The Green-function transform and wave propagation

*Correspondence: Colin J. R. Sheppard, Department of Nanophysics, Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy e-mail: [email protected] We review Fourier methods used in the disciplines of electromagnetism and signal processing, with a view to reconciling differences in approach. In particular, Fourier methods well known in signal processing are applied to three-dimensional wave propagation problems. The Fourier transform of the Green function, when written explicitly in terms of a real-valued spatial frequency, consists of homogeneous and inhomogeneous components. Both parts are necessary to result in a pure out-going wave that satisfies causality. The homogeneous component consists only of propagating waves, but the inhomogeneous component contains both evanescent and propagating terms. Thus, we make a distinction between inhomogeneous waves and evanescent waves. The evanescent component is completely contained in the region of the inhomogeneous component outside the k-space sphere. Further, propagating waves in the Weyl expansion contain both homogeneous and inhomogeneous components. The connection between the Whittaker and Weyl expansions is discussed. A list of relevant spherically symmetric Fourier transforms is given.