The Milne problem with Fresnel reflection

The Milne problem, with internally reflecting boundary conditions as required by Fresnel theory, is solved exactly by the method of Wiener and Hopf. In addition to the specular case we also solve for diffuse reflection, since this provides a useful comparison. The results are also compared with those of Aronson (1995 J. Opt. Soc. Am. 12 2532) and excellent agreement is obtained. The Wiener-Hopf procedure makes it possible to find an expression for the value of refractive index such that the extrapolated endpoint no longer exists. Tables for this critical value of refractive index are given for both specular and diffuse reflection. The reflected and transmitted surface angular distributions are also given for a range of refractive indices. Some comments are made about the nature of the solution as the reflection becomes perfectly specular. The equivalent diffusion theory results have also been obtained and it is seen that this approximation is reasonable for very small absorption, but rapidly becomes inaccurate for large absorption and large refractive index.