The far field in terms of the spatial Fourier transform of the sources and its implications on the inverse problem

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Abstract
It has been known for some time that the range and phase normalized far field is proportional to the three-dimensional spatial Fourier transform of the source distribution, taken on the surface of the Ewald sphere at a fixed frequency. Unfortunately, however, this very useful relationship is neither widely known, nor does there even exist for it a good, short, simple, and direct derivation in the open literature (thus making references to the relationship difficult). The purpose of this paper is to alleviate these difficulties by presenting a brief derivation of the relationship for both the scalar acoustic as well as the vector electromagnetic cases. A brief discussion of the implications of this relationship on the inverse source and inverse scattering problem is also presented.

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