The Stochastic Caustic

Abstract

The propagation of a high frequency initially plane wave through a homogeneous and isotropic random medium with small, order $O(\sigma )$, fluctuations in index of refraction is investigated using geometrical optics. It is shown that caustics occur along every ray in a distance scale of order $O(\sigma ^{ - 2/ 3} a)$ where a is the correlation length of the medium. On this scale the ray angle deviations are small, of order $O(\sigma ^{2/ 3} )$, while the ray position deviates $O(1)$ from its deterministic value. Furthermore, it is shown that if $t = \gamma ^{1/3} s$ where s is arclength along a ray, \[ \gamma = 2\int_0^\infty dr \left( \frac{1}{r}\frac{\partial }{\partial r} \right)^2 R(r), \] and $R(r)$ is the correlation function of the medium, then as a function of t the probability density of distance to first caustic is as $\sigma \to 0$ a universal curve, with no free parameters and thus does not depend on the detailed statistics of the random medium. For small values of t this density is given by the asymptotic formula $f(t) \approx (a_1 /t^4 )\exp \{ - a_2/t^3 \} $, with $a_1 = 1.7399^ + ,a_2 = .6565^ + $ These results parallel results of V. Kulkarny and B. S. White for two-dimensional random media, where the analogous small t formula has been shown to be valid into the initial region of caustic formation, and may thus be used to determine, in an experimental situation, whether or not caustic formation is likely.