Average transmittance of polarized diffuse light across a dielectric surface is calculated in both two and three dimensions. The incident light in both cases is confined to an angular range measured from the surface normal. Limiting values in three dimensions correspond to known results for two cases, (i) normal incidence, and (ii) diffuse light incident from a 180° cone. The two-dimensional formulation is solvable in terms of elliptic functions and incomplete elliptic integrals of the first, second, and third kinds. Results are displayed graphically for values of transmittances in refraction in the range m = 1.0 to m = 2.6.
William A. Allen, "Transmission of isotropic light across a dielectric surface in two and three dimensions," J. Opt. Soc. Am. 63, 664-666 (1973)