Random Rays and Stochastic Caustics
I review a theory of high frequency random wave propagation which is obtained by applying a stochastic limit theorem to the equations of ray theory in a random medium. The method is applicable in the “strong fluctuation region” when large amplitudes occur randomly. If index of refraction fluctuations are small, of order a, large amplitudes are accompanied by the occurrence of caustics which appear after long propagation distances of order a correlation lengths of the random medium. Under broad hypotheses it is shown that the region of random caustics is given, up to a single distance scale parameter, by a universal curve for the probability of caustic formation along a ray as a function of propagation distance. In this region wavefront geometry is in some sense preserved despite the order 1 random wanderings of the rays, since to leading order the random displacement of the rays is along the wavefronts.
On this scale limit equations can be derived for the joint distribution of an arbitrary number of ray positions and associated raytube areas. The transformation from ray coordinates to physical coordinates is effected by deriving expressions for quantities which are transported along a ray, when summed over the random number of randomly chosen rays which pass through a fixed point in physical space. Application is given to computation of the correlation function of intensity.
KeywordsPropagation Distance Random Medium Scintillation Index Universal Curve Functional Central Limit Theorem
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