Compositional reflectance and transmittance model for multilayer specimens

We propose a compositional model for predicting the reflectance and the transmittance of multilayer specimens composed of layers having possibly distinct refractive indices. The model relies on the laws of geometrical optics and on a description of the multiple reflection-transmission of light between the different layers and interfaces. The highly complex multiple reflection-transmission process occurring between several superposed layers is described by Markov chains. An optical element such as a layer or an interface forms a biface. The multiple reflection-transmission process is developed for a superposition of two bifaces. We obtain general composition formulas for the reflectance and the transmittance of a pair of layers and/or interfaces. Thanks to these compositional expressions, we can calculate the reflectance and the transmittance of three or more superposed bifaces. The model is applicable to regular compositions of bifaces, i.e., multifaces having on each face an angular light distribution that remains constant along successive reflection and transmission events. Kubelka's layering model, Saunderson's correction of the Kubelka-Munk model, and the Williams-Clapper model of a color layer superposed on a diffusing substrate are special cases of the proposed compositional model.