Analytical solution of the vector radiative transfer equation for single scattered radiance

In this paper, derivation of the analytical solution of the vector radiative transfer equation for the single scattered radiance of three-dimensional semi-infinite media with a refractive index mismatch at the boundary is presented. In particular, the solution is obtained in the spatial domain and spatial frequency domain. Besides the general derivation, determination of the amplitude scattering matrix, which is required for the analytical solution, is given in detail. Furthermore, the incorporation of Fresnel equations due to a refractive index mismatch at the boundary is presented. Finally, verification of the derived formulas is performed using a self-implemented electrical field Monte Carlo method based on Jones formalism. For this purpose, the solution based on Jones formalism is converted to Stokes-Mueller formalism. For the verification, spherical particles are assumed as scatterers, whereby arbitrary size distributions can be considered.