Transient excitation of a layered dielectric medium by a pulsed electric dipole: spectral representation

Spectral methods are the obvious choice for modeling the transient excitation of a continuously layered, plane-stratified dielectric halfspace. Such methods typically involve an inverse spatial Fourier transformation and the evaluation of the constituents. In this paper, we consider the spectral representation. The idea is to normalize the spatial wavenumber with respect to frequency. Compared with the Cagniard-De Hoop method, our approach is different in the sense that we keep the frequency real, and allow the time variable to become complex. In this respect, our work also resembles she spectral theory of transients. We restrict the temporal Fourier inversion to nonnegative frequencies by expressing the time-domain signal as the real part of a dual analytic signal. Reversing the order of the temporal and spatial Fourier inversions then leads to the so-called time-domain Weyl representation for the reflected field. In this representation, accumulated guidedwave poles give rise to an additional branch cut. The representation thus obtained is used to derive a suitable combination of Gaussian quadrature rules for the evaluation of the spectral integral