2.3 Anderson Localisation
2.3.6 Transverse Confinement
e−n
2π2D(t)t
L2 (2.52)
A deviation from the exponential decay is now introduced with the time depend-ent diffusion coefficidepend-entD(t), but neglecting the position dependence. However, the diffusion coefficient has still to be defined. The use of the time dependent diffusion coefficient of eq. (2.36) is not applicable, as for our strongest scattering samples the times would beα1∗ >100 ns andτD ≈8 ns, being too long to measure.
Additionally an expression at shorter times is missing. Zhang et al. [75] proposed, that we should use the calculatedD(t) from [69,89,90], which is not feasible for our system, as these calculations were done for one dimensional systems. The dependence on the dimensionless conductanceg would have to be a fit para-meter and would give the formD(t)/DB=A−B t, which does not fit our data.
Instead we use an empiric approach. The basic idea is that after a certain time, the localisation timeτloc, there will be a deviation from diffusive behaviour. The diffusion coefficient will decrease according to the following equation [47]:
D(t)=DB τloca
¡τmloc+tm¢ma (2.53)
Here a new parameter is introduced, the localisation exponenta. If there is only pure diffusionabecomes zero giving the classical description of diffusion, as in eq. (2.22). In the case of full localisation the parameter should be a =1. The valuemhas to be large enough to have a fast enough crossover from diffusion to localisation, setting it tom=10 satisfies a fast crossover. Further this approach seems feasible as former simulations done by Lenke et al. have shown a similar behaviour [91].
2.3.6 Transverse Confinement
An alternative way of studying the effect of Anderson localisation in transmission was proposed by N. Cherroret et al. [92]. They suggest to examine the transmis-sion profile instead of the overall intensity.
They use the self-consistent equations to derive the transverse confinement of a monochromatic, continuous beam and a short pulse focused to a point at the
Chapter 2. Theory
Sample
σ L
T(ρ)
Figure 2.9: The theory is assuming a spot like light source, thus the wave has to be focused onto the sample surface. The transmitted profileT(ρ) is observed and the mean square widthσ2evaluated.
surface of a sample in slab geometry. Therefore they introduce the mean square widthσ2.
σ2=
Rρ2T(ρ,t)d2ρ
RT(ρ,t)d2ρ (2.54)
2.3.6.1 Continuous Waves
For a detailed derivation see the PhD thesis of N. Cherroret [63], we will only focus on the results.
At first the case of static transmission shall be analysed. Therefore it is neces-sary to set the frequency in the self-consistent equations (2.33) and (2.34) equal to zero (Ω=0). As a consequence the time dependency of equation (2.54) disap-pears as well. Further only large samples, withLÀl∗, are considered.
In the case of pure diffusion (kl∗À1) the dependence of the diffusion constant on the position can be neglected and one can set:
D(z, 0)=DB µ
1− 1 (kl∗)2
¶
(2.55)
Solving the self-consistent equations then gives the mean square sizeσ2diff: σ2diff'2L2
3 (2.56)
27
Anderson Localisation
As expected, and typical for diffusion,σis proportional toL. The derivation for the localised regime (kl∗<1) is not as simple. The result forσ2loc reads as fol-lows:
σ2loc'2Lξ (2.57)
In contrast to diffusionσ2locdepends on the localisation lengthξand sample size L. At the mobility edge (kl∗=1) the mean square widthσ2mebecomes:
σ2me=3L2
8 (2.58)
Compared to the diffusive case there is only a difference in the pre-factor and still qualitatively different to the localised case. As a next step absorption will be included, but only for the diffusive case. In the limit of weak absorption (LaÀL) σ2changes to:
σ2diff,LaÀL=2L2 3 −2L2
45 µ L
La
¶2
(2.59)
And for strong absorption the mean square size will be:
σ2diff,La¿L=2LLa (2.60)
In the case of weak absorption there are only small additional corrections. How-ever, for strong absorption we can see that the absorption lengthLa plays sud-denly the same role as the localisation lengthξin equation (2.60). These results show that is is not possible to differentiate between localisation and absorption in a static experiment.
2.3.6.2 Short Pulses
A way to overcome the problem of distinguishing between absorption and local-isation is a time resolved measurement. A suitable way is the use of short pulses and examination of the dynamics. The mean-square width is now used in its time dependent form (eq. (2.54)). By doing so absorption (exp(−t/τa)) cancels out and will not influence the width. An intuitive explanation is possible if we are looking at the times the waves needed to travel through the sample. All waves detected at a certain time have spend, of course, the same amount of time in the sample, thus contributing to the profile width and being equally affected by absorption.
Nevertheless, absorption lowers the signal.
Unlike for the stationary case there is no simple analytic solution or approxim-ation forD(z,Ω) at arbitrary frequencies. That is why N. Cherroret solved the
Chapter 2. Theory
Figure 2.10: Numerical result forσ2(t) of the self-consistent equations are shown.
Diffusion (solid black curve), the mobility edge (solid red curve) and localisation (solid green, blue and pink curves) are shown. In the case of localisation a saturation can be observed. The plot was taken from the PhD thesis of N. Cherroret [63].
self-consistent equations numerically to computeσ2(t). The results of these cal-culations are shown in fig. 2.10. They are done for all three regimes for a fixed slab of the sizeL. The solid black curve shows diffusion, the solid red curve rep-resents the mobility edge and the localising case is shown with the solid green, blue and pink curves.
We will examine the long time evolution of the mean square width. In the case of pure diffusion, as an exception, an analytical solution exists.
σ2diff(t)'4DB µ
1− 1 (kl∗)2
¶
t (2.61)
As expected the spread is linear in time, which is also represented in the calcu-lations. Thekl∗ dependence is only a small correction and can be neglected.
The solution in the localised regime is made under the assumption of an infinite medium and was developed up to the first order inLξ ¿1.
σ2∞,loc=2Lξ µ
1−ξ L
¶
(2.62)
This result, in contrast to diffusion, has no more time dependence. This means that the spread is confined within a certain length given by the sample size and the localisation length. As can be seen, for example in the pink curve in fig.2.10, this results in a plateau at long times. The dashed line is a comparison with a
29
Anderson Localisation
numerical simulation, indicating that the result should be valid in the limit of big samplesLÀξ. Approaching the mobility edge (L≈ξ) is problematic, because equation (2.62) then becomes zero. At the mobility edge saturation also sets in, as can be seen in fig. 2.10. The asymptotic value for the mean-square size was found to be:
σ∞,ME≈L (2.63)
The short-time behaviour can be found in the inset of figure 2.10. The mean square width is growing in the limit of the mobility edge and the localisation re-gime withσ2(t)∝tα, withα'0.5. This finding is surprising, as one would expect at short times normal diffusive behaviour, because the wave has not exhibited any localisation effects.