Analytic Solution of the Nonlinear Radiative Diffusion Problem in a One-Dimensional Purely Scattering Medium. I

Abstract

This paper is the second part of the author’s previous work. Its purpose is to illustrate the efficiency of using the previously introduced auxiliary functions known as linear images (LI) of the reflection-transmission function with simple examples of nonlinear problems for a one-dimensional purely scattering medium. First, an explicit analytic solution is obtained for the nonlinear “direct” problem of determining the fields of the emerging radiation from a “composite” medium consisting of a reflecting surface and a layer of finite thickness with known reflecting-transmitting properties. Then solutions are obtained for the nonlinear “inverse” problems of determining: (a) the external exciting fields based on data from the radiation emerging from the medium, (b) the fields at an “inaccessible” boundary of the medium based on observations of the light regime at one of its boundaries, (c) the intensities of the radiation traveling in one direction based on measurements of the fields in the opposite direction, (d) some characteristics of the medium based on measurements of the intensities of radiation incident on it from outside and emerging through one of its boundaries, and (e) fields inside the medium based on measurements of the bidirectional radiation fields at just one boundary of the medium. Finally, it is shown that in the nonlinear problem of illumination of an semi-infinite medium the phenomenon of “bleaching” of the medium does not occur; the light regime coincides with the solution of the linear problem, both outside and inside the medium.