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.

Analytical Approaches to Model Light Transmission through Turbid Media

“…Note that equations for transmittance such as Eqs. 14and (16) may be found in numerous publications, e.g., by Stokes (1862), Ambartsumian (1942), Tuckerman (1947, Sobolev (1957), Ueno (1961), Hébert and Machizaud (2012), Sokoletsky et al (2014a), and Sokoletsky andBudak (2016a, b, 2017). However, to the best of our knowledge, a presentation of ef in the form of…”

“…was considered in only few publications, e.g., by Ambartsumian (1942), Sobolev (1957), Ueno (1961), Carder et al (1999), Hébert and Machizaud (2012), Sokoletsky et al (2014a), and Sokoletsky andBudak (2016a, b, 2017). In the other publications, instead of…”

Assessing the capacity of two-flux models to predict the spectral properties of layered materials

“…Experiments with stacks of nonscattering colored acetate films, illuminated by collimated light according to their normal direction, have already be presented in [13] and [8]. Other experiments based on transparency films printed in inkjet, with different reflectances on the inked and noninked faces due to a bronzing effect [14], have also been presented in [15]. In these studies, other mathematical methods than the present one were used to derive the reflectance and transmittance equations of stacks of films, which were however mathematically equivalent to equations (8).…”

“…The following nonscattering materials were tested: sheets of cyan acetate (1 to 5 sheets, similar as the ones used for [13]), and transparency films printed by inkjet with a green or a magenta color (1 to 16 sheets, similar as the ones used for [15]). Their reflectances and transmittances, displayed in Figure 2-a, respectively Figure 2-c, were measured by using the X-rite Colori7 spectro-photometer based on the diffuse:0° geometry.…”

Color and Spectral Mixings in Printed Surfaces

“…At least, the mixture has an interface with air whose optical effect cannot be neglected [3][4], and it is generally deposited on a support, specular or diffusing, whose internal reflectance has obviously capital impact of the final color. The refractive index of the layers may also have significant impact, as observed by stacking colored films on a white background with or without optical contact [5][6][7]. Lastly, the Beer-Bouguer-Lambert law is restricted to non-scattering media; extension to scattering media is possible but more difficult, except when scattering is sufficiently strong so that the Kubelka-Munk model applies [8][9].…”