# Modelling of dispersion and reflection of light on paper surface.

1. INTRODUCTION

Paper is today, despite the increasing of virtual mass-media, still one of the favorite media for information distribution. It is used in almost all areas of work and industry. In graphic technology its optical properties are very important, as they often identify types of paper according to its purpose.

The motivation of this simulation was to describe and anticipate the transport of light through a media such as paper and to see the impact the dispersion has had on the optical dot gain. The tone resolution and a reproduction characteristic for all graphics paper products is significantly conditioned by the way that the way the light scatters in a paper. The light that enters the paper between the raster elements can be scattered in the paper in all directions, including below raster element where it remains absorbed which causes the gray scale becoming darker than originally required. To manufacture a paper that successfully fulfils the needed optical properties, it is necessary to understand the physical principles of the structure and composition of a paper sheet. This theoretical work was performed within frame of Monte Carlo method which describes the transport of light. This is a numerical method for solving mathematical problems based on a random sampling from well defined probability distributions. Starting from real physical assumptions the subsurface light scattering in a substrate with a complex structure was modeled (Kubelka, 1931, Emmel, 1999, Mourad, 2002). For this kind of problem, where statistics approach offer the best insight and approximation for exact results, a method as Monte Carlo offers a more flexible approach to the transport of photons in a medium such as paper. Even though this model has pure stochastic nature it makes feasible quasi experimental approach in optical dot gain studying.

2. THE MONTE CARLO METHOD

Monte Carlo method was originally designed for needs of the studying of the interaction of elemental particles (neutrons, mesons...) with matter, but is also used to solve problems of radiation transport in high energetic physics, analysis of nuclear reactors, calculation of protection from harmful radiation, in treatment of cancer cells with radiation, etc. (Kalos (1986)). For implementation of this method a stochastic model is constructed in which expected variable value (or combination of variables), is equivalent to the physical value that needs to be determined. The expected value is defined by medium value of multiple independent samples that represent this random variable. We use random generated numbers, which follow the prior selected natural distribution, to construct the desired array of independent samples.

The Monte Carlo method simulations of this type are based on the macroscopic optical properties for which it is assumed that they prevail over small parts of the volume of paper (e.g. cellulose fibers, fillers, adhesives, etc). (Veach (1997)).

We would like to present a typical trajectory of a single photon and the method describes the local rules of propagation of photons. According to well known procedure each step size between photon--substrate interaction positions is variable and equals

-ln[xi]/([[mu].sub.a] + [[mu].sub.s]) (1)

where [xi] is a random number and [[mu].sub.a] and [[mu].sub.s] are the absorption and scattering coefficients, respectively. Every photon has assigned statistical weight which decrease from an initial value of 1 as it moves through the substrate, and equals [a.sup.n] after n steps, where a is the albedo:

a = [[mu].sub.s]/[[mu].sub.a] + [[mu].sub.s] (2)

Once the photon packet has been moved the photon packet is ready to be scattered. There will be a deflection angle, [theta] [member of] [0, [pi]), and an azimuthal angle, [psi] [member of] [0, 2[pi]> to be sampled statistically. The probability distribution for the cosine of the deflection angle, cos[theta], is described by the scattering function that Henyey and Greenstein (1941) originally proposed for galactic scattering:

p(cos[theta]) = 1 - [g.sup.2]/2[(1 + [g.sup.2] - 2g cos[theta]).sup.3/2] (3)

where the anisotropy, g, equals <cos[theta]> and has a value between -1 and 1(details in Modric, 2007).

When the photon strikes the surface, a fraction of the photon weight escapes as reflectance and the remaining weight is internally reflected and continues to propagate. Eventually, the photon weight drops below a threshold level and the simulation for that photon is terminated. In this example, termination occured when the last significant fraction of remaining photon weight escaped at the surface at some position which differs from incoming point. To satisfy energy conservation law we used photon packet of hundred photons.

[FIGURE 1 OMITTED]

We calculated numerous photon trajectories (104 to 106) to yield a plausible statistical description of photon distribution in the medium.

It is to emphasize that our simulation does not consider the wave nature of light, and that ignores values such as phase or light polarization. Inside the paper as an extremely complex media photons experience multiple scattering and the phase and polarization randomizes very quickly so that initially they don't affect much the energy transport.

3. MODELLING AND SOLVING THE PROBLEM

In this work we've approached the problem of light scattering in the paper in the way that describes the real situation in a more physical way. Our simulation offers a flexible and yet rigorous approach to the problem of transporting light in a media such as paper.

The complexity of the paper surface structure encouraged us to improve the models with the aim of approaching a more realistic description of the surface. In our first approximation we assumed that our paper surface was perfectly flat and that stretches infinitely in the X and Y plane. Above presented scattering profile is averaged overall scattering angle distribution realized with light beam which lighten whole paper sheet (Modric, 2007).

Fig. 2. confirms the initial idea that increasing of paper roughness spreads distribution of scattered light due to the additional randomization of initial direction of incoming light.

[FIGURE 2 OMITTED]

By varying parameter [[??].sub.m] we can also predict light behavior for calendared papers or papers with some additional surface superstructure. As it could be seen the most spread distribution is for papers with parameter [[??].sub.m] = 53[degrees] which was presented only for illustration because paper with such surface doesn't have any commercial significance (Modric, 2007).

4. CONCLUSION

The contribution of this work is clearly connected with the demands of an optimal reproduction and print quality. All paper components such as mechanical and/or chemical pulp, whiteners, fillings, adhesives etc., affect the way the light scatters in paper as well its surface properties. It is evident that paper appearance is not consequence generally of its surface topography but also of its subsurface optical properties.

The light that enters the paper between the raster elements can be scattered in the paper in all directions, including below, where the dot element remains absorbed which causes the gray scale becoming darker than originally required. Starting from real physical assumptions the undersurface light scattering in a substrate with a complex structure was modelled.

Our model offers the possibility of "experimenting" with various combinations of paper components to verify some ideas without the long-lasting and expensive realizing an actual paper. Given the combination and variation possibilities of the composition share of each component of a paper, this model can study the optical properties of any type of paper, including the recycled ones, where as one component appears the particle remains of the dyes and the treated components of previous paper.

We expect that improvement and optimization of our model will lead to its implementation in manufacturing process. Our future efforts will be directed to improve our model of paper (with generalization on all substrates) and its interactions with dye, caused with various printing techniques and dyes, and to examine potential relationships between optical and mechanical properties of substrate. Beside paper--dye interaction (mainly, cross-section profile of raster element) our interest will also be focused on investigation of influence of anisotropy factor in our model which could be important for optimization of paper components initial mixture.

It should be pointed that we haven't include wavelength dependence of scattered light because there are no data in literature for scattering and absorption coefficients of every component. However, implementation of such dependence in model is trivial.

5. REFERENCES

Emmel P. and Hersch R.D., Towards a color prediction model for printed patches, IEEE Comp. Graphics and Appl. 19 (1999), 54-60.

Kalos, M.H., Whitlock P.A., (1986). Monte Carlo Methods, I: Basics. John Wiley & Sons, Inc.

Kubelka P., Munk F. (1931), Ein Beitrag zur Optik der Farbanstriche, Z.Tech.Phys., 11a (1931), pp 593-601.

Modric, D., Mikac Dadic, V., Dzimbeg-Malcic V., Light Scattering Numerical Modeling Compared with Kubelka Munk Method, Proceedings of the 11th International Conference on Printing, Design and Graphic Communications, Zadar, (2007) 107-111

Modric, D. (2007), PhD Thesis, University of Zagreb

Mourad M.S., Color predicting model for electrophotographic prints on common office paper, (2002), Ph.D Thesis, Ecole Polytechnique Federale de Lausnne.

Veach E. (1997), Robust Monte Carlo Methods for Light Transport Simulation. PhD thesis, Stanford University