Fast time domain integral equation solver for dispersive media with auxiliary Green functions

We describe elements and representative applications of a fast time domain integral equation solver based on the AIM type spatial matrix compression, applicable to problems involving interaction of wide-band pulses with 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 to (i) the Fourier transform of the dispersive medium Green function, (ii) the Fourier transform of the product of the dispersive medium Green function with the frequency dependent dielectric permittivity, and, (iii) 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 discuss details 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. The full Galerkin discretization is used with RGW spatial basis functions and band limited temporal basis functions. Our formulation constitutes the basis for accurate numerical simulation framework for a variety of problems involving wideband pulse propagation, including wideband antenna pattern simulation or propagation of narrow pulses through dispersive media.