Beam-series representation and the parabolic approximation: the frequency domain

Abstract

The frequency-domain wave equation is shown to be asymptotically replaceable by a discrete collection of parabolic equations formulated along ray-centered coordinates. This approximate formulation replaces the original boundary-value problem with a considerably simpler set of initial-value problems whose exact solutions are the well-understood Gaussian beams. The basic question addressed herein pertains to the methodology by which the initial field conditions, specified on a prescribed surface, can be represented by elementary beam constituents. An exact procedure based on the Gabor representation is proposed, leading to the so-called beam-series expansion. The beam series constitutes a two-dimensional superposition of appropriately weighted Gaussian beams. A simple and numerically efficient method for evaluating the expansion coefficients, the beam spectrum, is a direct by-product of the proposed scheme. Three distinct advantages are notable. First, since the proposed representation is intrinsically discrete, no further discretization is required in the implementation stage. Second, the elementary beam-field components are nonsingular. Thus foci and caustic regions do not require special attention. Third, the truncation rules of the beam series are well established and sharply defined. The basic beam-series expansion is extended herein to encompass curved initial surfaces, basic Gaussian beam components whose waists are freely located (not necessarily confined to the initial surface), and inhomogeneous media. The relationship between the Gabor and Gauss–Hermite expansions is also discussed.