An approximate method for solving the vector radiative transfer equation in discrete random media

Analytical expressions for the specific intensity of circularly polarized optical waves propagating in discrete random media are obtained by approximately solving the vector radiative transfer equation. The proposed method extends the solution for scalar waves to include polarization effects and facilitates calculations of multiply scattered incoherent intensities over a wide range of optical depths. To derive simple analytical solutions for given particle sizes, we first separate the first-order solution or small-angle solution from the solution to be obtained. We then expand the remainder, which includes the higher order multiple scattering, in terms of the generalized spherical functions. The first two terms of the expansion are retained, and the resulting vector diffusion equation is solved analytically. Approximate solutions for both the copolarized and cross-polarized specific incoherent intensities are shown to be in good agreement with numerical solutions for almost arbitrary optical depths.