Abstract
The problem considered is that of calculating the moments of waves propagating in a medium with random refractive index fluctuations and deterministic focusing. Focal regions produced by deterministic refractive index variations are defined within the ray approximation and classified into two types: the caustic and the cusp.It is shown that the problem of waves propagating in a random medium can be approximated by a phase screen problem in which the waves freely propagate to a plane, where they are perturbed in phase and then propagate freely to the observation point. This is done first within the ray approximation by considering higher order terms in an expansion about a determinstic ray. The approximation is also derived using the path integral solution to the parabolic wave equation; this derivation of the approximation gives an improved estimate of the range of validity of the phase screen approximation.A phase screen model with deterministic focusing is defined which includes as special cases the phase screen approximations to propagation in random media. An additional focal type, perfect focusing with a Gaussian amplitude window at the screen, is also allowed. Non-dimensionalizing the integral expressions for the field beyond the screen produces dimensionless parameters. The strength parameter can be identified with the strength parameter in a propagation problem in an extended medium, but the diffraction parameter for the approximating phase screen is a generalization of the usual diffraction parameter for propagation problems.A Monte-Carlo simulation is used to estimate the moments of the field in several cases of interest. The moments are found to depend on the strength of the fluctuations in a complicated way when the generalized diffraction parameter is near unity. In particular, the scintillation index is large at local intensity minima even for relatively weak fluctuations. Fluctuation parameters for focal regions produced in underwater sound propagation are estimated.