Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

In certain layered electromagnetic media, one can construct a waveguide that supports a harmonic electromagnetic field at a frequency that is embedded in the continuous spectrum. When the structure is perturbed, this embedded eigenvalue moves into the complex plane and becomes a “complex resonance” frequency. The real and imaginary parts of this complex frequency have physical meaning. They lie behind anomalous scattering behaviors known collectively as “Fano resonance”, and people are interested in tuning them to specific values in optical devices. The mathematics involves spectral theory and analytic perturbation theory and is well understood [16], at least on a theoretical level for deterministic (fixed coefficient) media.

This dissertation is a study of how random fabrication errors in the waveguide structure affect this kind of resonance. The governing equations are the free harmonic Maxwell equations of electromagnetics, which reduce to a system of ODEs in layered media. The material coefficients (dielectric permittivity ϵ(z) and magnetic permeability μ(z)) are considered to be random variables depending only on the z-variable, with small variance σ(z), and we are interested in how the real and imaginary parts of the complex resonance ω=ω^+iω behave as random variables depending on these coefficients.

The first main theorem of this thesis states that, if ϵ(z) and μ(z) are stationary random variables with respect to the variable z, then the variances of ω^ and ωˇ can be computed by considering only events in which ϵ and μ are random variables that are constant in z.

The significance of this theorem is that it allows one to compute the mean and variance of ω, to leading order in σ, by means of (deterministic) sensitivity analysis of ω as a function of ϵ and μ. One can consider ω to be a complex-analytic function of ϵ (and μ) if ϵ=ϵ^+iϵˇ, where both parts are Hermitian matrices. The Cauchy-Riemann equations, together with known properties of complex resonances allow one to make interesting and useful conclusions about how the frequency of a resonance and the amount of radiation losses depend on the real and imaginary (lossy) parts of the material coefficients.

Date

11-13-2017

Committee Chair

Shipman, Stephen P.

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