The recently formulated stochastic geometrical theory of diffraction combines localization and transport of high-frequency fields and their statistical measures along the geometrical rays in deterministic background. Conventionally, these statistical measures are associated with the average statistical moments of the field, thereby restricting the propagation process to one-directional propagation and not permitting the consideration of double-passage effects. This limitation is now removed by introducing paired random functions in which the random information along the propagation path is preserved. To render these statistical measures physically meaningful and analytically tractable, the choice of appropriate propagators plays a central role. Here a two-scale expansion asymptotic procedure is applied to solve the partial differential equations for the corresponding propagators in a statistically homogeneous random medium. The resulting solutions are presented in a multiple-integral form and contain enough spectral information to reproduce the expressions obtained from the two-scale solutions of the equations for the averaged statistical moments. Moreover, the fact that the solutions obtained here contain the information about the variation of the random part of the refractive index makes them suitable for application to backscattering problems. These solutions may be tested by applying them to a canonical double-passage problem of a point source and a point scatterer. Together with reciprocity conditions they permit the choice and demonstration of an appropriate propagator for a simple example of plane-wave reflection from a plane mirror.
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