Beam Summation Theory for Waves in Fluctuating Media. Part I: The Beam Frame and the Beam-Domain Scattering Matrix

We present a novel beam summation (BS) formulation for tracking wavefields in fluctuating media. This formulation utilizes the ultrawideband phase-space BS method, which is structured upon a windowed Fourier transform (WFT) frame expansion of the sources and expresses the field as a discrete phase-space sum of beam propagators. This paper extends the frame concept, proving that this beam set constitutes a frame not only in the source domain, where it reduces to the conventional WFT, but everywhere in the propagation domain. This 'beam frame' provides a self-consistent framework for tracking wavefields through scattering media in which the local interaction of the incident beams with the medium is expanded using the same set of beams. The resulting beam-To-beam (B2B) scattering coefficients depend on the local spectral properties of the medium. The overall B2B scattering matrix is therefore compact, coupling only beams that are adjacent in phase space, given the fluctuation properties. The new formulation expresses the entire scattering problem in terms of coefficients dynamics in the phase space. As demonstrated, the formulation is computationally efficient and captures all the relevant phenomenology. Part II extends this formulation for stochastic fields in random fluctuating medium characterized by the medium statistics.