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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
87
1
1
( )
(
) ( )
(1 ) .
N
i
i
F x F x x F x R
R
(16)
Note, there is the enhancement of
F
−
(
x
) with the increment of Δ
x
. The more size Δ
x
and
N
are, the more
F
−
(
x
) is formed due to the backscattering of
F
+
(
x
). The sum in
brackets is the geometric progression describing the decrement of
F
+
(
x
) inside Δ
x
. It
can be calculated directly:
1
1
(1 ) 1 (1 ) 1
(1 )
.
(1 ) 1
N
N
N
i
i
R
R
R
R
R
(17)
Hence,
(
) ( ) ( )[(1 ) 1],
N
F F x x F x F x R
(18)
and finally:
( )
( ),
dF x SF x
dx
(19)
that is similar to Eq. (15). Thus, we found a set of two coupled linear differential
equations of the first-order describing the 1D pure scattering problem with the use of
SSA:
( )
( );
( )
( ).
dF x SF x
dx
dF x SF x
dx
(20)
At boundary condition
x
= 0:
F
+
(0) =
F
0
;
x
=
H
0
:
F
−
(
H
0
) = 0, one can solve the system
with the output:
0
0
0
( )
; ( )
(
).
Sx
Sx
SH
F x F e
F x F e
e
(21)
For professionals such a result is expectable and near trivial. Nevertheless, it is
important to understand better all our next results. For example, we can note, that
scattering exponential laws in Eqs. (21), similar to the Bouguer's law, is appeared here
under the SSA as the direct consequence of SSA. Together with the obtained expres-
sion for the scattering coefficient at SSA
ln(1 )
S
R
(22)
it forms our first main output.
Multiple scattering approach.
The case of multiple scattering is more interesting.
For multiple scattering approach (MSA), we need to take into account all multiple
reflections (re-reflections) between all scatterers
r
i
inside the medium. Multiple
scattering will lead to a phenomenon that every flux will be reduced in Δ
x
due to the
reflections (scattering) on boundaries of inhomogeneities, as it was at single