Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to a polarized laser beam: spatially varying reflection matrix

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to a polarized, Gaussian laser beam directed perpendicular to the surface is studied. The focus of this investigation is the 4×4, spatially varying reflection matrix that can be used to determine the normally backscattered radiation when the polarization of the incident radiation is specified. An inverse integral transform is used to construct the spatially varying reflection matrix from the generalized reflection matrix found in a previous study. The elements of this matrix depend on location specified by optical radius and azimuthal angle. The azimuthal variation is found by performing part of the inverse transform analytically, while the radial variation is described by five functions that are calculated numerically via an inverse Hankel transform. Benchmark numerical results for these five functions are presented, and the effects of beam radius and particle concentration are discussed. Expressions that describe the behavior of the reflection functions at small and large optical radii are developed, and comparisons are made to the one-dimensional and scalar situations. The scalar approximation fails to predict the three-dimensional effects produced by the polarized beam, and even when the incident radiation is unpolarized, the error in the scalar reflection function can be as high as 20%.