Diffraction by Arrays of Complex Source Point Beams

Abstract

Scattering by an object depends not only on the shape of the object but also on the source of the incident field. Most analytical solutions (eg. Bowman et al, 1987), are for plane wave incidence; that is a source so distant that its directivity has no effect. Or, if the source is local, it is omnidirectional. Here a numerical procedure for extending local omnidirectional source solutions to those for local extended sources at any range is described. It can be applied to both low and high frequency scattering solutions with an accuracy dependant only on the number and accuracy of the basis source solutions used. A superposition of solutions for omnidirectional sources closely spaced in the aperture with amplitudes corresponding to the aperture distribution could provide a correct near field scattering solution but more efficient solutions will require larger source spacings. Then it is necessary to use beam rather than omnidirectional sources and arrange the beam sources in Gabor lattice as described by Einziger at el (1986). Both radiative and reactive aperture fields may then be represented to any accuracy at any range. Gabor (1946) proposed a series of time and frequency shifted Gaussian functions as an alternative to Fourier analysis in signal processing. Most of the impediments which delayed the implementation of Gabor analysis appear to have been overcome and its application to aperture radiation has been reviewed by Bastiaans (1998). The translated and phase shifted Gaussian functions of signal analysis become translated and directionally shifted Gaussian beams in aperture analysis. But Gaussian beams are approximate solutions to the wave equation and here it is found preferable to use complex source point (CSP) beams, which rigorously satisfy the wave equation. Then exact scattering solutions for local omnidirectional sources can be converted to exact CSP beam solutions by substituting appropriate complex coordinates for the real source coordinates. These then become the basis functions for extended source scattering solutions. Complex source point beam are paraxially Gaussian and thus fit well into the framework of Gabor analysis. When many are used with the same amplitude coefficients CSP and Gaussian beams provide virtually identical results in aperture analysis.