Coherent Backscattering from Multiple Scattering Systems - KOPS ...

2 Theory
Figure 2.5: Random walk. The vector⃗r denotes the displacement of a photon after M
steps ∆ri ⃗ , where θ i is the angle between two consecutive steps ∆ri−1 ⃗ and ∆ri ⃗ .
characterize the spreading of a cloud of photons that started off at a certain point and a certain
time that we set⃗r = 0 and t = 0 we use the mean square displacement
〈r 2 (t)〉 = 1 N
( )
N M(t) 2
∑ ∑
⃗∆r m,n
n=1 m=1
where M(t) is the number of steps ⃗ ∆r the photons have traveled after a certain time t, and N
is the number of photon paths considered (fig. 2.5). For large M(t) this is [38]
〈r 2 (t)〉 ≈ M(t) 〈 ⃗ ∆r
2
〉 + 2 〈 ⃗ ∆r〉
2
M(t) 〈cos θ〉
1 − 〈cos θ〉
where θ is the angle between two photon steps, and 〈cos θ〉 expresses the anisotropy of the
scattering.
For an exponential step length distribution p(∆r) = 1 ∆r
l
e− l we obtain 〈∆r〉 = l and 〈∆r 2 〉 = 2l 2 ,
and
〈r 2 (t)〉 = 2 M(t) l ·
l
1 − 〈cos θ〉 ≡ 2 M∗ (t) l ∗2 ≡ 2s(t)l ∗ (2.4)
Hence the mean square displacement can be expressed with the help of three different length
scales.
The first length scale is the scattering mean free path l, which characterizes the distribution of
the physical step lengths from one scattering site to the next.
The angular distribution of single scattering given by Mie theory correlates the directions of
two successive photon steps. In the mean square displacement this correlation is expressed
by an additional factor (1 − 〈cos θ〉) −1 . The transport mean free path l ∗ l
= and the
1−〈cos θ〉
effective number of photon steps M ∗ = M(t)(1 − 〈cos θ〉) incorporate this anisotropy factor. l ∗
is therefore the distance after which the photons have lost the memory of their initial direction
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