Modal theory for the two-frequency mutual coherence function in random media

Abstract

Short pulse propagation in random media is mainly determined by the two-frequency mutual coherence function which is governed in the multiple scattering regime by a parabolic equation. In this paper the modal expansion theory is presented as a new analytical approach for media which are statistically isotropic and homogeneous. By performing a separation of variables, the problem of the 3D partial differential equation is reduced to solving a one-dimensional eigenvalue problem. The full expansion theorem is presented applicable for any initial source configuration. For media characterized by a quadratic structure function, the eigenvalue problem is exactly solvable. The two-frequency coherence function is obtained as a modal series for the three most important source configurations, namely the plane wave, the point source and the beam wave. By Poisson's theorem, the series is summed up into a closed form expression and is shown to yield the known solutions in the literature. In this paper, we only present the general modal expansion theorem and the exact solution for a beam in a quadratic medium.