Mid Sweden University

The Kubelka-Munk scattering and absorption coefficients of a set of paper samples are assessed using reflectance measurements in d/0 geometry. The coefficients display the widely studied dependence between light scattering and light absorption, since the light scattering coefficient decreases in regions of high absorption. It is shown using general radiative transfer theory that part of this dependence can be explained and eliminated by taking into account the geometry of the d/0 instrument and the single scattering anisotropy, thus capturing the angular variations of the light reflected from the samples. These findings allow the papermaker to better predict the reflectance from mixtures of pulps, fillers, dye, and FWA, and to better evaluate bleaching efforts. 

The angle resolved reflectance factor of matte samples is measured with a goniophotometer and simulated using radiative transfer theory. Both measurements and simulations display the same characteristic dependence of the reflectance factor on the observation angle. The angle resolved reflectance spectra are translated to CIELAB color coordinates and the angular color differences are found to be surprisingly large. A chromatic adaptation that is dependent on the observation angle is suggested, in which a nonabsorbing opaque medium is used as the reference white, and the angular color differences are then reduced. Furthermore, the use of an undyed paper as the reference white is evaluated. The angular lightness differences are then reduced further, but the angular differences in chroma are still large. It is suggested that smaller variations in perceived color could be explained by angle dependent chromatic adaptation and a limited sensitivity of the human visual system to changes in chroma.

It is shown that the intensity of light reflected from plane-parallel turbid media is anisotropic in all situations encountered in practice. The anisotropy, in the form of higher intensity in large polar angles, increases when the amount of near-surface bulk scattering is increased, which dominates in optically thin and highly absorbing media. The only situation with isotropic intensity is when a non-absorbing infinitely thick medium is illuminated diffusely. This is the only case where the Kubelka-Munk model gives exact results and there exists an exact translation between Kubelka-Munk and general radiative transfer. This also means that a bulk scattering perfect diffusor does not exist. Angular resolved models are thus crucial for a correct understanding of light scattering in turbid media. The results are derived using simulations and analytical calculations. It is also shown that there exists an optimal angle for directional detection which minimizes the error introduced when using the Kubelka-Munk model to interpret reflectance measurements with diffuse illumination.

The anisotropic reflectance from turbid media predicted using the radiative transfer based DORT2002 model is experimentally verified through goniophotometric measurements. A set of paper samples with varying amount of dye and thickness is prepared and their angle resolved reflectance is measured. An alleged perfect diffusor is also included. The corresponding simulations are performed. A complete agreement between the measurements and model predictions is seen regarding the characteristics of the anisotropy. They show that relatively more light is reflected in large polar angles when the absorption or illumination angle is increased or when the medium thickness is decreased. This is due to the relative amount of near-surface bulk scattering increasing in these cases. This affects the application of the Kubelka-Munk model as well as standards for reflectance measurements and calibration routines.

Point spreading is investigated using general radiative transfer theory. We find that the single scattering anisotropy plays a significant role for point spreading together with the medium mean free path, single scattering albedo and thickness. When forward scattering dominates, the light will on average undergo more scattering events to give a specific optical response in reflectance measurements. This will significantly increase point spreading if the medium is low absorbing with large mean free path. Any fundamental and generic model of point spreading must capture the dependence on all of these medium characteristics.

The modulation transfer function (MTF) of 22 paper samples is computed using Monte Carlo simulations with isotropic or strongly forward single scattering. The inverse frequency at half maximum of the MTF (kp) is found inappropriate as a single metric for the MTF since it is insensitive to the shape of the modeled and simulated MTF. The single scattering phase function has a significant impact on the shape of the MTF, leading to more lateral scattering. However, anisotropic single scattering cannot explain the larger lateral scattering observed in paper. It is argued that the directional inhomogeneity of paper requires a light scattering model with both the phase function and scattering distances being dependent on the absolute direction.

The L&W Elrepho d/0 and the Spectrolino 45/0 instruments are examined using paper samples with different properties. External factors that influence the measurements such as the sample background, the instrument calibration and the sample inhomogeneity are studied, and a methodology for their minimization is presented. Experimental measurements show that such external factors, if not minimized by proper routines, affect the inter-instrument differences far more (up to 4-5 Delta E-ab(star)) than the instrument geometry (the effect of which is small and of order 0.1 Delta E-ab(star)). The DORT2002 radiative transfer model is used to simulate differences caused by instrument geometry. The simulated and measured differences are found to agree in magnitude, and the differences are mapped against sample properties. It is observed that the 45/0 instrument detects higher reflectance from paper samples with negligible absorption and transmittance. When there is considerable absorption (dyed samples) or transmittance (thin samples), the d/0 instrument detects higher reflectance. The physical mechanism behind this behavior is studied and explained using DORT2002, and the instrument differences are shown to depend on the anisotropy of the reflected light. The model/measurement agreement is satisfactory as the characteristic behavior is captured in almost all cases studied. This new understanding is important for facilitating accurate data exchange between the paper and graphic arts industries, but also for interpretation of reflectance measurements in general.