Beam Summation Theory for Waves in Fluctuating Media. Part II: Stochastic Field Representation

In Part I of this two-part sequence, we introduced a novel beam summation (BS) scheme for tracking waves through fluctuating media. A key step has been the proof that the set of beam propagators constitutes a frame everywhere in the propagation domain. This beam frame (BF) provides a self-consistent framework for wave tracking, where the field is expanded using the BF and the local scattering of each beam is reexpanded using the same beam-set and expressed as beam-to-beam (B2B) scattering coefficients. This theory is used here to derive a BS representation for the stochastic-field for cases where the medium fluctuations are random with a given statistics. The stochastic B2B scattering moments are therefore expressed in terms of the local spectral statistics of the medium projected onto a phase space window formed by the intersection of the excitation and the scattered beams. Since the medium statistics is typically smooth, unlike a single realization, the resulting stochastic B2B scattering matrix is compact and smooth and may actually be calculated analytically. It is demonstrated, via numerical examples and a comparison with Monte Carlo simulations, that the formulation is not only computationally efficient, but also provides a compact representation for the scattering phenomenology.