Coherent Backscattering from Multiple Scattering Systems - KOPS ...

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