Mid Sweden University

Modeling light scattering is important in diverse reasearch fields such as paper and print, optical tomography, remote sensing and also in computer rendering of im­ages. Particularly in paper and printing industry light scattering simulations play a significant role in understanding the optical response of paper in relation to its properties. Light scattering models are used in paper and print for improving the paper making process, designing new paper qualities, and evaluating printing tech­niques. The models most widely used for light scattering calculations in the paper and printing industry are based on the Kubelka-Munk theory. The theory proposed by Kubelka and Munk, a special case of radiative transfer theory, has several limi­tations and it can only be applied to homogeneous media with isotropic scattering and diffuse illumination. Real paper and print in particular do not satisfy these as­sumptions. These limitations of the Kubelka-Munk model encouraged scientists to develop models based on angle-resolved geometry to account for anisotropic scat­tering of light in paper and print, but in a single spatial dimension. To correctly represent spatial inhomogeneities like ink dots which spread as a function of depth, length and width of the paper, one-dimensional (lD) models are insufficient. In addi­tion to angle-resolved geometry, multi-dimensional models are necessary to analyze light scattering effects in a printed paper.

The method used in this thesis, unlike the Kubelka-Munk method employs gen­eral radiative transfer formulation to obtain the reflectances of paper with inhomo­geneities like ink dots. These ink dots printed on plain sheet of paper are consid­ered to spread not only as a function of depth but also as a function of length or width of the paper. First, a numerical solution method comprising of a combination of discrete ordinates and finite differences is developed to solve the general two­dimensional (2D) radiative transfer equation (RTE) with the two dimensions repre­senting the depth and length of the paper. The solver is validated by comparing the results obtained with Monte Carlo simulations adapted to suit paper optics and DORT2002. For isotropic scattering, and for angles close to the normal direction, good agreement is observed among all the three solvers. As the anisotropy factor increases, the present solver needs higher number of radiation streams for conver­gence.

The 2D radiative transfer (RT) solver is then applied to printed paper and re­flectances obtained are analyzed. The ink distribution is considered to be non-uniform such that the density of ink decreases linearly with depth. The dots are separated by a distance to study the interference pattern of the intensity distribution which is use­ful in understanding defects like print mottle, print density and optical dot gain. The reflectances obtained are analyzed based on medium parameters such as thickness of the paper sample, its optical parameters and assymetry factor. The illuminating and viewing angles and the depth of ink penetration also influence the optical response and appearance of print. It is observed that the reflectance of dots largely depends on the illuminating and viewing angles with an apparent increase in the size of the dots seen more prominently when viewed across the line.

A 2D RT solver is superior in understanding the interference pattern of radiation as observed in the results presented in this thesis, when compared to a lD RT solver. A lD RT solver uses independent columns to approximate the radiation in the lateral direction. It also assumes that the layers in the lateral direction are homogeneous and the radiation from the columns do not interfere with each other. The independent column approximation pays little attention to the lateral variations in intensity.