Topics in Material Science: Mathematics of Waveguides

Math 7384-1
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
Telephone: 225/578-1674
Email: shipman@math.lsu.edu
Office Hours: Monday and Wednesday 9:30-11:30 or by appointment

Course Description

(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.

Prerequisite

Course Topics

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. Amplitude enhancement.

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.

Periodic waveguides.

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.

Nonlinear waveguides.

The nonlinear Schrödinger equation and coupling of modes. Second-harmonic generation. Bistability.

Simple models with nonlinearity only in the scatterer. Weak nonlinearity and weak coupling --> pronounced nonlinear effects.

Literature

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.

Assignments

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

Evaluation of performance in the course is based on performance on the assignments

Lecture Notes and Problem Sets

Bibliography

1. 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.
2. 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.
3. A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, §36: The Stieljes integral, 1968 (translation by Silverman, Dover 1975).
4. Gerald B. Folland, Introduction to Partial Differential Equations, Second Edition, Princeton University Press, 1995. Excerpt: Chapter 0.
5. G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1973.