Coherent Backscattering from Multiple Scattering Systems - KOPS ...

5 Experiments
space with the energy inside an absorbing cut-out of the half-space at the position of the
sample:
j with losses (⃗r ⊥ , t) = A(t) · j lossless (⃗r ⊥ , t) = j lossless (⃗r ⊥ , t) · ∫
∫
finite sample
infinite half-space
ρ τ (⃗r, t) d⃗r
ρ τ→∞ (⃗r, t) d⃗r
This ignores the fact that the photon density distribution is altered by the sample boundaries,
but if the sample is large enough, only a few photons reach those boundaries anyway, so that
the presence of the boundaries is of nearly no consequence for the density distribution.
As solving eqns. 5.5 and 5.3 for the photon density distribution of the infinite half-space raises
considerable algebraic and numeric difficulties, we fall back to the density distribution in free
space, where the solution is easier to obtain. Fig. 5.2 shows that the albedos A(t) are similar
enough to make this assumption, although the actual photon density distributions look quite
different (fig. 5.3). The only difference in the albedos occurs when the photon cloud reaches
the boundaries, which for the teflon sample in the figure happens at photon travel times
around 10 to 100 ns. Even at this point the error is only a few percent, which presumably is
not larger than the error that is already made by ignoring the altered density distribution.
(5.5)
Evaluating eqn. 5.5 gives the albedo
∫ L
1
A(τ, L, R) =
π(l ∗ +z 0 )
0
(√
− K 0
K 0
(√
4(l ∗ +z 0 ) 2 +z 2
Dτ
z 2
Dτ
)
− 2K 0
(√
)
+ 2K 0
(√
) (√
R 2 +z 2
Dτ
+ K 0
R 2 +4(l ∗ +z 0 ) 2 +z 2
Dτ
)
− K 0
(√
)
2R 2 +z 2
Dτ
−
2R 2 +4(l ∗ +z 0 ) 2 +z 2
Dτ
)
dz (5.6)
where the K n (x) are modified Bessel functions of the second kind. The last step of integration
is performed numerically.
The transport mean free path l ∗ is the result of the evaluation of the backscattering data, for
which the albedo is an input parameter. Therefore the correct value for l ∗ is not yet known
when the albedo is calculated. Instead, we use l ∗ = 3D v
≈ 3n effD
c 0
(see sec. 2.3), which should at
least have the right order of magnitude.
The resulting albedos for the samples examined in the backscattering experiments are given in
tab. 5.1. One immediately observes the striking difference between the albedos of the titania
samples, which are in excess of 99%, and the albedo of teflon, which is about 90%. The
accuracy of the experiments therefore depends strongly on the accuracy of the teflon albedo.
It is however nearly impossible to give an error for the calculated albedos, as the validity of the
various assumptions is not known. Altogether, the error of the albedo mismatch is probably
of the order of one or two percent.
5.1.3 The correct theory of coherent backscattering
The first steps towards a theoretical description of coherent backscattering have already been
taken in sec. 2.1, where the intensity in a multiple scattering medium is presented as the
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