Consistent Prony’s Approximation for FDTD Modeling of Dispersive Media Characterized by Completely Monotone Susceptibilities

A finite-difference time-domain (FDTD) scheme for simulating wave propagation inside Cole–Cole, Davidson–Cole, Havriliak–Negami, and Raicu dispersive media is proposed. The polarization relations of these media are fractional integro-differential equations and their FDTD modeling exhibits significant difficulties. However, a common property of these media is the fact that their time-domain susceptibility functions are completely monotone. By exploiting this significant property, the approximation of the susceptibility by means of sums of decaying exponentials is proposed. In particular, the approximants are derived by Prony’s method. In contrast to previously reported studies, which also adopt Prony’s method, the required consistency of the derived exponential terms is considered. Actually, it is shown that the parameters of the Prony approximants should fulfill specific restrictions, which are equivalent to approximating the media by sums of Debye terms only. The proposed FDTD scheme is based on discretizing the convolutional polarization relation by use of recurrence formulas, while the singularity of the original susceptibilities at the time origin is appropriately circumvented. Comparisons between analytical and FDTD results for the reflection and transmission coefficients as well as the transfer functions inside the dispersive media illustrate the efficiency of the proposed scheme, over a wideband frequency range.

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