Advanced Saddle-Point Methods for the Asymptotic Evaluation of Single Contour Integrals

Abstract

The integral representation developed in the previous chapters is an exact, formal solution to the problem of electromagnetic pulse propagation in homogeneous, isotropic, temporally dispersive media filling all space. However, an exact analytic evaluation of the resulting integral is not possible for realistic initial pulse shapes. Consequently, an approximate evaluation of the integral representation for a given initial pulse envelope is necessary in order to determine the behavior of the phenomena of primary interest, e.g., the properties of the precursor fields, the arrival of the main signal, and the evolution of the pulse. There are two possible approaches to accomplish this approximate evaluation. The first is a direct numerical evaluation of the integral representation; such a numerical integration is difficult to accurately do, however and can be done for only one initial pulse envelope and carrier frequency and one set of medium parameters at a time. Since the dependance of the propagation characteristics on these parameters is complicated, such an approach would require many calculations in order to obtain a general knowledge of dispersive pulse-propagation phenomena. The alternate approach is an asymptotic analysis of the integral representation for any given initial pulse envelop. This approach is also difficult to accomplish, but it yields analytic approximations for the propagated field that display clearly all of the basic features of the propagation phenomena as a function of the medium parameters and the applied signal frequency.