On the Mathematics of Slow Light

211 Pages·2011·1.18 MB·English

UNIVERSITY OF CALIFORNIA,
IRVINE
On the Mathematics of Slow Light
DISSERTATION
submitted in partial satisfaction of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
in Mathematics
by
Aaron Thomas Welters
Dissertation Committee:
Professor Aleksandr Figotin, Chair
Professor Svetlana Jitomirskaya
Professor Abel Klein
2011
Chapter 2 ⃝c 2011 Society for Industrial and Applied Mathematics. Reprinted with
permission. All rights reserved.
All other materials ⃝c 2011 Aaron Thomas Welters.
DEDICATION
To Jayme
ii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS v
CURRICULUM VITAE vi
ABSTRACT OF THE DISSERTATION ix
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Electrodynamics of Lossless One-Dimensional Photonic Crystals . . . . . . . 3
1.2.1 Time-Harmonic Maxwell’s Equations . . . . . . . . . . . . . . . . . . 3
1.2.2 Lossless 1-D Photonic Crystals . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Maxwell’s Equations as Canonical Equations . . . . . . . . . . . . . . 4
1.2.4 Definition of Slow Light . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1
2 Perturbation Analysis of Degenerate Eigenvalues from a Jordan block 12
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The Generic Condition and its Consequences . . . . . . . . . . . . . . . . . . 22
2.3 Explicit Recursive Formulas for Calculating the Perturbed Spectrum . . . . 23
2.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Spectral Perturbation Theory for Holomorphic Matrix Functions 46
3.1 On Holomorphic Matrix-Valued Functions of One Variable . . . . . . . . . . 46
3.1.1 Local Spectral Theory of Holomorphic Matrix Functions . . . . . . . 50
3.2 On Holomorphic Matrix-Valued Functions of Two Variables . . . . . . . . . 63
3.3 On the Perturbation Theory for Holomorphic Matrix Functions . . . . . . . 68
3.3.1 Eigenvalue Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.2 Eigenvector Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.3 Analytic Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . 79
1
The contents of this chapter also appear in [61]. Copyright ⃝c 2011 Society for Industrial and Applied
Mathematics. Reprinted with permission. All rights reserved.
iii
4 Canonical Equations: A Model for Studying Slow Light 93
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.1.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Canonical ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.2 Energy Flux, Energy Density, and their Averages . . . . . . . . . . . 102
4.2.3 On Points of Definite Type for Canonical ODEs . . . . . . . . . . . . 103
4.2.4 Perturbation Theory for Canonical ODEs . . . . . . . . . . . . . . . . 105
4.3 Canonical DAEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3.1 The Correspondence between Canonical DAEs and Canonical ODEs . 108
4.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.5 Auxiliary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Bibliography 196
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ACKNOWLEDGMENTS
I would like to begin by thanking my advisor, Professor Aleksandr Figotin. His insight,
guidance, and support have been invaluable. The efforts he has put in helping me develop as
a mathematician and a researcher went above and beyond the usual duties of an advisor. The
depth and breadth of his knowledge in mathematics and electromagnetism and his passion
for research is a continual source of inspiration for me. He has profoundly influenced my life
and my research and for all of this I am eternally grateful.
To the members of my thesis and advancement committees, Professor Svetlana Jitomirskaya,
Professor Abel Klein, Professor Ozdal Boyraz, and Dr. Ilya Vitebskiy, I would like to thank
them for their time and thoughtful input.
Thank you also to Adam Larios, Jeff Matayoshi, and Mike Rael, for many useful and enjoy-
able conversations.
A special thanks to Donna McConnell, Siran Kousherian, Tricia Le, Radmila Milosavljevic,
and Casey Sakasegawa, and the other staff at the UCI math department, for handling the
massive amount of paperwork and administrative details that allow me the time to concen-
trate on research. Their hard work behind the scenes helps make the UCI Mathematics
Department the excellent place that it is.
I would also like to thank my family. My parents, Tom and Kathy Welters, and my brothers,
Matt and Andy Welters, for being great role models on how to work hard and achieve
while still balancing ones life with respect to family, fun, and career and for instilled in
me the importance and joys of family. Thank you so much for all the love, support, and
encouragement over the years. I would like to thank my son, Ashton Welters, for making my
life happier than I thought possible. I would especially like to thank my wife and best friend,
Jayme Welters. I could not have made it without her encouragement, support, commitment,
friendship, love, and laughter. Thank you all that you have done for me over the years, it is
appreciated more than I can possibly say in words.
The work on the explicit recursive formulas in the spectral perturbation analysis of a Jordan
block was first published in SIAM Journal of Matrix Analysis and Applications, Volume 32,
no. 1, (2011) 1–22 and permission to use the copyrighted material in this thesis has been
granted.
I am thankful to the anonymous referees for helpful suggestions and insightful comments on
parts of this research. I am thankful for the University of California, Irvine, where this work
was completed.
This work was supported in part by the Air Force Office of Scientific Research (AFOSR)
under the grant FA9550-08-1-0103.
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CURRICULUM VITAE
Aaron Thomas Welters
PARTICULARS
EDUCATION
University of California Irvine Irvine, CA
Ph. D. in Mathematics June 2011
Advisor: Dr. Aleksandr Figotin
St. Cloud State University St. Cloud, MN
B. A. in Mathematics May 2004
Magna Cum Laude
Advisor: Dr. J. -P. Jeff Chen
RESEARCH INTERESTS
My research interests are in mathematical physics, material science, electromagnetics, wave
propagation in periodic media (e.g., photonic crystals), spectral and scattering theory. I spe-
cialize in the spectral theory of periodic differential operators, nonlinear eigenvalue problems,
perturbation theory for non-self-adjoint matrices and operators, linear differential-algebraic
equations (DAEs), block operator matrices, boundary value problems, and meromorphic
Fredholm-valued operators. I apply the mathematical methods from these areas to study
spectral and scattering problems of electromagnetic waves in complex and periodic struc-
tures.
ACADEMIC HONORS
∙ Von Neumann Award for Outstanding Performance as a Graduate Student, UC Irvine, 2010
∙ Mathematics Department Scholarship, SCSU, 2003
PUBLICATIONS
2. A. Welters, Perturbation Analysis of Slow Waves for Periodic Differential-Algebraic of Defi-
nite Type. (in preparation)
1. A. Welters, On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan
Block, SIAM J. Matrix Anal. Appl., 32.1 (2011), pp. 1–22.
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TALKS
9. UCLA PDE Seminar, University of California, UCLA, CA, February 2011.
8. 2011 AMS/MAA Joint Mathematics Meetings, New Orleans, LA, January 2011.
7. MGSC (Mathematics Graduate Student Colloquium), University of California, Irvine, CA,
November 2010.
6. UCI Mathematical Physics Seminar, University of California, Irvine, CA, November 2010.
5. MAA Southern California-Nevada Section Fall 2010 Meeting, University of California, Irvine,
CA, October 2010.
4. Arizona School of Analysis with Applications, University of Arizona, Tucson, AZ, March
2010.
3. 2010 AMS/MAA Joint Mathematics Meetings, San Francisco, CA, January 2010.
2. UCI Mathematical Physics Seminar, University of California, Irvine, CA, November 2009.
1. SCSU Mathematics Colloquium, St. Cloud State University, St. Cloud, MN, November 2009.
RESEARCH EXPERIENCE
∙ Graduate Student Researcher (GSR), UCI, September, 2007 - April, 2011. The GSR
was supported by the AFOSR grant FA9550-08-1-0103 entitled: High-Q Photonic-Crystal
Cavities for Light Amplification.
TEACHING EXPERIENCE
TEACHING ASSISTANT
∙ Math2D: Multivariable Calculus. University of California, Irvine, Summer 2006
∙ Math184: History of Mathematics. University of California, Irvine, Spring 2006
∙ Math146: Fourier Analysis. University of California, Irvine, Spring 2006
∙ Math2B: Single Variable Calculus. University of California, Irvine, Fall 2005
∙ Math2B: Single Variable Calculus. University of California, Irvine, Summer 2005
∙ Math2A: Single Variable Calculus. University of California, Irvine, Summer 2005
∙ Math2J: Infinite Series and Linear Algebra. University of California, Irvine, Spring
2005
∙ Math2B: Single Variable Calculus. University of California, Irvine, Winter 2005
∙ Math2A: Single Variable Calculus. University of California, Irvine, Fall 2004
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INSTITUTIONAL SERVICE
∙ Co-Organizer. “Math Graduate Student Colloquium,” University of California, Irvine;
2010-2011
∙ Graduate Recruitment Speaker. “Mathematics Graduate Recruitment Day Event,”
University of California, Irvine; April 2010.
COMPUTATIONAL SKILLS
Programming: C/C++, MATLAB, and Maple.
Markup Languages: LaTeX and Beamer.
Operating Systems: Linux/Unix based systems and Windows.
viii
ABSTRACT OF THE DISSERTATION
On the Mathematics of Slow Light
By
Aaron Thomas Welters
Doctor of Philosophy in Mathematics
University of California, Irvine, 2011
Professor Aleksandr Figotin, Chair
This thesis develops a mathematical framework based on spectral perturbation theory for
the analysis of slow light and the slow wave regime for lossless one-dimensional photonic
crystals.
Electrodynamics of lossless one-dimensional photonic crystals incorporating general bian-
isotropic and dispersive materials are considered. The time-harmonic Maxwell’s equations
governing the electromagnetic wave propagation through such periodic structures is shown
to reduce to a canonical system of period differential-algebraic equations (DAEs) depending
holomorphically on frequency which we call Maxwell’s DAEs. In this context, a definition
of slow light is given.
We give a detailed perturbation analysis for degenerate eigenvalues of non-self-adjoint matri-
ces. A generic condition is considered and its consequences are studied. We prove the generic
condition implies the degenerate eigenvalue of the unperturbed matrix under consideration
has a single Jordan block in its Jordan normal form corresponding to that eigenvalue. We
find explicit recursive formulas to calculate the perturbation expansions of the splitting eigen-
values and their eigenvectors. The coefficients up to the second order for these expansions
are given.
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