# Phase-Space Sampling in Asymptotic Diffraction Theory

## Abstract

In this paper, the discretisation of 2-dimensional wavefields is studied from two points of view: discrete Fourier synthesis and discrete Gaussian synthesis (the Gabor representation). In high-frequency fields localisation of the wave occurs, summarised *in extremis* by the laws of geometrical optics and GTD. Efficient numerical methods should incorporate this localisation if at all possible. In the case of Fourier synthesis, a wave is represented as a superposition of plane waves in arbitrary directions; high-frequency localisation occurs through the principle of stationary phase, wherein at a given point only a small part of the spectrum makes the dominant contribution, the remainder of the spectrum being self-cancelling in superposition due to destructive interference. The directions of dominant plane waves are those closest to the GO ray directions through the point of observation. In the Gaussian synthesis case, localisation occurs in the spectral elements themselves. Under certain circumstances, which can be determined rigorously, both the plane wave and Gaussian syntheses can be discretised, and comparison between them becomes feasible. In order to carry this out, consistent definitions of localisation are required. We show that spatial localisation is most conveniently expressed, in both cases, in terms of the unit of length L=(2π|R|/k)^{1/2} and n=2π/L=(2πk/|R|)^{1/2}, where R is the radius of curvature of the wavefront at the observation point of interest. In ray terms, L is the width of the spatial zone around any given ray which influences strongly the field on that ray, and for that reason it is called the *spatial influence zone.*; the parameter Ω plays the same role in the spectral domain and is called the *spectral influence zone.*

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