Fast Time Domain Integral Equation Solver for Dispersive Media

Abstract

We describe elements and representative applications of a new time domain integral equation solver applicable, in particular, to problems involving interaction of wide-band pulses with dispersive media.

We discuss our new analytical formulation of integral equations specially tailored to problems involving dispersive media. The formulation is both general and significantly simpler than the conventional approaches: instead of using the customary integral equation operators involving the Green function and its derivatives, we construct effective integral equation operators equal (i) to the Fourier transform of the dispersive medium Green function, (ii) to the Fourier transform of the product of the dispersive medium Green function with the frequency dependent dielectric permittivity, and, (iii) to the Fourier transform of the product of the dispersive medium Green function with the inverse of the dielectric permittivity. An important benefit of such an approach is that the resulting integrals involve only single (and not double) time convolutions. The formulation is applicable to systems involving bulk dispersive regions and thin dispersive sheets represented as interfaces.

We present results of complete analytical calculations and of corresponding numerical procedures for the evaluation of matrix elements of the integral operators, executed in the framework of the full Galerkin scheme in space and time variables, for the “conductive Debye medium” (i.e., for a medium with the electric permittivity given by the Debye formula supplemented with a term responsible for the medium conductivity). The procedure employs a suitable contour integration around singularities of the effective Green function operators in the complex frequency plane.