Per Edström

A Fast and Stable Solution Method for the Radiative Transfer Problem

Radiative transfer theory considers radiation in turbid media and is used in a wide range of applications. This paper outlines a problem formulation and a solution method for the radiative transfer problem in multilayer scattering and absorbing media using discrete ordinate model geometry. A selection of different steps is brought together. The main contribution here is the synthesis of these steps, all of which have been used in different areas, but never all together in one method. First, all necessary steps to get a numerically stable solution procedure are treated, and then methods are introduced to increase the speed by a factor of several thousand. This includes methods for handling strongly forward-scattering media. The method is shown to be unconditionally stable, though the problem was previously considered numerically intractable.

Examination of the revised Kubelka-Munk theory: considerations of modeling strategies

The revised Kubelka-Munk theory is examined theoretically and experimentally. Systems of dyed paper sheets are simulated, and the results are compared with other models. The results show that the revised Kubelka-Munk model yields significant errors in predicted dye-paper mixture reflectances, and is not self-consistent. The absorption is noticeably overestimated. Theoretical arguments show that properties in the revised Kubelka-Munk theory are inadequately derived. The main conclusion is that the revised Kubelka-Munk theory is wrong in the inclusion of the so-called scattering-induced-path-variation factor. Consequently, the theory should not be used for light scattering calculations. Instead, the original Kubelka-Munk theory should be used where its accuracy is sufficient, and a radiative transfer tool of higher resolution should be used where higher accuracy is needed.

Anisotropic reflectance from turbid media. I. Theory

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 at 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. Angle-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 that minimizes the error introduced when using the Kubelka-Munk model to interpret reflectance measurements with diffuse illumination.

A two-phase parameter estimation method for radiative transfer problems in paper industry applications

A two-phase method for estimation of the scattering and absorption coefficients and the asymmetry factor (σ s , σ a and g) in the radiative transfer problem is presented. The first phase parameterizes σ s and σ a through g via a simplified model and performs–at a relatively low cost–a scalar optimization over g. It is shown that this gives such a good starting point that the second phase can be accurately performed by a simple Gauss-Newton method. It is also shown that as a part of the first phase can be used on its own when only σ s and σ a are wanted, and it is noted that this gives higher accuracy than the commonly used Kubelka–Munk method when using standardized paper industry reflectance factor measurements. The parameter estimation problem is shown to be non-trivial and ill-conditioned, and its character is analysed. It is discussed that as standard optimization methods are so sensitive to the choice of starting point for this problem that it is hard to find a starting point that gives convergence at all. The new two-phase method is illustrated by application to relevant paper industry problems, and efficiency and sensitivity measures are given.

Anisotropic reflectance from turbid media. II. Measurements.

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 amounts 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 at 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.

Determination of quantum efficiency in fluorescing turbid media

A method is proposed to estimate the optical parameters in a fluorescing turbid medium with strong absorption for which traditional Kubelka-Munk theory is not applicable, using a model for the radiative properties of optically thick fluorescent turbid media of finite thickness proposed in 2009 [J. Opt. Soc. Am. A26, 1896 (2009)]. The method is successfully applied to uncoated papers with different thicknesses. It is found that the quantum efficiency of fluorescent whitening agents (FWAs) is nearly independent of the fiber type, FWA type, FWA concentration, and filler additive concentration used in this study. The results enable an estimation of the model parameters as function of the FWA concentration and substrate composition. This is necessary in order to use the model for optimizing fluorescence in the paper and textile industries.

Levenberg–Marquardt methods for parameter estimation problems in the radiative transfer equation

A discrete ordinate method is developed for solving the radiative transfer equation, and the corresponding parameter estimation problem is given a least-squares formulation. Two Levenberg–Marquardt methods, a feasible-path approach and an sequential quadratic programming-type method, are analysed and compared. A sensitivity analysis is given, and it is shown how it can be used for designing measurements with minimal impact of measurement noise. Numerical experiments are performed to exemplify the usefulness of the theory.

Point spreading in turbid media with anisotropic single scattering.

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.

Soap-film coating: High-speed deposition of multilayer nanofilms

The coating of thin films is applied in numerous fields and many methods are employed for the deposition of these films. Some coating techniques may deposit films at high speed; for example, ordinary printing paper is coated with micrometre-thick layers of clay at a speed of tens of meters per second. However, to coat nanometre thin films at high speed, vacuum techniques are typically required, which increases the complexity of the process. Here, we report a simple wet chemical method for the high-speed coating of films with thicknesses at the nanometre level. This soap-film coating technique is based on forcing a substrate through a soap film that contains nanomaterials. Molecules and nanomaterials can be deposited at a thickness ranging from less than a monolayer to several layers at speeds up to meters per second. We believe that the soap-film coating method is potentially important for industrial-scale nanotechnology.

Simulation and modeling of light scattering in paper and print applications

When developing and applying models to light scattering problems, things usually turn very mathematical. This is all in good order, but it may also be a hindrance for a broader audience to gain insight into the overall issues. This chapter aims at discussing a range of light scattering simulation and modeling issues with a minimum of mathematics involved, and with the specific perspective of paper and printing industry applications. Shorter sections of mathematical content are included, but the mathematically interested reader is here pointed to selected references and other chapters in this volume.