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A New Look at Fundamentals of the Photometric Light Transport and Scattering Theory
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 5
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heterogeneities inside the medium (Rayleigh scattering, Mie scattering, etc. [4]).
In these cases, the diffraction technique is also used, but together with the statistical
formalism. Scattering cross-section as an average value over particles and (or) over
scattering angles is usually acted as the scattering properties of such a medium.
However, this complex computational formalism has no clear expression by any
simple formula. Moreover, the electromagnetic scattering cross-section is not
identical to the photometric scattering coefficient. Often, for the turbid media of
discrete particles the scattering coefficient is defined as an integral over the solid angle
of the differential scattering cross-section for a single particle multiplied by average
density of particles inside the medium [5, 6]. Nevertheless, in our opinion, this
diffraction-based approach with the use of diffraction integrals cannot be adopted as
the ''first principle'' for the photometric LT&ST. The scattering coefficient
s
, as it was
initially involved in RTE, was the original turbid medium property, which was
introduced purely heuristically, without the use of any items of electromagnetisms.
Therefore, there is the need to study this problem in detail. Therefore, the issue of the
scattering coefficient formation in the pure photometric theory will be our main
objective for the first part of the article.
One dimensional scattering problems.
Let us start with the simplest one-
dimensional (1D) scattering problem. Although, the 1D model seems to be very far
from the reality, it is the basis of the Bouguer's law, of the Schuster — Schwartzchild
approximation [7, 8], Kubelka — Munk (KM) approximation [5, 9], and of many
other approaches, so the selection of such elementary model is not accidental, but is
determined by a series of essential advantages. First, the simple 1D consideration of
light ray absorption and scattering temporarily avoids complications with the
definition of phase scattering functions, and therefore, allows us to concentrate on the
phenomenological fundamentals of the scattering coefficient definition. Second, the
1D model is the simplest approximation of RTE. Having the precise and analytical
solutions for all basic tasks, it opens a very powerful and convenient way to compare
these solutions with any other results based on other approaches. Third, in full
formulation with multiple scattering and absorption, the 1D model, being known
more as the two-flux Kubelka — Munk (KM) model, has totally accepted opinion
about its inaccuracy, as well as about disharmony of the results based on KM model
and results based on RTE. Especially it concerns the compliance between scattering
and absorption coefficients in KM approximation and in the general RTE. Thus, the
study of the basic 1D scattering problems as fundamentals of LT&ST is of great
importance, in our opinion.
Preliminary remarks.
Our experience on publications as well as conference
presentations shows that professionals do not always understand 1D formulation of
the scattering problem in the same way. Therefore, there is a need to clarify our 1D
approach in the beginning of the section. In many publications, 2D or 3D radiative
transfer problem is considered. Usually, the formulation of the problem looks like it is
shown in Fig. 3 in the ''flat'' (plane layer) multidimensional formulation [10].