Spectral reflectance and transmittance of stacks of nonscattering films printed with halftone colors

Abstract

This paper combines and extends two optical models based on a two-collimated-flux approach that we previously proposed for the reflectance and transmittance of nonscattering elements, i.e., stacked nonscattering plastic films on the one hand, and films printed in halftone on the other hand. Those two models are revisited and combined by introducing different reflectances and transmittances on the two sides of a printed film, a common situation in practice. We then address the special case of stacks of identical films for which we obtain closed-form expressions for the reflectance and transmittance of the stacks as functions of the number of films. Experimental testing has been carried out on several different films printed with an inkjet printer. The accuracy of the model is good up to 16 films in most cases, despite a slight decrease in the case of yellow ink, which is more scattering than the other inks. By transposing the model to thin diffusing layers and considering diffuse fluxes instead of collimated ones, the closed-form expressions yield the well-known Kubelka-Munk reflectance and transmittance formulas. When these stacks of films are backed by a colored specular reflector, the reflectance is in certain conditions independent of the number of films.