Compositional model for predicting multilayer reflectances and transmittances in color reproduction

The color of materials such as paints, prints and glass may be characterized by a reflectance or a transmittance spectrum. Modeling their reflectance and their transmittance requires describing the interaction of light, from the light source to the observer, across the different layers and interfaces. Each layer and interface behaves as a light reflector and transmitter, and is given the generic name of "biface". Multilayer specimens, called "multifaces", result from the superposition of various bifaces between which light is subject to multiple reflections and transmissions. We establish a multiple reflection-transmission model which describes the transfers of fluxes between the different bifaces using the basic laws of geometrical optics. This approach is valid for multilayer specimen composed of strongly scattering and/or non-scattering layers and flat interfaces. Weakly scattering layers and rough interfaces are allowed if they are surrounded by strongly scattering layers. We first develop the multiple reflection-transmission model in a general manner, i.e. regardless to the specific optical properties of the bifaces. The light multiple reflection-transmission process is represented by a Markov chain. The well established mathematical tools provided by the Markov theory enable deriving the formulae for the reflectance and transmittance of superposed bifaces. Then, we show how the multiple reflection-transmission formulae are applied for a specific multiface and for a specific measuring geometry. We retrieve as special cases of our general model the Kubelka model for stacked intensely scattering layers, the Williams-Clapper model for a diffusing background coated with a non-scattering layer, the Saunderson correction, and the Clapper-Yule model for high quality halftone prints. We finally explore new possibilities offered by the multiple reflection-transmission model, both for developing new reflectance or transmittance models and for checking the relevance of parameters deduced from measured data. We develop a method for characterizing papers independently of the measuring geometry by modeling two superposed sheets of paper and draw the bases of a reflectance and transmittance prediction model for recto-verso halftone prints.