Beam summation theory for waves in fluctuating media. Part I: The beam-to-beam scattering matrix

We present a novel beam summation (BS) formulation for tracking wave fields in weakly fluctuating media. The overall strategy is to express the field in the medium as a discrete phase-space spectrum of beam waves, and to expand the local scattering excited by the interaction of each beam with the medium using the same beam set. Since the beams are localized in the spatial-spectral phase-space, the resulting beam-to-beam (B2B) scattering coefficients depend directly on the local spectral properties of the fluctuations, as a local alternative to the conventional plane wave scattering operator. This strategy is implemented by introducing the novel concept of propagating beam frame (PBF), a phase-space set of beam propagators that can be used as an over complete frame for a local expansion of the field and its interaction with the medium. This set constitute, in fact, a novel extension of the windowed Fourier Transform (WFT) frame. A key feature of this method is that it utilizes the same beam skeleton for all frequencies in the band. In this paper we derive this formulation and in particular we derive the closed form expressions for the B2B scattering matrix.