Three-dimensional radiative transfer for anisotropic scattering medium with refractive index greater than unity

Abstract

An exact mathematical formulation is developed for three-dimensional radiative transfer in a finite medium which scatters anisotropically. The scattering medium is finite in the z-direction and infinite in both the x- and y-directions. The scattering phase function is expressed as a finite series of Legendre polynomials, and the refractive index of the medium is greater than unity. The upper boundary is exposed to radiation for which the directional variation and the spatial distribution are separable, while the lower boundary is not exposed to radiation. A double Fourier transform is used to reduce the transport equation to a one-dimensional form. Using superposition, a two-dimensional linear integral equation is developed for the back-scattered intensity. The kernel for this integral equation consists of the bidirectional reflectance for a medium with refractive index of unity and the interface reflectance. The bidirectional reflectance is related to the back-scattered intensity from a two-dimensional rectangular medium subject to cosine varying collimated radiation, and the interface reflectance is found from Fresnel's relations. The situation of a medium exposed to a cylindrically symmetric laser beam is also studied. The Gaussian beam is collimated and normal to the upper boundary. Again, the results are expressed in terms of a two-dimensional rectangular medium with refractive index of unity.