The k-space formulation of the scattering problem in the time domain

Abstract

The arbitrary direct scattering problem is solved numerically in closed form in the time domain and spatial Fourier transform space. This solution consists of casting the general basic global laws (i.e., the second-order partial differential wave equation or its integral representation) as a local algebraic equation in the spatial Fourier transform space, and leaving the specific local constitutive equations (i.e., the algebraic boundary conditions, which specify a given structure, which are conventionally imposed on the differential or integral representation of the general basic global wave equation) as a local algebraic equation in real space, thereby reducing the scattering problem to a statement of two simultaneous local algebraic equations in two unknowns (the fields and the induced sources) in two spaces connected by the spatial Fourier transform. A temporally local representation in both spaces is obtained with the aid of an introduced auxiliary field and two propagators. By virtue of causality, a numerically efficient closed form solution to this set of equations is obtained that utilizes the fast Fourier transform algorithm as the transformations between the two spaces. By virtue of the numerically efficient fast Fourier transform algorithm and the local algebraic representations, the number of required complex multiply-add operations and storage allocation is of the order of N log2 N and N per temporal discretization, respectively (where N is the number of spatial cells into which the scattering problem is descretized). It is shown that the solution is only of the order of log2 N slower than an ideal solution. The solution is thus practical for very large one-, two-, and three-dimensional scattering problems. Numerico-experimental results for a variety of refrective index profiles, including perfect reflectors, are presented, thus verifying the presented algorithm.