2.4 Approximate Methods for Solving the Transport Equation
19
where mean free path of photon is
l ph =
1
μ a + μ s
Since
∞
0
p(L)d L = 1.
For the calculation of the mean free path we take random number ξ ∈ (0, 1):
ξ =
L
0
p(l)dl.
The number ξ , which is uniformly distributed in the interval (0, 1), is given as
computer generated random number.
Thus, the free path of a photon is:
L = −l ph ln(l − ξ).
After that one models the interaction of a photon with a particle of the medium, which
can be either absorbing or scattering center. The probability of photon scattering on
the particle is
p s =
μ s
μ s
+ μ a ,
The probability of absorption is:
p a =
μ a
μ s
+ μ a = 1 − p s .
If the generator produces a random number in the range (0, p), then the photon is considered to be scattered, otherwise it is absorbed. The total layer of the medium along
the z-axis virtually divided into a number of thinner layers having equal thickness
to which data arrays correspond. In each array the number of absorbed or scattered
photons is recorded. Thus, the spatial resolution of the depth of the sample is
1
L ave
.
If the photon is scattered, its new direction and coordinates are calculated with the
following formulas::
x = x 0 + L sin θ cos ϕ,
y = y 0 + L sin θ sin ϕ,
19
where mean free path of photon is
l ph =
1
μ a + μ s
Since
∞
0
p(L)d L = 1.
For the calculation of the mean free path we take random number ξ ∈ (0, 1):
ξ =
L
0
p(l)dl.
The number ξ , which is uniformly distributed in the interval (0, 1), is given as
computer generated random number.
Thus, the free path of a photon is:
L = −l ph ln(l − ξ).
After that one models the interaction of a photon with a particle of the medium, which
can be either absorbing or scattering center. The probability of photon scattering on
the particle is
p s =
μ s
μ s
+ μ a ,
The probability of absorption is:
p a =
μ a
μ s
+ μ a = 1 − p s .
If the generator produces a random number in the range (0, p), then the photon is considered to be scattered, otherwise it is absorbed. The total layer of the medium along
the z-axis virtually divided into a number of thinner layers having equal thickness
to which data arrays correspond. In each array the number of absorbed or scattered
photons is recorded. Thus, the spatial resolution of the depth of the sample is
1
L ave
.
If the photon is scattered, its new direction and coordinates are calculated with the
following formulas::
x = x 0 + L sin θ cos ϕ,
y = y 0 + L sin θ sin ϕ,