Three-dimensional radiative transfer with polarization in a multiple scattering medium exposed to spatially varying radiation

Abstract

The present investigation of three-dimensional radiative transfer accounts for polarization and multiple scattering by using the vector transport equation. The intensity vector is comprised of the four Stokes parameters, which are functions of three coordinates describing position and two angles specifying direction. Scattering is characterized by a general phase matrix for randomly oriented particles with a plane of symmetry. The medium is finite in the z-direction and infinite in the x- and y-directions. The incident radiation is polarized and spatially varying on the upper boundary. The vector transport equation is converted, using a double Fourier transform, to an ordinary integro-differential equation with a complex coefficient. The problem of collimated radiation incident on the upper boundary of a cylindrical medium is shown to be of fundamental importance. The analysis is greatly simplified by expressing the transformed intensity vector in a form that depends on only one transform variable. Using superposition, a linear Fredholm integral equation of the second kind is developed for the 4 × 4 generalized source matrix. The bidirectional reflectance and transmittance of a scalar analysis are replaced by 4 × 4 reflection and transmission matrices. The emergent radiation can be determined from these matrices for incident radiation of any polarization. Symmetry relationships are developed using successive orders of scattering and symmetry properties of the phase matrix.