Beam wave two-frequency mutual-coherence function and pulse propagation in random media: an analytic solution

Pulse propagation in a random medium is determined by the two-frequency mutual coherence function which satisfies a parabolic equation. In the past, numerical solutions of this equation have been reported for the plane wave case. An exact analytical solution for the plane wave case has also been reported for a Gaussian spectrum of refractive-index fluctuations. Using the same approximation, an exact analytic solution for the more general case of an incident beam wave is presented. The solution so obtained is used to study the propagation characteristics of the beam wave mutual coherence function at a single frequency as well as at two frequencies. Simple expressions are obtained which qualitatively describe the decollimating and defocusing effects of turbulence on a propagating beam wave. The time variation of the received pulse shape, on and away from the beam axis, is studied when the medium is excited with a delta function input. The results are presented for both collimated and focused beams.