Coherent Backscattering from Multiple Scattering Systems - KOPS ...

2.4 The influence of boundaries
Figure 2.7: Scattering in the presence of two parallel boundaries. The presence of
two parallel boundaries with distance L creates two series of image points B
1 ′ , B′ 2 , B′ 3 , . . .
and B
1 ′′,
B′′ 2 , . . .. The effective sample thickness L′ = L + 2z 0 simplifies the series.
On the other hand, all photon paths that lead to B ′ contain at least one such point P, which
is located at an average distance z 0 from the boundary. The photon density that is missing
from the free space photon density in B due to the sample boundary is therefore the free
space photon density in B ′ , and the photon density distribution in the presence of a boundary
becomes
ρ(A → B, t) 1 surface =
= ρ(A → B, t) − ρ(A → B ′ , t)
= e− τ
t
√
(e ( − ⃗r ⊥,B −⃗r ⊥,A) 2 +(z B −z A ) 2
4Dt − e ( − ⃗r ⊥,B −⃗r ⊥,A) 2 +(−z B +2z 0 −z A ) 2
4Dt
3
4πDt
)
(2.9)
where we have also considered absorption by its τ-dependent exponential decay. The vector
⃗r ⊥ denotes the components of the position vector⃗r perpendicular to the z-axis and parallel to
the sample boundary. Note that z 0 and z B or respectively z A have different signs, as depicted
in fig. 2.6.
If a second surface opposite to the entrance surface is to be considered, things become a little
more complicated. In this case, the photon density in B is the free space photon density
distribution minus the photon densities at the two image points B ′ and B ′′ with respect to the
two mirror planes in front of the surfaces at z = z 0 and z = L + z 0 . These photon densities
again have to be calculated as the densities in the presence of the other surface, and so on.
13