Coherent Backscattering from Multiple Scattering Systems - KOPS ...
Coherent Backscattering from Multiple Scattering Systems - KOPS ...
Coherent Backscattering from Multiple Scattering Systems - KOPS ...
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2 Theory<br />
Figure 2.5: Random walk. The vector⃗r denotes the displacement of a photon after M<br />
steps ∆ri ⃗ , where θ i is the angle between two consecutive steps ∆ri−1 ⃗ and ∆ri ⃗ .<br />
characterize the spreading of a cloud of photons that started off at a certain point and a certain<br />
time that we set⃗r = 0 and t = 0 we use the mean square displacement<br />
〈r 2 (t)〉 = 1 N<br />
( )<br />
N M(t) 2<br />
∑ ∑<br />
⃗∆r m,n<br />
n=1 m=1<br />
where M(t) is the number of steps ⃗ ∆r the photons have traveled after a certain time t, and N<br />
is the number of photon paths considered (fig. 2.5). For large M(t) this is [38]<br />
〈r 2 (t)〉 ≈ M(t) 〈 ⃗ ∆r<br />
2<br />
〉 + 2 〈 ⃗ ∆r〉<br />
2<br />
M(t) 〈cos θ〉<br />
1 − 〈cos θ〉<br />
where θ is the angle between two photon steps, and 〈cos θ〉 expresses the anisotropy of the<br />
scattering.<br />
For an exponential step length distribution p(∆r) = 1 ∆r<br />
l<br />
e− l we obtain 〈∆r〉 = l and 〈∆r 2 〉 = 2l 2 ,<br />
and<br />
〈r 2 (t)〉 = 2 M(t) l ·<br />
l<br />
1 − 〈cos θ〉 ≡ 2 M∗ (t) l ∗2 ≡ 2s(t)l ∗ (2.4)<br />
Hence the mean square displacement can be expressed with the help of three different length<br />
scales.<br />
The first length scale is the scattering mean free path l, which characterizes the distribution of<br />
the physical step lengths <strong>from</strong> one scattering site to the next.<br />
The angular distribution of single scattering given by Mie theory correlates the directions of<br />
two successive photon steps. In the mean square displacement this correlation is expressed<br />
by an additional factor (1 − 〈cos θ〉) −1 . The transport mean free path l ∗ l<br />
= and the<br />
1−〈cos θ〉<br />
effective number of photon steps M ∗ = M(t)(1 − 〈cos θ〉) incorporate this anisotropy factor. l ∗<br />
is therefore the distance after which the photons have lost the memory of their initial direction<br />
10