As we have seen in the intuitive picture of Anderson localization, the diffusion coefficient has to become position/time dependent. The self-consistent theory originally developed by Vollhard und W¨olfle [VW80a], [VW80b] (for 2D-systems) introduces this property of the diffusion coefficient. S. Skipetrov et B. van Tigellen [SvT06] expanded the theory to open 3D systems and applied it to slabs of multiple scattering material. These equations read in Fourier space as follows:
[−iΩ− ∇ ·D(Ω,r)∇]C(r,r0,Ω) =δ(r−r0) (2.21) 1
D(Ω,r) = 1 D0
+ 12π
k2l∗C(r,r0,Ω) (2.22) The first equation is a diffusion equation5 with a position-, and frequency dependent diffusion coefficient D. The dependence of the position is im-portant for the theory to be able to include boundary effects. C(r,r0,Ω) is the intensity Green’s function. Its inversely Fourier transformed counterpart C(r,r0, t−t0) can be physically interpreted as density of energy atr at time t from a short pulse emitted at r0 at time t0.
v. Tiggelen et. al. calculated in [vTLW00] the renormalized diffusion coefficient to be
D =D0
1− 1 (kl∗)2
(2.23) for kl∗ ≈ 1, recovering the Ioffe-Regel criterion [IR60], i.e. the stopping of diffusion forkl∗ = 1.
5Transforming the time variable into the frequency domain, one can simplify the diffu-sion equationut=c2·uxxto be a differential equation in space only, since ˆut=iωu.ˆ
2.4 Self-Consistent Theory Theoretical Background
2.4.1 Time-resolved 1D Transmission through a Slab
We have discussed the transmission through a slab of multiple scattering media in section 2.1.2 in the case of pure diffusion. For describing a localizing sample, the diffusion coefficient has to become a function of time: D(t) and has be accounted for in eq. (2.14). Berkovits and Kaveh [BK87] calculate the intensity of transmission through a slab for this case to
I(t)∝exp In other words: any deviation from an exponential decay at long times in-dicate a temporally varying diffusion coefficient, as has been found e.g. by M. Stoerzer [SGAM06]. For a complete localization, where the spread of the photon cloud comes to halt, one can easily conclude from
r2
=D(t)t≡const. (2.25)
that the diffusion coefficient has to scale according to D(t)∝ 1
t (2.26)
which has also been shown analytically by Skipetrov et al. using the self-consistent theory [SvT06]. Intuitively, we have already introduced the length scale ξ necessary for the photons to “see” localization. This has been shown by Lenke et al. [LTM02] using simulations of a self-attracted random walk to be qualitatively correct.
2.4.2 2D-Transmission Profiles
As has been described by N. Cherroret in [CSvT10] and [Che09] the two-dimensional time-resolved spreading of the photon cloud should reveal signa-tures of Anderson localization. The idea is, that the mean square width σ2 of the photonic cloud, if the photons show diffusive behaviour, should evolve linearly in time, i.e. in path length, so that the cloud spreads as sketched with dotted lines in fig. 2.5. In contrast, if localization takes place, the spread of the cloud in the sample should stop at the localization length ξ, i.e., when the photons feel the interference effect, as being marked with the orange area in fig. 2.5. He calculated the width, being defined as
σ2 =
R ρ2T(ρ)d2ρ
R T(ρ)d2ρ (2.27)
ρ
Figure 2.5: Sketch of the steady state transmission through a slab of ma-terial. The difference of pure diffusion and localization is marked, and the transmitted 2D intensity is sketched with the relevant quantities, as used in the text. Sketch courtesy of T. Sperling.
of a static transmission 2D profile from a point-source through a slab of turbid media to be [Che09]
σ2 ≈ neglecting absorption. In the diffusive regime for weak absorption (La L), i.e. the sample size being smaller than the macroscopic absorption length, absorption yields the following: and in case of strong absorption (La L), i.e. the sample size being much bigger than the macroscopic absorption length:
σdiff2 ≈2LLa. (2.30)
This expression has exactly the same structure as σ2loc = 2Lξ from eq. 2.28 which indicates, that static transmission cannot distinguish between rather strong absorption and possible localization. As we will see, all samples avail-able for the experiments described here will fulfill the case of strong absorp-tion discussed here.
2.4 Self-Consistent Theory Theoretical Background However, using short light pulses and watching the width of the photonic cloud evolving over time, one can distinguish localization effects from ab-sorption similar to the total time resolved transmission [SGAM06]. For this dynamic case, σ2 reads as
σ2 =
R ρ2T(ρ, t)d2ρ
R T(ρ, t)d2ρ (2.31)
In this expression, absorption shows up in the nominator as well as in the denominator as a simple exponential factor exp−t/τabs, which cancels out, therefore the dynamic σ2 should not be affected by absorption. There is no easy analytical solution for eq. (2.31). However, to get a first impression, one can for a purely diffusive case take a 3D random walk through a slab, assuming the walks in the different coordinates being independent. Using this ansatz, one would expect a Gaussian profile in x and y direction exiting the sample opposite to the impinging surface. This can, however, only be a first order approximation, because all paths exiting the impinging surface are cut away from the problem, especially when dealing with interference effects as well.
N. Cherroret solved the self-consistent equations for this case numerically for a fixed L and gets the results shown in fig. 2.6 from [Che09]. For kl∗ above 1, the solid black curve shows increase of σ2/L2 as expected, at the mobility edge (red solid) the width increases at all times, but very slow, and below,σ2/L2 reaches a fixed value after some time.
For the diffusive regime (and only there) an analytical solution can be given for the long time behaviour.
σ2(t)≈4D0
1−(kl∗)−2
t (2.32)
i.e. recovering the renormalization of the diffusion coefficient as expected.
Additionally, the inset of fig. 2.6 shows that the short time behaviour of all σ2(t) roughly scales with √
t. This is a surprising result, since the intuitive expectation would be, that for short times, i.e. normal diffusion, the width σ2 should roughly be proportional to a 1D random-walk mean square dis-placement (e.g. in x direction of the exiting plane) as has been motivated before in the simplified picture, therefore scale linearly witht. A very simple simulation of a 3D random walk as presented in appendix B considering the loss of photons through the entrance surface for particles also confirms the scalingσ2 ∝t within errors (see fig. B.1 on page 75).
Finally, adding absorption to this formalism shall be briefly discussed:
absorption should enter with exp−t/τabs in the nominator as well as in the denominator of eq. (2.31) and therefore should cancel out, which means,
4.3. Transverse localization of short pulses 55
Figure 4.4: Numerically calculated mean-square width σ2(t) of the time-dependent transmission coefficientT(ρ, t) of a disordered slab in the diffusive regime (kℓ= 3), at the mobility edge (kℓ = 1), and in the localized regime (L/ξ = 1, 2, 4). The dashed lines show σ2(t) = 4DB[1−(kℓ)−2]t for kℓ = 3 and σ∞2 = 2Lξ(1−ξ/L) for L/ξ = 4. The thickness of the slab is L = 100ℓ. Time is given in units of tD =L2/(π2DB). The inset shows the same results in the log-log scale, the dashed line isσ2(t)∝t1/2.
4.3.2.1 Long-time behavior
First, in the diffusive regime kℓ > 1 one readily obtains σ2(t) ≃ 4Dt, where D ≃ DB[1−(kℓ)−2] (this is the only regime where an analytical result can be found for σ). For kℓ= 3 this result is shown in Fig. 4.4 by a dashed line which is indeed very close to the result of the numerical calculation (solid black line). The difference between diffusive and localized regimes at long times is manifest in Fig. 4.4. The rise ofσ2(t) with time is unbounded for kℓ >1, whereas σ2(t) saturates at a finite value σ∞2 for kℓ < 1 (solid green, blue and pink curves). The latter result can be understood from the following approximate calculation which, however, turns out to be quite adequate. Instead of using the self-consistent equation forD(r,Ω), let us simply setD(z,Ω) =−iΩξ2, which is the solution of self-consistent model in the infinite medium in the limit Ω→0 (see Eq. (2.23)). A straightforward calculation then yields σ2∞ = 2Lξ(1−ξ/L) up to the first order in ξ/L ≪ 1. This equation, shown by a dashed horizontal line in Fig. 4.4, falls fairly close to the numerical result atL/ξ= 4, allowing us to conjecture that it might be a good estimate ofσ2∞ in the limit ofL≫ξ.
Our calculation suggests that the saturation of σ2(t) at a constant level takes place not only in the localized regime, but also at the mobility edge (solid red curve).
Figure 2.6: Time evolution of the width of a photon cloud for different local-ization and diffusion lengths. Figure and the original caption directly taken from [Che09] (adapted to nomenclature in this work): numerically calculated mean-square widthσ2 of the time-dependent transmission coefficient T(ρ, t) of a disordered slab in the diffusive regime (kl∗ = 3), at the mobility edge (kl∗ = 1), and the localized regime (L/ξ = 1, 2, 4). [...] The thickness of the slab is L = 100l∗. Time is given in units of tD = L2/(π2D0). The inset shows the same results in the log-log scale, the dashed line isσ2 ∝√
t.
20
2.5 Coherent Backscattering Theoretical Background