Two-dimensional radiative transfer in a cylindrical layered medium with reflecting interfaces

Abstract

A system of exact linear integral equations for the source function, intensity, and flux is presented for a twodimensional cylindrical medium consisting of up to four layers with reflecting interfaces between the layers. Properties that may change from layer to layer are the single scattering albedo, optical thickness, and refractive index. The incident radiation is coUimated and has a Bessel function distribution. The Bessel function boundary condition reduces the two-dimensional problem to a one-dimensional problem. Superposition is then used to derive the solution for any other boundary condition that is Hankel transformable. A special case of a Gaussian distribution that models a laser beam is presented. Some one-dimensional numerical results are presented for the source function and intensity within one- and two-layer media. Two-dimensional results are presented for back-scattered intensity due to the laser-beam boundary condition. Only the conservative case, optical thicknesses of 2.0 and 5.0, and refractive indices of 1.00 and 1.33 are considered. Nomenclature A, = multiple reflection coefficient, defined in the Appendix B{J = function defined in the Appendix DN — function defined in the Appendix E = component of kernel function for source function integral equation Giy = function defined in the Appendix g = Hankel transformed function /, = intensity of radiation in layer i /+ = intensity in layer i in + TZ direction It~ = intensity in layer i in — rz direction I0 = magnitude of incident intensity outside the medium IT = intensity transmitted across an interface /, = magnitude of incident intensity J0 = zeroth order Bessel function of first kind Kfj = function defined in the Appendix W = total number of layers HIJ = ratio of layer i refractive index to that of layer ; P = scattering phase function Qij = function defined in the Appendix qzi = z direction flux in layer i Ri = function defined in the Appendix r = radial distance from center of medium r0 = laser beam radius 5, = source function in layer i Scio = source function lead term, defined in the Appendix T = transmission function, defined by Eq. (32) Ti = function defined in the Appendix tfj = transmission through the interface between layer i and / x = dummy integration variable