High-Frequency Directed Wave Propagators: A Path Integral Derivation

In this paper, we derive an approximate high-frequency Green's function (propagator) for directed waves scattered in an inhomogeneous medium and governed by a standard parabolic wave equation. The analysis is based on a path integral formalism incorporating the ray concept into an approximate description of the diffraction effects. Although this propagator was obtained earlier by using asymptotic expansions applied to the analysis of an appropriate differential equation, the procedure based on the path integral technique is conceptually simpler, and also clarifies the physical nature of the approximations performed in the derivation of the final result. It is shown that the propagator obtained belongs to a family of well known straight-line approximations. All of them can be presented in the form of ordinary or paired Fresnel transforms of a complex exponential (coined here a “Radon hologram”), which encodes the scattering potential of the object. Applications of the high-frequency propagators to solving both direct and inverse problems of wave scattering are briefly discussed.