Transport theory of a random planar waveguide with a fixed scatterer: Mode theory

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Abstract
Based on the unified theory of random medium and random boundaries introduced in a previous paper [Furutsu, J. Opt. Soc. Am. A2, 913 (1985)], an exact version of the (normal) mode theory of both the coherent wave and the mutual coherence function is given without assuming a particular model. A mode equation for the second-order Green function is obtained from the governing (Bethe-Salpeter) equation based on a Maclaurin expansion at the set of poles of the first-order (renormalized) Green function, and is eventually given in the form of an equation of radiative transfer. The expansion would not be possible with the ‘‘bare’’ Green function. The overall unitarity of the Bethe-Salpeter equation is investigated in particular detail, and the involved optical relations are shown, not only of the entire system but also of the medium and each of the boundaries (of intrinsically dispersive property) separately. An exact theory of a fixed scatterer embedded in the waveguide is given with several expressions of the solution, including that with the conventional form in scattering theory of a coherent wave, in terms of an effective cross section having negative values in the shadow direction and its neighborhood. A detailed structure of the power equations, constructed by both coherent and incoherent waves in a complex way, is shown in terms of two optical relations for the scatterer’s two basic quantities that change the original Bethe-Salpeter equation. Whenever possible, the equations are so written in a general form that they hold true for a wide class of random systems with a fixed scatterer. Specific examples are given.