2.3 Anderson Localisation
2.3.4 Coherent Backscattering
Coherent backscattering is an experimentally very useful tool to determine the mean free scattering pathl∗of a sample. To obtain this effect the medium does not need to be localising, which is very similar to the mechanism of Anderson localisation.
If we look at a scattering medium in reflection we will see a speckle pattern, ori-ginating from the random interferences of the back scattered waves. We want to consider the introduced idea of Anderson localisation. The basic idea behind co-herent backscattering is that it is possible to have counter propagating paths in back reflection, which is called weak localisation. These paths are picking up the same phase shift leading to constructive interference in backscattering direction, see fig.2.8. It is similar to interference from double slits with the slit distance be-ing the start and end point of the pathρ. Such an interference pattern will have a contribution of 1+cos(qρ). The contributions to constructive interference of all double slits are adding up to a contribution only in direct backscattering dir-ection, leading to an enhancement of a factor of two. This signal survives all averages in the medium, unlike the speckle pattern, which will give a mean in-tensity. Weighted with probability of a distanceρto occur we get the coherent backscattering cone:
α(q)= Z
p(ρ) cos(qρ)dρ (2.37)
The solution for the shape of the backscattering cone can be found in e. g. in [39,77–80]. For a more detailed description of coherent backscattering as here presented see for example [39,79,80].
From now on we will follow the description of Akkermans and Montambaux [79].
A plane and uniform wave illuminates the multiple scattering medium perpen-dicular to its surface. The wave should have infinite spatial and temporal coher-ence and the medium should not be absorbing and semi-infinite. The backs-cattered wave can be described with an incoherent partαd, called ‘Diffuson’, and a coherent partαc, called ‘Cooperon’. Single scattering is not included in this description. The expression for the incoherent part is:
αd= 3 4πµ
µz0 l∗+ µ
µ+1
¶
(2.38)
Withµ=cos(θ) andz0=23l∗being the average penetration depth5. The coherent
5In [79] it is mentioned thatz0'0.710l∗is the exact solution of the Milne problem, but this is not consistent with the diffusion approximation.
Chapter 2. Theory
0 0
1 1
2 2
θ 0
rxy I(θ)
Figure 2.8: The principle of coherent backscattering is shown. Counter propagat-ing paths interfere similar to a double slit with the distanceρ(rxy). The dotted part corresponds to a longer path that includes internal reflec-tions. In the left arc three generic interference patterns are shown. All these contributions will sum up to the coherent backscattering cone, right arc. The result is an enhancement of a factor of two in direct backscattering direction. Figure taken from [49] (modified).
part is described by:
αc= 3 8π
1
³
k⊥l∗+µ+12µ
´2
µ1−e−2k⊥z0 k⊥l∗ + 2µ
µ+1
¶
(2.39)
Here isk⊥=k|sin(θ)|. What we want to note is that in exact backscattering dir-ection (α=0) the ‘Diffusion’ and the ‘Cooperon’ are equal to one, giving a backs-cattered intensity enhanced by a factor of two.
α(θ)=αd(θ)+αc(θ) (2.40)
α(0)=2αd (2.41)
Short paths correspond to big angles and long paths to small angles on average.
Without any cut-off the cone tip would be triangular, but absorption and local-isation introduce cut-off lengths, reducing longer paths. The result is a rounding of the cone tip. The mean free path is determinig the width of the cone. Ifl∗ be-comes smaller the cone is getting wider, the relation is: FWHM−1≈kl∗. Internal reflections at the sample boundaries can extend diffusive paths, which would lead to an overestimation ofkl∗. The corrected FWHM was calculated by Zhu et
23
Anderson Localisation
al. [81], with the reflectivityR(see paper for definition):
FWHM−1=
In the case of internal reflections also the penetration depthz0has therefore to be corrected [81]:
The term for absorption can be introduced with the simple substitution of:
k⊥→
Combining internal reflections and absorption, under the assumption of weak absorption (l∗ ¿la =p
DBτa), which should be valid for all our samples, the
‘Cooperon’ then yields6: αc= 3
The latter derivation does not include energy conservation. This aspect is taken into account in Fiebig et al. [83]. They derived a correction to the ‘Cooperon’αe
that ensures that energy conservation is fulfilled. The correction factor is propor-tional to−(kl∗)−2and is given by:
αe' − 1.15 (kl∗)2
µ
µ+1 (2.47)
The factor of 1.15 is in this case an approximation, matching the experimental parameters and does not ensure energy conservation for other parameters. To have energy conservation for all situations we have to solve the integral over the corrected cone:
Z
(αc+αe) sinθdθ=0 (2.48)
6In the PhD thesis of S. Fiebig the ‘Cooperon’ (eq. (2.18)) is given without the assumption of l∗¿la=p
DBτa, using an other normalisation [82]. In the given equation it was not explicitly mentioned to usez0with internal reflections, which should be done.
Chapter 2. Theory
By introducing the parameteraintoαeas done in [82], replacing the factor 1.15, we can ensure energy conservation.
αe= − a (kl∗)2
µ
µ+1 (2.49)
If we use this expression ofαewe get, by solving fora, a general expression that gives energy conservation7:
a=
Rπ/2
0 αcsinθdθ Rπ/2
0 1 (kl∗)2
µ+1µ sinθdθ (2.50)
Recently there was an alternative derivation published for the energy conserva-tion in coherent backscattering, putting in quesconserva-tion the calculaconserva-tion made in [83].
They found a correction of ln(kl∗)/(kl∗)2[84,85]. However, this result is equal to the form of the previous correction, if we seeaas a fit parameter:
ln(kl∗) (kl∗)2 = a
(kl∗)2 (2.51)
We want to introduce briefly the concept of the coherent forward scattering cone [86,87]. In addition to coherent backscattering, as the name indicates, there should also be a cone in forward scattering direction. The idea behind the for-ward scattering cone is that if a wave-package launched inside a random poten-tial the coherent backscattering cone is complemented by a forward cone. The authors state, that the forward cone should only appear in the regime of Ander-son localisation and thus providing a useful tool to proof the existence of loc-alisation. The reason why we mention this interesting effect is that we tried to measure the forward cone, but did not succeed and will not go into any detail in this work. It failed most probably because we are not able to launch the wave in-side our scattering medium. The best candidate to measure this effect is a matter wave experiment where a laser creates a speckle potential, which can be switched on at any time (see section3.2). This way the disorder can be turned on to have the wave package inside the speckle potential.