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A Unifying Formulation of Material Dispersion in Finite-Difference Time-Domain Simulations of Optical Problems

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Abstract

Methods of solving Maxwell’s equations on a discrete grid in space and time show promise in predicting the time-evolution of ultrashort pulse propagation in optical devices [1,2]. Finite-Difference Time-Domain (FDTD) algorithms developed for microwave problems [3,4] discretize Maxwell’s curl equations and consider the permittivity ε and permeability μ of a material to be constant. Over the bandwidth of short optical pulses, the material dispersion cannot be ignored. Recently, various authors [5-11] have attempted to model this dispersion by adding finite-difference approximations of the constitutive equations to the algorithm. Two competing approaches with very different motivations have emerged; comparisons between them have been in the form of lengthy numerical simulations. We formulate the problem of modeling dispersive materials as a filter design problem in signal processing. The formalism of discrete-time linear systems then enables us to examine the difference equation as an approximation to the desired continuous filter, and to equate the design of an optimal filter to the solution of a constrained minimization problem. We show that the previously used methods of approximating dispersion are analogous to the matched and bilinear approximation methods of discrete filter design theory. We compare the exact frequency response of existing discretizations to the desired response for Debye and Lorentzian media.

© 1993 Optical Society of America

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