Date of Award

Fall 2004

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Gregory A. Kriegsmann

Second Advisor

Demetrius T. Papageorgiou

Third Advisor

John Charles Hensel

Fourth Advisor

David C. Stickler

Fifth Advisor

Jonathan H.C. Luke

Abstract

Using a scattering matrix approach we analyze and study the scattering and transmission of waves through a two-dimensional photonic crystal which consists of a periodic array of parallel rods with circular cross sections. Without making any assumptions about normal incidence, single mode propagation, and sufficient inter-scatter separation in the direction of propagation, we show how to compute the transmission and reflection coefficients of these periodic structures. The method is based on the computation of a generalized scattering matrix for one column of the periodic structure.

We also develop an analytical method to analyze and to study the scattering and transmission of waves through a two-dimensional photonic crystal which consists of a periodic array of parallel metallic rods with rectangular cross sections. The method is based on the computation of generalized scattering matrices for several parts of the periodic entire structure, and their composition to form the scattering matrix for the structure. We derive an explicit formula for the reflection and transmission coefficients when we take into account only one propagating mode in a specific portion of the periodic structure.

Finally, we develop an analytical method to analyze and to study Rayleigh-Bloch surface waves propagating along a two-dimensional diffraction grating which again consists of a periodic array of rods with rectangular cross sections. The method is based on mode matching. By taking into account all propagating and only a finite number of evanescent modes in a specific portion of the waveguide we show that the surface waves correspond to the zeros of the determinant of a Hermitian matrix.

Included in

Mathematics Commons

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