Date of Award

Spring 2011

Document Type

Dissertation

Degree Name

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

Department

Mathematical Sciences

First Advisor

Peter G. Petropoulos

Second Advisor

Gregory A. Kriegsmann

Third Advisor

Jonathan H.C. Luke

Fourth Advisor

Shidong Jiang

Fifth Advisor

Gerald Martin Whitman

Abstract

This dissertation addresses electromagnetic pulse propagation through anomalously dispersive dielectric media. The Havriliak-Negami (H-N) and Cole-Cole (C-C) models capture the non-exponential nature of such dielectric relaxation phenomena, which is manifest in a variety of dispersive dielectric media. In the C-C model, the dielectric polarization is coupled to the time-dependent Maxwell's equations by a fractional differential equation involving the electric field. In the H-N case, a more general pseudo-fractional differential operator describes the polarization.

The development and analysis of a robust method for implementing such models is presented, with an emphasis on algorithmic efficiency. Separate numerical schemes are presented for C-C and H-N media. A straightforward numerical implementation of these models using finite-difference time-domain (FD-TD) techniques is expected to be second order accurate in both space and time. However due to the singular nature of the kernels appearing in the fractional convolution operators, the standard C-C implementation, produces first order accuracy in time. As we show, this method is equivalent to most approaches presented in the current literature, which implies that they are also first order. The desired accuracy is instead achieved by applying multistep methods to the fractional differential equation; however multistep methods are unnecessary in the H-N implementation to preserve the accuracy. Furthermore, the C-C model is a specific case of the H-N model and can therefore be constructed using the latter of these approaches.

The FD-TD methods are validated by evaluating the electric field for the signaling problem, using numerical quadrature to evaluate the integral form of the solution. This is accomplished using the Green's function of the dispersive medium; in addition, the behavior of pulse propagation is studied asymptotically using the Green's function, which further validates the observed results of the numerical experiments.

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Mathematics Commons

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