ELECTROMAGNETIC WAVE PROPAGATION IN ALMOST PERIODIC MEDIA Thesis by Alan R

186 Pages·2012·18.84 MB·English

ELECTROMAGNETIC WAVE PROPAGATION
IN ALMOST PERIODIC MEDIA
Thesis by
Alan R. Mickelson
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
1978
i i
ACKNOWLEDGEMENTS
For my scientific development during the early stages of my
research, I owe much to my original advisor, Professor Nicholas George.
While in Professor George•s optics group, I often worked with Dr. David
MacQuigg, a friend and colleague from whom I learned much. During this
period and following this period up to the present time, I would like to
acknowledge my friend and co-worker, Dr. de Logi, for stimulating
~~alter
discussions and collaboration on various projects.
This present 1t10rk stemmed from an idea of my present advisor,
Professor C. H. Papas, whose encouragement and assistance were instrumental
in all stages of this research. I also acknowledge Dr. Dwight L. Jaggard
with whom I worked closely on various aspects of this work.
The style and accuracy of this work improved markedly upon careful
proofreadings by my advisor, Dr. Jaggard, Dr. de Logi, and Randy Bartman.
The reference lists could never have been so complete without the expert
assistance of Electrical Engineering librarian Paula Samazan. Addi
tional references were obtained from the private library of Dr. Crockett
L. Grabbe.
The draft of this work was typed by my wife Susan, v1ho
,~ough
really deserves to be acknowledged for more than typing. The expert
typing of the final draft was a product of Ruth Stratton, Susan Dycus
and Pat Neill.
My financial support while at Caltech has come from Caltech, the
Air Force Office of Scientific Research and the National Science Founda
tion.
iii
ABSTRACT
The problem of electromagnetic wave propagation in almost periodic
media is investigated and a solution is obtained directly from Maxwell's
equations. Techniques to evaluate this solution are developed. These
techniques involve a generalization to almost periodic media of the
Brillouin diagram of periodic media. The method of invariant imbedding
is applied to the coupled mode equations which determine the Brillouin
diagram for the purpose of transforming them to coupled Riccati equations.
These coupled Riccati equations, when subjected to a single boundary con
dition, determine the solutions to both the periodic and alnost periodic
boundary value problems. These evaluation techniques are used to place
in evidence similarities and differences of wave propagation in periodic
and clmost periodic media. It is shown that although the periodic and
almost periodic theories agree in many cases of interest,there exist
cases in which distinct differences appear. In cases of multi-tone per
turbations, the almost periodic theory yields both simpler and more rea
sonable results than the periodic theory.
iv
TABLE OF CONTENTS
Page
I. Introduction 1-19
A. Statement of Purpose 1
B. The Theory of Almost Periodic Functions 1
1. Historical perspective 1
2. Important concepts involved in almost periodic
function theory 3
C. Why Bother to Study Wave Propagation in Almost
Periodic Media? 12
D. Preview of the Following Material 15
Bibliography 17
II. The General Theory of Wave Propagation in Almost
Periodic Structures 20-42
A. Longitudinal Media 20
Bibliography 41
III. Brillouin Diagrams in Almost Periodic Media 43-103
A. Basic Concepts 43
B. Brillouin Diagrams in Periodic Media - Information
Contained 50
1. Historical perspective 50
2. Use as a predictor 51
3. The Brillouin diagram as an ordering device 57
C. The Brillouin Diagram in Almost Periodic Media 66
D. The 6K + 0 Limit 78
v
TABLE OF CONTENTS
(continued)
Page
1. Need for a new perturbation theory 78
2. Almost periodic perturbation theory 83
3. Periodic perturbation theory 87
Bibliography 102
IV. The Reflection Coefficient l 04-151
A. Standard Perturbation Theory for the Reflection
Coefficient 104
1. Setting up the exact boundary value problem 104
2. Born-Neumann series 107
3. Secularity of Born-Neumann series at resonance 108
B. The Electromagnetic Riccati Equation 109
l. Invariant imbedding- historical perspective 109
2. Stokes• recursion relation and conservation
of energy 110
3. The electromagnetic Riccati equation 115
C. A Generalized Look at Invariant Imbedding 117
l. First order wave equations from particle
counting 117
2. The energy conservation relations 120
3. Invariant imbedding of coupled equations 123
D. Riccati Equations, Brillouin Diagrams and
Dispersion Matrices 130
1. Single harmonic perturbation 130
vi
TABLE OF CONTENTS
(continued)
Page
2. Almost periodic 0 solution 131
~K +
3. The periodic media small 6K limit 143
Bibliography 149
v.
Conclusions 152-159
A. Relationship of the Brillouin Diagram to the
to the Reflection Coefficient 152
B. How Periodic and Almost Periodic Media Differ 152
1. Phase relations 153
2. Band coalescence regime 153
C. The Basic Nature of Almost Periodic Structures 153
1. Multi-tone almost periodic perturbations 153
2. Almost periodic ensembles and stochastic
ensembles 154
D. Possible Research Directions 156
1. Extensions to more general geometries 156
2. Modeling of random perturbations 157
3. Extensions to active media- stability criteria 157
4. Experimental verification of the theory 157
Bibliography 159
APPENDIX A 160-167
Bibliography 167
APPENDIX B 168-171
APPENDIX C 172-175
vii
TABLE OF CONTENTS
(continued)
Page
Bibliography 175
APPENDIX 0 176-177
APPENDIX E 178-180
l.
Chapter I
Introduction
A. Statement of Purpose
To begin, we wish to enphasize that this report is an
atte~pt
to unify an abstract mathe!71atical theory with \;ave propagation
theory and not an attempt to solve a specific engineering problem.
!Je feel that the results of this report could become applicable to a
number of specific problems, but at the present tine we are not able
to predict in areas this work will becor.1e important. The in
~·thich
tent here is to present a theory of vJave propagation in this nev:ly
considered medium and to compare and contrast the electromagnetic
properties of this medium with the properties of better understood
media. In the process of the investigation, we intend to re-express
certain abstract mathematical concepts in engineering terminology as
well as to develop new analytical techniques with which to treat
wave-propagation phenomena. 'le hope the report achieves its purpose
and, moreover, provides an interesting treatise to the reader.
He v:i 11 novt proceed to give tile reader \"that fee 1 is the
lf!e
necessary rnather:1atical history and ba.ckground.
B. The Theory of /\lmost Periodic Functions
1. Historical rerspective
The theory of almost periodic functions is a rather recent
development. Although the theory of Fourier series was essentially
2.
completed by the 1820's with the work of Fourier, Poisson, Cauchy
anJ numerous others as footnoted in Hhittaker and Hatson's trea-
1
tise, it vias a century later v1hen Harald Bohr~t. -,~ found t::e alr.10st
periodic gene·ral ization of Fouder series, heavily on the
borrm·dn~
5
quasi-periodic theory of Boh 1 Llr and Esclangon . Inspired by Bohr's
6 8 9 10 11
set of review papers - , Besicovitch , Bochner and Favard found
various simplifications of the theory soon after its inception. In
12 13
1926 and 1927 , Favard extended t he theory by considering linear
differential equations with almost periodic coefficients. (He later
14
unified his research into a book on the topic.) Other authors took
a different direction from Favard, concentrating their efforts more
on lifting restrictions on the class of almost periodic functions.
15
Besicovitch was the first to publish a generalized theory, follm'led
16 17 18
by Stepanoff , P.esicovitch and Bohr , Linfoot and ~~iener19. Al
20
though Bohr was the first to publ ish a book on the topic, he was
21
closely followed by Besicovitch.
Following this opening era of almost per·iodic theo:--y of the
1920's and early 1930's, there was a three-decade period of little
activity in the field. f-Jowever, \·Jith the davm of the 1960's there
came a deluge of papers and books on the topic. Characteristic of
this period of rene\\•ed interest in the theory is Nicholas t1inorsky's
chapter on almost periodic solutions in his book on non-linear oscil
lations22. His interest in the theory lay in the relationship
betv.Jeen almost periodic solut·ions and system stability, a common
3.
reason for renewed interest in the topic. To completely review the
developnents in this field in the last decade and a half is beyond
the scope of this report as is evidenced by the length of the ex
tensive, but not exhaustive, bibioaraphy in fJ• • tl. Fink's set of
23
lecture notes on almost periodic differential equations . One
author's is of importance in t•elation to our study. That v1ould
~JOrk
24-25 . "d d h
be the two papers of Jacob Abel who f1rst cons1 ere t e almost
periodic Mathieu equation, the same equation to be considered in this
work. Abel ran across the equation while ennaginq in a study of
stability of elastic systems.
2. Important Concepts Involved in Almost Periodic Function Theory
tie begin this section Hith a short qualitative discussion of
almost periodic functions before jumping to the more abstract ele-
ments of the theory. Perhaps the most direct way of obtaining a
feeling for almost periodic functions is to compare theM Hith their
close relatives (actually a direct subclass), the periodic functions.
Consider the periodic function
fp(z) =COS TIZ +COS 3TIZ +COS 5TIZ (1)
and an almost periodic counterpart
1 7 5/3
cos TIZ + cos 3 _.- TIZ + cos D TIZ (2)
13
These functions are plotted in figure (1).
For the first few periods of (1), the functions don't seem to