In a previous paper, we derived expressions for the irradiance distribution of a finite beam. Here we derive expressions for the full coherence function. We find an asymptotic analytic expression for the coherence function for very large propagation distances. In this region, the on-axis angular spectrum of the finite beam is one fourth that of an initial plane wave and three fourths that of an initial spherical wave. When the scattering dominates diffraction, we find overlapping asymptotic representations for the coherence function for all distances. We present some explicit results for an initially coherent gaussian wave.
MARK J. BERAN and ALAN M. WHITMAN, "Asymptotic Theory for Beam Propagation in a Random Medium," J. Opt. Soc. Am. 61, 1044-1050 (1971)