . . . . . . . . . 186
Frequently Used Symbols 191
About The Author 192
Bibliography 194Chapter 1
Introduction
1.1 Motivation
Photonic crystals are materials, which are composed of two or more di erent di-
electrics or metals, and which exhibit a spatially periodic structure, typically at the
length scale of hundred nanometers. Depending on whether the periodicity extends
into one, two or three space dimensions, a photonic crystal is called one-, two-,
three-dimensional. Photonic crystals can be fabricated using nano-technological
processes such as photolithography or vertical deposition methods. They also oc-
cur in nature, e.g. in the microscopic structure of certain bird feathers, buttery
wings, or beetle shells (see e.g. [12], [46]).
A characteristic feature of photonic crystals is that they strongly a ect the
propagation of light waves at certain optical frequencies. This is due to the fact that
the optical density inside a photonic crystal varies periodically on the length scale
of about 400 to 800 nanometers. One nds that the so-called optical wavelengths
of light waves lie in precisely the same length scale. Light waves that penetrate a
photonic crystal, are therefore subject to periodic, multiple di raction, which leads
to coherent wave interference inside the crystal. Depending on the frequency of
the incident light wave this interference can either be constructive or destructive.
In the latter case the light wave is not able to propagate inside the photonic crystal
at all. Typically, this phenomenon only occurs for a bounded range of optical wave
frequencies, if it does occur at all. Such a range of inhibited wave frequencies is
called a photonic band gap. Light waves with frequencies inside a photonic band
gap are totally re ected by the photonic crystal. It is this e ect, which causes e.g.
to the iridescend colors of peacock feathers (see [54]).
Whether or not photonic band gaps occur, strongly depends on the geometric
structure of the photonic crystal, as well as on the contrast in optical density be-
tween the di erent materials the photonic crystal is built