2 / 17
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
79
coefficient as a native physical property of a light scattering (turbid) medium. In
addition, some phenomenological nature of RTE and adopted universal form of RTE
for all practical problems simplifies and distorts our perception of the complex
mathematical formulation of the equation(s). Therefore, a number of other problems
still remained in a shadow. These two statements we would like to prove from the
''first principles'' in the paper.
We will consider only stationary, time-independent problems, because they are
exactly the fundamentals of the phenomenological LT&ST. We work on biomedical
optics, so examples for turbid biological tissues are more habitual to us. We will use them
in this paper. However, our results are also applicable to other light scattering media —
clouds, milk glasses, turbid liquids, etc. We assume a reader is familiar with the classic
LT&ST. Therefore, there is no need to explain the basics of the theory, the basic items, and
the existing variety of approaches to solve the direct task — Kubelka — Munk (KM)
approximation, diffusion equation, Monte-Carlo simulation, etc., as well as to describe the
difference between them. Nevertheless, we should make two preliminary notes.
1. Our note 1 concerns the photometric and phenomenological foundations of the
theory. Fundamentals of the photometric LT&ST were developed as early as the 18th
century. That time no one understood in a proper manner the wave nature of light,
especially its electromagnetic nature. LT&ST was created by P. Bouger, J.H. Lambert,
et al., as the pure ray theory — the theory describing light beam in terms of its power or
energy as perceived brightness by human eyes. Neither diffraction nor interference were
known that time in applications solving photometric issues. Understanding the waves,
its phase, a power density as a square of the wave field came much later. In the original
photometry, light beam was described as a radiant flux (beam of radiant rays with
power) propagating and decreasing inside a continuous medium. The flux had an
amplitude and a direction, but it was not a vector, because two fluxes were summed in
the theory by amplitudes as scalars at intersection in a point of the medium, and not as
vectors, unlike to electromagnetic fields. Classic RTE, which arose somewhat later, also,
initially represented the pure photometric and phenomenological formulation of the
problem.
The main law of the photometry is the exponential law discovered by P. Bouger
in the beginning of 18th century, also known as Lambert — Bouguer law, Bouguer's
law, etc. It relates the attenuation of the radiant flux
F
to the absorption properties of
the medium's substance, as well as to a distance, which the light travels inside the
medium. Therefore, RTE is primarily based on the Bouguer's law. So, there is a sense
to introduce it before the main issue. The simplest way to reproduce the Bouguer's
law from the ''first principles'' is as follows. Assume that an external beam of light
(flux)
F
0
enters a material sample from the left side (Fig. 1). Determine the material
sample is not light scattering, i. e., assume the pure absorption medium. Define
X
as
an axis parallel to the direction of the light beam propagation. Inside the sample, one
can observe the radiant flux
F
(
x
) of the light as a function of the fluent coordinate
x
.
Select within the sample an element Δ
x
and define the decrement (increment) of the
flux
F
(
x
) inside Δ
x
, obtaining the flux
F
(
x +
Δ
x
).