approximation as discussed above, one gets an expression for the transmit-ted intensity of a point source through a slab of thickness L (M. Stoerzer:
[St¨o06])
One can see, that the long-time behaviour of thisI(t) reads as I(t)∝exp−
as the higher n terms in the sum do not contribute to sufficiently large times. This means, that for long times classical diffusion should be visible in a linear decay in a semi-log plot as can be seen in fig. 2.2. Measuring the time-dependent transmission in a Time-of-Flight experiment we are therefore able to determine D0 and τabs of a sample of thickness L. Introducing the diffusion time
τD:= L2
π2D0, (2.16)
one can also approximately calculate the maximum time of I(t), using only the n = 1 term of the sum and neglecting the other terms (see appending A), to: Given, that the sample propertiesD0 andτabs, being intrinsic, do not change with the thickness of a slab, we see that τmax is roughly proportional to τD, whereas the ln-term gives small corrections depending on the absorp-tion. However, the sample properties are comparable and ln varies slowly, therefore, we can in good approximation eliminate the dependence of the Time-of-Flight curves from L by normalizing the time axis toτmax.
2.2 Anderson Localization
The diffusion of photons as described before, does not take into account the wave character of light, but exactly this property of light enables photons to interfere constructively or destructively. The interference leads, as P. W.
Anderson stated 1958 in [And58] (for electronic systems), to a change from
Figure 2.2: Transmitted intensity through a slab of multiple scattering ma-terial, see text.
diffusive behaviour in a random lattice, i.e. in a random conformation of scatterers. The idea will be phenomenologically sketched in the following.
Imagine two photons diffusing in a multiple scattering sample on the same path, but in opposite direction, having originated from the same in-cident wave as depicted in fig. 2.3 on the left-hand side. If we can assume path-reversal invariance2 the two photons keep their ability to interfere con-structively, because they collect on their path the same phase changes. This interference effect is called weak localization and gives rise to e.g. Coherent Backscattering as will be explained in section 2.5. If we now let the two photons start from the same scatterer inside the medium as depicted in fig.
2.3 on the right hand side, we get a closed loop, on wich the photons interfere constructively. This might happen in any medium, but if we now increase the scatterer density to higher levels, at some point the probability of having such loops will be high enough to hinder normal diffusion of the photons, and the diffusion will finally break down when the probability for each pho-ton to have a counterpropagating phopho-ton reaches 1. This is called strong localization and has been first described by P.W. Anderson [And58] for an electronic system, where this effect would lead to a metal-insulator transi-tion. The Ioffe-Regel criterion [IR60] states, that this should be the case for kl∗ ≈13.The dimensionality of the system is important, as has been treated
2Normally, one speaks of time-reversal invariance, which is easier to understand, but, as we will see with non-linear effects, will give rise to much more puzzling in understanding Anderson Localization
3This criterion actually implies, that a wave will be scattered roughly 6 times during one period, which means that one cannot think of any plane wave, not to speak of geometrical rays as have been used to depict the idea here, for any real strongly scattering sample.
2.2 Anderson Localization Theoretical Background
Figure 2.3: Left-hand side: two counterpropagating photons from the same incident light wave in a multiple scattering (weak localization). Right-hand side: the same inside a sample originating from one scatterer (strong local-ization) - see text.
by Abrahams et al. [AALR79]: they show, that for 1D and 2D systems any wave will localize in presence of a disordered lattice of scatterers , whereas a phase transition from a diffusive to a localized state is to be expected only for 3D systems, as will be presented later in sec. 2.3. Following from that, 1D and 2D systems are interesting for technical applications, but from the physical point of view, only 3D systems are interesting for understanding the phenomenon.
From this picture it becomes clear, that in the case of a localizing sam-ple, additional quantities are necessary for characterizing the sample: the microscopic localization length lloc which is the path length necessary for the photons to experience the looping effect, connected to the macroscopic localization length ξ, i.e. the measurable length scale, via
ξ =p
l∗lloc/3 (2.18)
2.2.1 A Short Overview of Experiments on Anderson Localization
Since the mechanism of Anderson Localization requires only waves and a random scattering potential, different systems should exhibit Anderson Lo-calization. Therefore systems using different waves, such as microwaves, ultrasonic waves, matter waves (i.e. ultra-cold atoms) and light have been used to study the effect of Anderson Localization. We will shortly summarize the systems of the present state to our knowledge (without the claim to be fully complete).
1D-systems In the group of A. Genack, experiments with microwaves scat-tered at aluminum beads in a quasi-1D-waveguide were carried out. They have measured ToF, speckle statistics and transmission profiles. [CG01]
[CZG03] [ZHSG07] [BBF+08] [WG11]. Also, photonic bandgap materials have been used as 1D waveguides for light and strong localization has been shown by the group of F. Vollmer [TIV07]. Ultracold atoms interacting with a random speckle light field as scattering potential have been used by Billy et al. [BJZ+08] and Roati et al. [RDF+08] to show Anderson localization for matter waves.
2D-systems A photonic bandgap material study for 2D Anderson localiza-tion has been presented by Schwartz et al [SBFS07], although this experiment is, in fact, a 3D measurement since the localization takes places along the z-axis through a 3D crystal with rod-like structure. Also Topolancik et al.
[TIV07] published signatures of Anderson localization in such materials, as well as did later Riboli et al. [RBV+11].
3D-systems Wiersma et al. claimed 1997 [WBLR97] to have found Ander-son Localization of light in a 3D system, the results having been questioned by F. Scheffold et al. [SLTM99], since they could be explained simply by absorption. Recently, v. d. Beek et al. [vdBBJ+12] published a study of similar particles as were used for the experiment in discussion and they found measurement results consistent with a purely classical description with an absorption length proposed by Aegerter et al. in [AM09]. M. Stoerzer et al. have reported Anderson localization [SGAM06] using time of flight measurements with the setups being described in this work.
Kondov et al. [KMZD11] report on Anderson Localization with ultracold atoms interacting with random speckle patterns. Also, Hu et al. have ex-amined [HSP+08] ultrasound scattered at alumina beads4 in the group of J.
Page finding signs of Anderson Localization using time of flight experiments.
As we have seen in the phenomenological picture, the diffusion coefficient should become dependent on the path length of a photon through the sample, as the forming of the interference loops will need a certain spatial extend.
Since the path length of a given photon path corresponds to the traveling time through the sample, the change of the diffusion coefficient due to localization
4It should be mentioned, however, that the samples used for ultrasonic studies are rather thin in units of l∗, which leaves the question if those samples are large enough to allow for closed loops to be formed. The impedance mismatch, and therefore internal reflections might be the reason for the sample being virtually sufficiently large enough.