Analysis of the Phase Function and Its Saddle Points

Abstract

In preparation for the asymptotic analysis of the integral representation given in either equation (4.3.31) or (4.3.28) for the propagated field in a temporally dispersive medium, it is necessary to first determine the topography of the real part X(ω),θ) of the complex phase function φ(ω,θ), defined in (4.37), in the complex ω-plane. In particular, the location of the saddle points of φ(ω, θ), the value of φ( ω, θ) at these points, and the regions of the complex ω-plane wherein X((ω, θ) is less than the value of X(ω, θ) at the dominant saddle point for a given value of θ are all required. This chapter is devoted to that purpose. The analysis presented here is involved because the behavior of the complex-valued phase function is complicated. Specifically, for a Lorentz medium with a single-resonance frequency, φ(ω,θ) possesses four saddle points and four branch points, and its topography evolves with the space-time parameter θ = ct/z in a complicated manner.