Beam summation theory for waves in fluctuating media. Part II: Stochastic fields

Abstract

In Part I of this two-part paper we presented the “propagating beam frame” (PBF) concept, which provides a self-consistent framework for wave tracking through a fluctuating medium. The field is expanded using the BPF and the local scattering of each beam by the medium is re-expanded using the same beam-set and expressed as beam-to-beam (B2B) scattering coefficients. The entire scattering problem is thereby described in terms of the coefficients dynamics in the phase-space. In the present paper, we use the theory of Part I to derive a beam summation (BS) representation for the stochastic-field moments (observables) for cases where the medium fluctuations are expressed as a random process with given statistics. We derive closed form approximations for the stochastic B2B scattering moments, which are expressed in terms of the local spectral statistics of the medium projected on phase-space windows formed by the intersection of the excitation and the scattered beams. Since the medium statistics is typically smooth, unlike its realization, the resulting stochastic beam-to-beam (B2B) scattering matrix is compact and smooth. The stochastic observables are fully described in terms of the local dynamics of the B2B scattering moments as the wave propagates through the medium. It is demonstrated that the formulation computationally efficient and provides a compact representation for the scattering phenomenology.