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Topics in Material Science: Mathematics of Waveguides
Louisiana State University
Spring Semester, 2012
Prof. Stephen Shipman
Place: Lockett 132
Time: 1:40-3:00 Tuesday and Thursday
Office: Room 314 of Lockett Hall
Office Hours: Monday and Wednesday 9:30-11:30 or by appointment
Problems of the guiding of electromagnetic, acoustic, and elastic waves by material structures pervade mathematical physics, and their analysis penetrates a rich swath of classical and modern mathematics. This course will concentrate on a number of specific problems for which techniques in complex and functional analysis will be developed and applied.
(1) A waveguide containing an obstacle.
The spectral theory for self-adjoint operators in Hilbert space is realized in a concrete way in this problem. The continuous spectrum is associated with extended fields--those resulting from the scattering of a guided wave by the obstacle in the guide. The eigenvalues are frequencies of trapped modes--those fields whose energy is exponentially trapped around the obstacle. Eigenvalues embedded in the continuous spectrum are unstable with respect to perturbations of the structure. This instability causes anomalous scattering behavior, which we will analyze by means of complex-analytic perturbation techniques.
(2) A bifurcated waveguide. Exact solutions for waves in split waveguides can be obtained by sophisticated methods of complex variables called Wiener-Hopf techniques. They involve Tauberian and Abelian theorems of Fourier analysis.
(3) Periodically layered waveguides. The monodromy matrix, which transfers field data across one period, is the fundamental object of analysis. Through it, one develops the spectral theory for periodic media, known as the Floquet theory, which is the Fourier transform of the subgroup of translations in the real line. The salient phenomenon is the existence of stop-bands--frequency intervals in which the coherent scattering by the periodic structure prohibits the propagation of waves. Localized defects admit trapped modes with frequency in a stop-band.
(4) Nonlinear waveguides. High resonant amplification of fields makes the study of nonlinearities necessary. We will see how even weak nonlinearity has pronounced effects, such as bistability, in the presence of resonance.
The prerequisite is a good graduate analysis course and basic complex variables.
Waveguide with an obstacle--acoustic and Maxwell.
Weak formulation with sources contained in a bounded set; the radiation condition.
The scattering problem and its spectral theory: Scattering states and bound states (trapped modes).
Formulation in a truncated domain; accessing embedded eigenvalues.
Unitary groups of operators, Stein's theorem, and application to the waveguide problem.
Analyticity in frequency and other parameters.
The scattering matrix; Scattering data, as the transmission coefficient.
Complex "dispersion" relation between frequency and a material parameter.
Analysis of the scattering matrix near the dispersion relation.
Unstable trapped modes (with resp. to the material parameter).
The transmission coefficient near an unstable mode; anomalies.
Some open problems.
A bifurcated waveguide.
Wiener-Hopf techniques in complex variables: Matching of analytic functions in the UHP and LHP.
Fourier analysis and Tauberian/Abelian theorems.
Mode matching: Infinite matrix equations.
The Floquet theory and monodromy matrices.
Spectral theory for periodic media; pass bands and stop bands.
Trapped modes in locally defective periodic media with frequency in a stop band.
The nonlinear Schrödinger equation and coupling of modes.
Simple models with nonlinearity only in the scatterer.
Weak nonlinearity and weak coupling --> pronounced nonlinear effects.
We will not use any one source as a text book. My lectures will draw from my notes and several references.
There is a bibliography of relevant works below, with links to some PDF files of excerpts.
I will assign problems periodically. At the end of the course, I may give a longer and somewhat comprehensive set of problems.
Students may discuss problems with each other and other people (including me, of course) and consult other literature; in fact students are encouraged to search the literature and discuss ideas. However, all work that is turned in must ultimately be that of the submitter alone. If a student receives aid on an assigned problem from discussions with people or other sources, he or she must begin from scratch in writing the solution so that the result is the product of his or her own understanding alone.
Evaluation of performance in the course is based on performance on the assignments
Lecture Notes and Problem Sets
Notes 1: Functions of bounded variation
Notes 2: Integral representation theorems in complex variables
Notes 3: Unbounded self-adjoint operators in Hilbert space
Notes 4: The spectral theorem for self-adjoint operators
Notes 5: The spectral theorem and the Fourier transform
Notes 6: Spectral theory for the wave equation in 1D
Notes 7: Scattering by an obstacle in a string: Generalized Fourier integrals
Notes 8: Compact operators; eigenvalues for a bounded domain
Notes 9: Scattering by an obstacle in a cylindrical waveguide
Notes 10: A nonlinear eigenvalue problem for trapped modes
Notes 11: Nonlinear Schrödinger Equation
Problems 1-3 Due Jan. 31
Problems 4-6 Due Feb. 9
Problems 7-10 Due Feb. 23
Problem 11 Due Mar. 8
Problems 12-15 Due Apr. 3
Problems 16-17 Due May 3
Last Assignment: Redo any previous assignment. Due May 9
Michael Reed and Barry Simon,
Methods of Modern Mathematics:
Vol. I Functional Analysis,
Vol. II Fourier Analysis and Self-Adjointness,
Vol. III Scattering Theory,
Vol. IV Analysis of Operators,
excerpts from Vol. I (spectral theorem),
Vol. II (self-adjoint extensions),
Academic Press, 1980.
N. I. Akhiezer and I. M. Glazman,
Theory of Linear Operators in Hilbert Space,
Dover, 1993, replication of the edition of F. Ungar Publishing, NY, 1961, 1963.
A. N. Kolmogorov and S. V. Fomin,
Introductory Real Analysis, §36: The Stieljes integral, 1968 (translation by Silverman, Dover 1975).
Gerald B. Folland,
Introduction to Partial Differential Equations,
Second Edition, Princeton University Press, 1995.
Excerpt: Chapter 0.
G. B. Whitham,
Linear and Nonlinear Waves,