Two-frequency radiative transfer and asymptotic solution.

Two-frequency radiative transfer (2f-RT) theory is developed for classical waves in random media. Depending on the ratio of the wavelength to the scale of medium fluctuation, the 2f-RT equation is either a Boltzmann-like integral equation with a complex-valued kernel or a Fokker-Planck-like differential equation with complex-valued coefficients in the phase space. The 2f-RT equation is used to estimate three physical parameters: the spatial spread, the coherence length, and the coherence bandwidth (Thouless frequency). A closed-form solution is given for the boundary layer behavior of geometrical radiative transfer and shows highly nontrivial dependence of mutual coherence on the spatial displacement and frequency difference. It is shown that the paraxial form of 2f-RT arises naturally in anisotropic media that fluctuate slowly in the longitudinal direction.