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 />
waves enter and leave the loop. As the light waves are always in phase at this point, constructive<br />
interference leads to an enhanced photon density compared to the normal diffusive<br />
behavior. In strongly scattering media the probability of photons running on closed loops is<br />
enhanced, so that a macroscopically reduced diffusion can be observed. Eventually this leads<br />
to a complete breakdown of photon transport and a transition to a localizing state, which is<br />
called Anderson localization [13].<br />
The critical amount of disorder for this phase transition can be expressed by the criterion<br />
proposed by Ioffe and Regel [27], namely that kl ∗ ≈ 1. The width of the steady-state coherent<br />
backscattering cone, which is inversely proportional to this quantity, is therefore an important<br />
experimental measure for the disorder in the sample [12].<br />
2.7 The theory of coherent backscattering<br />
To develop a theory for the steady-state coherent backscattering cone, the easiest case to consider<br />
is a uniform plane wave with infinite spatio-temporal coherence impinging perpendicularly<br />
on the surface of a non-absorbing multiple scattering medium. In this case the photon<br />
flux distribution has the simple form given in eqn. 2.11, and furthermore is translationally<br />
invariant, i.e. j back (⃗r ⊥,A ,⃗r ⊥,B , θ) becomes j back (⃗r ⊥ , θ) with⃗r ⊥ =⃗r ⊥,B −⃗r ⊥,A .<br />
We define the cooperon α c (θ) as the coherent and the diffuson α d (θ) as the incoherent addition<br />
of the flux emerging <strong>from</strong> the time-inverted paths in the sample:<br />
∫<br />
α d (θ) =<br />
∫<br />
j back (⃗r ⊥ , θ) d⃗r ⊥<br />
j back (⃗r ⊥ , θ = 0) d⃗r ⊥<br />
and α c (θ) =<br />
∫<br />
j back (⃗r ⊥ , θ) · e i ⃗ k⃗r⊥<br />
d⃗r ⊥<br />
∫<br />
(2.13)<br />
j back (⃗r ⊥ , θ = 0) d⃗r ⊥<br />
where⃗ k is the wave vector of the emitted light wave. The diffuson is sometimes also referred<br />
to as the incoherent background of the backscattering cone.<br />
If single and low order scattering can be blocked completely, the backscattered intensity measured<br />
in an experiment is the sum of these two contributions. Otherwise, the height of the<br />
cooperon compared to the diffuson is reduced, as single scattering contributes to the incoherent,<br />
but not to the coherent addition of the photon flux [38].<br />
Evaluating the above equations yields [10]<br />
( )<br />
α d (θ) = µ z0<br />
l<br />
+ µ<br />
∗ µ+1<br />
z 0<br />
l<br />
+ 1 ∗ 2<br />
(2.14)<br />
and<br />
α c (θ) =<br />
2 ( z 0<br />
l ∗ + 1 2<br />
1−e −2qz 0<br />
ql ∗ + 2µ<br />
µ+1<br />
) ( ql ∗ + µ+1<br />
2µ<br />
) 2<br />
(2.15)<br />
20