Scaling Theory - Anderson Localisation - The Experimental Search for Anderson Localisation of L

2.3 Anderson Localisation

2.3.1 Scaling Theory

As already mentioned the Ioffe-Regel criterion is only valid for infinite systems, but experimentally only open systems are accessible. D. J. Thouless described localisation in finite open systems [61], which developed to the scaling theory of localisation [62]. On short time scales waves propagate like in an infinite me-dium. If the waves are starting to ‘feel’ the effects of the boundaries, these can not be neglected any more. An extended (diffusive) state spans (extends) over the the entire sampleL^d. The Thouless criterion distinguishes localised and ex-tended states by their sensitivity to boundary conditions.δωrepresents the shift in frequency when the boundary conditions are changed from symmetric to anti-symmetric. This frequency width is due to the Thouless timeτT ∼1/δω, which is needed for the wave to react to the change in the boundary conditions.∆ωis the average frequency separation between neighbouring states, which is inverse pro-portional to the density of states, also called the Heisenberg timeτH∼1/∆ω. The dimensionless Thouless conductance can be defined byg =δω/∆ω. If the Thouless conductance becomesg >1 the states are diffusive (extended) and in the case ofg <1 they are localised.

In the next step we will examine what happens, if small samples are coupled to-gether to produce a larger one. In the case of diffusive systems, when the states overlap in frequency (δω>∆ω, see fig. 2.6(left)), the Thouless time becomes δω∼τ⁻¹_T =τ⁻¹_D =DL⁻². The Heisenberg time then reads as∆ω=1/ρL^d ∝L^−d.

Chapter 2. Theory

Therewith the dimensionless conductance becomes:

g= δω

∆ω (2.29)

∝L^d⁻² (2.30)

For a localised system the states are separated in frequency (δω<∆ω). Here an additional length scale will be introduced, the localisation lengthξ. It describes the minimum extend a medium has to have to localise. The Thouless time be-comesδω∼τ⁻_T¹∼e⁻^L^ξ and the Heisenberg time stays the same. So the dimen-sionless conductance becomes:

g ∝e⁻^L^ξ forL>ξ (2.31)

These results are giving different behaviours for the dimensionless conductance for different system dimensions. In a three dimensional systemg increases with the system sizeLfor diffusive states, but on the other hand in a localised stateg decreases withL. In lower dimensions (1D and 2D) the dimensionless conduct-ance always decreases withL. This means there is no transition in such systems, they are always localising. The scaling of the dimensionless conductance with L can be described with the scaling functionβ(g) [62]:

β(g)=d lng

d lnL (2.32)

If the scaling function is larger than zero the states are extended, in the case of localisationβis smaller than zero, see fig2.7. It follows that the transition from diffusion to localisation is atβ(g)=0. The effect of changing the disorder can be compensated by changing the system sizeL, asg depends on both.

The predictions that can be made with the scaling theory are that there is only a transition from extended to localised states in a three dimensional system. This statement was recently limited in the case of light scattering by Skipetrov and Sokolov [64]. They state that in a random ensemble of point scatterers, which is the case for Rayleigh scattering, there is no Anderson Localization. In one and two dimensional systems all states are localised, no matter what the degree of disorder is, the sample has only to be made large enough.

2.3.2 P´ olya’s Random Walk Theorem

We want to make a short excursion to a more mathematical way of looking at the problem. Long before the scaling theory was developed there was a math-ematician called Georg Pólya who investigated the probabilityp(d) of a random walk on addimensional lattice to come back to its origin [65].

Anderson Localisation

Figure 2.7: The scaling function β(g) in dependence of lng is shown. In one and two dimensional systems localisation is always present (β(g)<0).

Only for three dimensional systems a transition from diffusion to loc-alisation atβ(g)=0 is present. Figure taken from [62].

A particle (the random walker) is located at a certain point of the integer lattice Z^d, withdbeing the dimension of the lattice. The particle is jumping periodically to a random neighbouring lattice point. The probability to jump into any direc-tion is equal. The particle is performing the random walk on the latticeZ^d, as discussed earlier on multiple scattering in section2.2. In his work Pólya showed that in one and two dimensions the probability of a random walk to return to its origin is one (p(1)=p(2)=1). For higher dimensionsd >2 he could show that the return probability is smaller than one (p(d>2)<1). This result is equal to the statement of the scaling theory. The particle in 1D and 2D is not able to leave, the probability to escape is zero (p_esc=0), meaning the random walker is recurrent (localising). Whereas the escape probability ind>2 is larger than zero (p_esc>0).

Later it was shown that the probability of a random walk in three dimension to return to its origin isp(3)=0.34 [66].

2.3.3 Self-Consistent Theory

The scaling theory is not the only theory that established to describe the phe-nomenon of localisation. The self-consistent theory will be introduced, which was developed by D. Vollhardt and P. Wölfle in 1980 [67] and later published in the book ‘Self-consistent Theory of Anderson Localization’ [68]. In their work

Chapter 2. Theory

only low dimensional (d≤2) and infinite media were considered. B. van Tiggelen et al. generalised the theory to media of finite size [69].

Later in 2006 the self-consistent theory was expanded by S. Skipetrov and B. van Tiggelen to three dimensional open systems in slabs [70], and was also derived microscopically [22]. The self-consistent equations then read as follows:

(−iΩ− ∇rD(r,Ω)∇r)C(r,r⁰,Ω)=δ(r−r⁰) (2.33) 1

D(r,Ω)= 1

D_B +12π

k²l_eC(r,r,Ω) (2.34)

The first equation (2.33) is the self-consistent solution of the diffusion equation using the intensity Green’s functionC(r,r⁰,Ω), which characterises in the time domain the density of wave energy at a given point r at the time t of a wave packet emitted at the pointr⁰at time t⁰. The new concept is that the diffusion coefficient in eq. (2.34) is position and time dependent (by Fourier transform-ation). The self-consistent equations are in accord with the super-symmetric field theory for finite media [71]. The position dependent diffusion coefficient was numerically confirmed for wave-guides [72,73], and later also experiment-ally shown [74]. With these dependencies of the diffusion coefficient it is possible to include boundary effects.

Skipetrov and van Tiggelen further calculated the time dependent diffusion coef-ficientD(t), whose rough dependency for timestÀτD and in the diffusive re-gimekl^∗À1 is:

D(t)

DB ∼1− 1

(kl^∗)² (2.35)

This results implies that it would be experimentally very challenging to meas-ure, since the effect is very small⁴. For the localising regime with kl^∗ <1 the parameterα^∗=D_Bξ⁻²exp(−Lξ) is introduced, with the requirement that the loc-alisation length is much smaller than the sample size ξ ¿L. Then for times τD¿t<_α¹∗ the time dependent diffusion coefficient becomes:

D(t) D_B ∼τD

t (2.36)

We want to briefly note, that it is discussed that the self-consistent theory fails at long times (for quasi 1D media), where long-lived modes are dominating [75,76]

(and note in [72]).

4In fact we do not see any deviation for pure diffusive samples.

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

r_xy 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(r_xy). 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

k_⊥l^∗+^µ+1_2µ

´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

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^∗ ¿l_a =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+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:

(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^∗¿l_a=p

D_Bτa, 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+1 (2.49)

If we use this expression ofαewe get, by solving fora, a general expression that gives energy conservation⁷:

R_π/2

0 αcsinθdθ R_π/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.

2.3.5 Localisation in Slab Geometry

Before introducing the concept of Anderson localisation we already gave a de-scription how waves are transmitted through a slab in the diffusive case. The

7In the work of S. Fiebigadoes not have the same normalisation used in the work [82], which should be corrected if used.

Anderson Localisation

concept of localisation shall be combined with equation (2.22) to allow for a de-scription in the localised regime. In the previous chapter on the self-consistent theory, we have seen that the diffusion coefficient should become time depend-ent, which has to be accounted for. This was done by R. Berkovits and M. Kaveh [88], who give the following description of time dependent transmission through a slab:

I(t)∝e⁻^τa^t X∞

(−1)ⁿ⁺¹n² µD(t)

D_B

¶ 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 be_α¹_∗ >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)=D_B τ_loc^a

¡τ^m_loc+t^m¢_m^a (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)=D_B µ

1− 1 (kl^∗)²

(2.55)

Solving the self-consistent equations then gives the mean square sizeσ²_diff: σ²_diff'2L²

3 (2.56)

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 forσ²_loc reads as fol-lows:

σ²_loc'2Lξ (2.57)

In contrast to diffusionσ²_locdepends on the localisation lengthξand sample size L. At the mobility edge (kl^∗=1) the mean square widthσ²_mebecomes:

σ²_me=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 (L_aÀL) σ²changes to:

σ²_diff,L_a_À_L=2L² 3 −2L²

45 µ L

L_a

¶2

(2.59)

And for strong absorption the mean square size will be:

σ²_diff,L_a_¿L=2LL_a (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 lengthL_a 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.

σ²_diff(t)'4D_B 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.

Im Dokument The Experimental Search for Anderson Localisation of Light in Three Dimensions (Seite 22-0)