Analytical solution of the polarized photon transport equation in an infinite uniform medium using cumulant expansion

An analytical solution for time-dependent polarized photon transport equation in an infinite uniform isotropic medium is studied using a circular representation of the polarized light and expansion in the generalized spherical functions. We extend our cumulant approach for solving the scalar (unpolarized) photon transport equation to the vector (polarized) case. As before, an exact angular distribution is obtained and a cumulant expansion is derived for the polarized photon distribution function. By a cutoff at the second cumulant order, a Gaussian analytical approximate expression of the polarized photon spatial distribution is obtained as a function of the direction of light and time, whose average center position and half-width are always exact. The central limit theorem claims that this spatial distribution approaches accuracy in detail when the number of collisions or time becomes large. The analytical expression of cumulants up to an arbitrary high order is also derived, which can be used for calculating a more accurate polarized photon distribution through a numerical Fourier transform. Contrary to what occurs in other approximation techniques, truncation of the cumulant expansion at order n is exact at that order and cumulants up to and including order n remain unchanged when higher orders are added, at least as applied in our photon transport equation.