FDTD Schemes for Wave Propagation in Davidson-Cole Dispersive Media Using Auxiliary Differential Equations

Two finite-difference time-domain (FDTD) schemes based on the use of auxiliary differential equations are proposed for modeling time-domain wave propagation in dispersive Davidson-Cole media. The main difficulty in FDTD implementations for such media is the appearance of fractional derivatives in the time-domain polarization relation. To circumvent this difficulty, the relative complex permittivity of the medium is approximated by Padé approximants resulting in auxiliary differential equations of integer order. Moreover, it is proven that under certain conditions the Padé approximation is equivalent to one given by a sum of Debye terms. Hence, an alternative set of auxiliary differential equations is derived. Over a wideband frequency domain, the comparisons between analytical and calculated values of the relative complex permittivity as well as of the transfer function inside the dispersive medium illustrate the efficiency of both FDTD schemes.