Anderson Localization of Light in the Presence of Non-linear Effects

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Anderson Localization of Light in the Presence of Non-linear Effects

Dissertation zur Erlangung des akademischen Grades Dr. rer. nat.

vorgelegt von Wolfgang B¨uhrer an der

Naturwissenschaftliche Sektion, Fachbereich Physik Tag der m¨undlichen Pr¨ufung: 24. Oktober 2012 1. Referent: Prof. Dr. Georg Maret 2. Referent: Prof. Dr. Frank Scheffold

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Chapter 1 Introduction

Anderson Localization was first described theoretically by P. W. Anderson in his famous paper “Absence of diffusion in certain random lattices” [And58].

Even though Anderson received the 1977 Nobel Price partly for this work and Localization was heavily investigated over the last five decades, it is still not fully understood.

An intuitive explanation for this effect is based on the fact that waves can interfere with each other. Generally, if one has many scatterers, a wave is transported through such a sample diffusively, changing its direction randomly when hitting a scatterer on the length scale of the mean free path. But, if one imagines two waves travelling the same path through a distribution of scatterers in opposite directions, these waves will interfere constructively assuming that all phase shifts accumulated over the path are independent of the travelling direction. The shorter the mean free path, the more likely waves share part of their paths and thus eventually stopping the expansion of the wave within the medium. Originally treated for electrons in matter described as matter waves, many systems using different kind of waves and scatterers have been explored for the experimental search for Anderson localization in 1D, 2D and 3D. This is in order to circumvent the problem, that electrons inherently carry charge and thus interact not only by matter wave interference, but by Coulomb potentials as well.

It has been shown [AALR79], that in 1D and 2D a wave interacting with randomly distributed scatterers will always localize if the system is large enough, whereas in 3D a phase transition takes place. Experimentally, reaching this transition and unambiguously identify signs of Anderson localization is difficult, as on one hand the randomness of the system has to be high enough, on the other hand one has to exclude any other effects giving rise to a measurable transport velocity reduction or change from purely diffusive to non-diffusive behaviour.

A non-classical diffusion associated with Anderson Localization has been found in open 3D TiO₂ samples in slab geometry for visible light by M.

St¨orzer [SGAM06]. The aim of this work is to further characterize this non- classical diffusion for a more thorough understanding. In particular, it will

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be argued that the intrinsic non-linearity of the scattering medium can be used to show, that counterpropagating paths become increasingly populated when reaching the transition to Anderson Localization in such systems. Fur- thermore, we test a theoretical suggestion by N. Cherroret [CSvT10] that Anderson Localization should be visible in the expansion behaviour of the photon cloud exiting the sample slab. This method is, according to the underlying theory, independent from absorption present in any real sample.

In the following chapter, the basic theoretical aspects of this work will be explained briefly, and a short overview over the different experiments and systems used for characterization of Anderson Localization so far will be given.

The third chapter will introduce the samples of the system used in this work, i.e. Titania (TiO2) particles as powder of nano particles packed to high densities in order to increase the scattering ability or as sponge-like structures. These samples are created in slab geometry, which can be illuminated from one side while the exiting light on the opposite side can be characterized.

For characterizing the samples and their non-diffusive behaviour, two techniques have been used: the first is time-resolved single photon counting using a pulsed laser as light source, the second method uses an ultrafast gateable high rate intensifier enabling us to watch the light exiting from the sample with temporal resolution in 2D. Their experimental realizations are described in chapter four, including data analysis procedures and character- izations of the abilities and limitations of these setups. Additionally, the coherent backscattering cone setup used here solely for characterizing the turbidity of the sample will be described.

In chapter five, the results of the experiments will be shown and discussed, before they are summarized in chapter six.

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Chapter 2 Theoretical Background

2.1 Principles of Light Scattering

2.1.1 Single Scattering

In this work we will discuss multiple scattering of light. Scattering of light takes place, whenever light ”sees” a change in the optical refractive index of the material. Depending on the spatial extenda of this index variation and the wavelength λ of the interacting light, different scattering mechanisms take place:

Rayleigh Scattering When the wave length of the interacting light is much larger than the size of the scatterer (λ a) we are in the regime of Rayleigh-Scattering [Ray71]. The dependence of the scattered intensity

I_sc∝ a⁶

λ⁴ 1 + cos²θ

(2.1) in proportion to 1/λ⁴ is a (simplified) reason for e.g. the cloudless sky being blue.

Mie Scattering When the size of the scatterer is comparable to the wavelength of the incident light (λ ≈ a) the scattering mechanism for spherical particles can be described by Mie scattering [Mie08]. In contrast to Rayleigh scattering, the scattering cross-section varies strongly with λ and the size of the scatterer, see figure 2.1. For certain wavelengths, so-called Mie reso- nances can show up which strongly increase the scattering. As we will see, the scattering in the samples used in this work are dominated by Mie scattering.

Geometrical Optics Limit When the scatterer is much larger than the wavelength (λa) we are in the limit of geometrical optics and Snell’s law can be used to describe the scattering. Here, the scattering cross-section is twice the geometrical cross-section of the scatterer.

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0 , 1 5 0 , 2 0 0 , 2 5 0 , 3 0 0 , 3 5 0 , 1

0 , 2 0 , 3 0 , 4 0 , 5 0 , 6 0 , 7

scattering cross section (µm2)

s i z e (µm )

n = 2 . 7 λ = 5 9 0 n m

3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0

0 , 0 0 , 1 0 , 2 0 , 3 0 , 4

scattering cross section (µm2)

w a v e l e n g t h ( n m )

n = 2 . 7 d = 2 4 5 n m

Figure 2.1: Scattering cross section for Mie scatterers illuminated with a plane wave, calculated with [Val98]. As can be seen, the scattering cross section varies strongly with λ and diameter. The parameters have been chosen to closely represent R700, a sample used widely in the experiments described in this work.

All the processes above describe elastic scattering, i.e. there is no loss in energy during scattering, which means, that the frequency stays constant.

2.1.2 Multiple Scattering

We have discussed single scattering phenomena in the previous section. If we now consider a sample made of many randomly positioned single scatterers, we come to multiple scattering samples. Since we now cannot observe each and every scattering event any more, we need to have a set of physical quantities being able to characterize the accumulated process of light propagating through such a sample. These are:

thickness of the sample L

elastic mean free path l (path between two scattering events)

transport mean free pathl^∗ (path length on which orientational corre- lations are lost)

microscopic absorption length l_a, i.e. the path length on which the intensity is attenuated by a factor of 1/eand connected to the measurable absorption time la=τabs ·c0/neff

macroscopic absorption length L_a=p l^∗l_a/3¹

1This expression follows from the definitionLa :=√

Dτabs and eq. 2.12.

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2.1 Principles of Light Scattering Theoretical Background Within such samples, we can distinguish three regimes:

L < l (ballistic regime) i.e. most of the photons propagate through the sample without being scattered.

L ≈ l which translates to the fact, that most of the photons traveling through the sample will be scattered once, so that we can apply the described scattering processes before (sec. 2.1.1).

L l. In this case, any photon travelling through the sample will un- dergo many scattering events. The transport of light through the sample can then be described by a random walk of the photons through the sample:

with a stepsize ∆r (in one dimension) during a time step ∆t, the spread of the photons after N steps can be calculated to

r²(ti)

= 1 N

N

X

j=0

r²_j(ti) (2.2)

Rewriting this equation with r(ti) = r(ti−1)±∆r and doing an ensemble average, the term linear inrhas to vanish, since the symmetry of the random walk will keep the photon cloud centered and one gets

r²_j(t)

= t

∆t∆r² (2.3)

This is directly connected to the Boltzmann diffusion coefficient defined by D0 = ∆r²/(2∆t), adding up the three independent random walks for each dimension of a 3D system yields

r²

= r_x²

+ r²_y

+ r²_z

= 6D0t (2.4)

Assuming photon conservation (which will of course normally not be the case, the term for absorption will be introduced later), the equation of continuity with a photon fluxj for the photon density ρ in a volume element,

∂tρ=−∇j (2.5)

has to be fulfilled. Introducing the term 1/τabsρ for absorption and using Fick’s law j=−D0∇ρ one ends up with the diffusion approximation

∂_tρ=D₀∆ρ− 1 τabs

ρ (2.6)

being solved by a Gaussian in the following form for an infinite medium without boundary conditions for dimensionality d:

ρ(r, t) = 1

(4πD0t)^(d/2) exp

− r²

4D0t − t τabs

(2.7)

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The characteristic length of the random walk l, i.e. the path between two scattering events includes the details of the independent scattering process via the scattering cross section σ and density ρs of the scatterers:

l= 1 σρs

(2.8) In the samples used in this work, typically the scatterers are Mie scatterers with an enhanced probability to scatter in forward direction, therefore l has to be replaced by the transport mean free pathl^∗ using the anisotropy factor [LM00]

hcosθi=

R cosθ σ(θ)dΩ

R σ(θ)dΩ (2.9)

where θ is the angle between incoming and scattered light beam and σ(θ) the angle-dependent scattering cross section. With thisl can be corrected to

l^∗ = l

1− hcosθi (2.10)

The random walk model yields furthermore, assuming an exponential step length distribution for the step length ∆r, p(∆r) = 1/l^∗exp(−∆r/l^∗) which leads toh∆ri=l^∗ (i.e. recovering a mean step size of l^∗) and h∆r²i= 2(l^∗)², and therefore

r²

= 2sl^∗ (2.11)

for a given photon path length s = ml^∗, where m is the number of steps having lead to that path. Using this in eq. (2.3) and setting vE = s/t one gets

D= vEl^∗

d (2.12)

A quantity for comparing, how strong different multiple scattering samples scatter an incident wave is the product of the magnitude of the wave- vector and l^∗

kl^∗ = 2π

λ l^∗ (2.13)

thus directly comparing the length scale of the used wave and the transport mean free path.

As has been stated above, the so-called diffusion approximation is valid for an infinite medium with a spreading photon cloud inside the medium.

Experimentally, we will have finite systems and the photons need to be in- jected from outside. In this work, we will use slabs of multiple scattering medium and measure the transmission through the slab. Using the image

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2.2 Anderson Localization Theoretical Background point method, as has been done by R. Lenke [LM00] and using the diffusion approximation as discussed above, one gets an expression for the transmitted intensity of a point source through a slab of thickness L (M. Stoerzer:

[St¨o06])

I(t) =−2D0exp

− t τabs

X

n

(−1)ⁿl^∗ L

nπ L

2

exp

−n²π²D0

L² t

(2.14)

One can see, that the long-time behaviour of thisI(t) reads as I(t)∝exp−

1 τabs

+ π²D0

L²

t (2.15)

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:

τ_max= τD

3 ln

"

16 + 4_τ^τ^D

abs

1 + _τ^τ^D

abs

#

(2.17) 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 absorption. 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

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Figure 2.2: Transmitted intensity through a slab of multiple scattering material, 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 incident 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 constructively, 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 photon to have a counterpropagating photon 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 transition. 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.

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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 localization) - 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 sample, 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).

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1D-systems In the group of A. Genack, experiments with microwaves scattered 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 localization 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.

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2.3 Scaling Theory Theoretical Background

Figure 2.4: β-function as explained in the text for 1D, 2D, and 3D. For 3D, β changes from negative to positive values indicating a phase transition from diffusive to localized behaviour. Figure taken from [AALR79].

effects should become visible in a time-of-flight measurement. We will briefly discuss theoretical models for Anderson Localization in the next sections and include them into the formalism given so far to show that this is indeed the case. A very nice detailed overview of the theoretical descriptions of Anderson Localization can be found in the dissertation of N. Cherroret [Che09]. Based on his work, the main ideas and results for the theories used for this work will be summarized in the following sections.

2.3 Scaling Theory

One major theoretical approach published 1979 by Abrahams, Anderson, Lic- ciardello and Ramakrishnan [AALR79] is the scaling theory of localization.

The basic idea behind a scaling ansatz is, that the overall physical behaviour is independent from the scale one is looking at, or in other words, one can generalize from the behaviour of a small volume of a sample to the behaviour of the whole sample. They introduce the ’dimensionless conductance ’ g as the only parameter necessary to describe the transmission through a multiple scattering sample and that

β = dlng

dlnL (2.19)

is dependent only on g. This function β sets the criterion for localization: if it changes from positive to negative values, the conductance will decrease for large enough L. Ohm’s law states, that g ∝ L^d−2 as can be seen intuitively

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from the conductance for a 3D system g3D = A

ρ_gL (2.20)

where ρg is the resistivity. In 3D, A ∝L², in 2D, A would become a width

∝Land for 1D, it is a constant. From this one can deduce, that for 1D and 2D the conductance for sufficiently large L will decrease and therefore, as stated before, these systems will always localize, whereas in 3D a transition is possible, as can be seen in fig. 2.4.

2.4 Self-Consistent Theory

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 important 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 diffusion equationut=c²·uxxto be a differential equation in space only, since ˆut=iωu.ˆ

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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

− t τ_abs

X

n

(−1)ⁿ⁺¹

D(t) D₀

2

n²exp

−n²π²D(t)t L²

(2.24) In other words: any deviation from an exponential decay at long times indicate 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 signatures 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)

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ρ

σ T(ρ)

ξ

sample

L

pure di ffusi

on localizing

Figure 2.5: Sketch of the steady state transmission through a slab of material. 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 ≈







2L²

3 =:σ²_diff for l^∗ L

2Lξ =:σ_loc² for l^∗ ξL i.e. far in the localized regime

3L²

8 =:σ²_ME for the mobility edge kl^∗ = 1

(2.28) neglecting absorption. In the diffusive regime for weak absorption (L_a L), i.e. the sample size being smaller than the macroscopic absorption length, absorption yields the following:

σ_diff² ≈ 2L²

3 − 2L² 45

L La

2

(2.29) and in case of strong absorption (L_a L), i.e. the sample size being much bigger than the macroscopic absorption length:

σ_diff² ≈2LLa. (2.30)

This expression has exactly the same structure as σ²_loc = 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 available for the experiments described here will fulfill the case of strong absorption discussed here.

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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 absorption 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,

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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) = 4D_B[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 t_D =L²/(π²D_B). The inset shows the same results in the log-log scale, the dashed line isσ²(t)∝t^1/2.

4.3.2.1 Long-time behavior

First, in the diffusive regime kℓ > 1 one readily obtains σ²(t) ≃ 4Dt, where D ≃ D_B[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 localization 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

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2.5 Coherent Backscattering Theoretical Background that the curves shown should not be affected by absorption. The physical explanation can be seen as follows: all photons being measured in the profile for a given traveling time, have spent the same time in the sample, therefore being equally affected by absorption. For different times, this holds, since the measured width is given only by the spread of the surviving photons.

2.5 Coherent Backscattering

In this work, Coherent Backscattering is used only to characterize the transport mean free path of the multiple scattering samples. Therefore just the main idea is given here. A more detailed description can be found e.g. in [Fie10].

As we stated already for the simple picture of Anderson Localization, there is a certain probability for two photons originating from the same wave- front to travel a path in reversed order (i.e. in phase). This will give reason to interference in backscattered direction, as is sketched in fig. 2.7. The interference pattern for a given path can be seen as a double slit experiment with slits in distanced, being the distance between start and end of the chosen loop. This will give an interference intensity of 1 + cosqrxy. Of course, all paths will contribute to the overall backscattered intensity, weighted with their probability to occur:

α= Z

p(rxy)·cos(qrxy)drxy (2.33) This integral has been solved and results for the FWHM of the cone [AWM86, vdMvAL88, LM00, AM09]

FWHM≈ 1

kl^∗ (2.34)

If the sample has a quite high refractive index, internal reflections, as depicted in figure 2.7 by the blue dotted line, can prolong the diffusive paths through the sample and therefore increased, which leads to an overestimation of kl^∗. Zhu et al. [ZPW91] have calculated the corrected kl^∗ to:

FWHM⁻¹ =

1 + 1 +R 1−R

2

3kl^∗ (2.35)

where R can be calculated from Fresnel’s equations

R(η) = R_k(η) +R_⊥(η)

2 =

( _tan2(θ−η)

2 tan²(θ+η) + _{2 sin}^sin²2^(θ−η)(θ+η) for η≤arcsin_n¹

eff

1 forη >arcsin_n¹

eff

(2.36)

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0 0

1 1

2 2

θ 0

r_xy

Figure 2.7: Principle of coherent backscattering: a given path with counterpropagating photons, where start and endpoint are separated with distance d will give rise to a corresponding interference pattern similar to a double slit experiment, depicted in the left-hand side of the arc. Contributions from different d will add up to an overall intensity, known as the coherent backscattering cone, shown in the right-hand side of the arc. For high refractive indices of the sample, internal reflection, as being sketched with the dotted line, will increase d and therefore decrease the width of the cone, which leads to overestimation of kl^∗, see text. Sketch courtesy of S. Fiebig.

withηbeing the angle of the incidence (within the sample) andθof the transmitted light beam (into the surrounding medium). These two are connected via Snell’s law:

sinθ =n_effsinη (2.37)

The total reflectivity is then obtained by R = 3C2+ 2C1

3C₂ −2C₁+ 2 (2.38)

whereC1 and C2 are defined as C1 =

Z π/2 0

R(η) sinηcosη dη C2 = Z 0

−π/2

R(η) sinηcos²η dη (2.39)

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2.6 Non-linear Effects in Random Media Theoretical Background

Term in P⁽²⁾ effect

∝χ⁽²⁾E₁²e^i2ω¹ + c.c. second harmonic generation (SHG)

∝χ⁽²⁾2E₁² or ∝χ⁽²⁾2E₂² optical rectification

∝2χ⁽²⁾E1E2e^i(ω¹⁻^ω²^)t+ c.c. difference frequency generation

∝2χ⁽²⁾E1E2e^i(ω¹^+ω²^)t+ c.c. sum frequency generation

Table 2.1: Non-linear optical effects described in the 2nd order term.

2.6 Non-linear Effects in Random Media

So far we only have considered linear media - i.e. we excluded any processes not following the superposition principle. However, media, with a power-dependent non-linear response in their polarizability are well known:

in general, the polarizability P can be expanded in a power series.

P(t)∝χ⁽¹⁾E(t) +χ⁽²⁾E²(t) +χ⁽³⁾E³(t) +. . . (2.40) where we can plug in a wave composition of e.g. two waves

E(t)∝E₁e^iω¹^t+E₂e^iω²^t+c.c. (2.41) Generally, χ will be a tensorial quantity, but for simplicity of the argument we treat it here as a scalar. If we just take the 2nd order term, we already get several two-frequency effects as shown in table 2.1. Of course, higher order terms can show interaction with more frequencies and therefore open a whole zoo of effects, which could occur in a non-linear medium.

Non-linear effects are not time-reversible, so the loop-picture would col- lapse if we postulated time-reversal invariance for the counterpropagating photons. However, if the non-linear effects collected on a given path are in sum path-reversal invariant, the picture holds in principle. If we assume the simplified picture of the interference loops to be valid within the localizing regime, quite high intensities of the E-field should be present within those loops, giving cause of non-linear effects of second or higher order to become significant, even if the non-linearity coefficients in the given medium are small.

In a multiple scattering random medium all possible effects may occur which unfortunately makes it very hard to create a closed theoretical framework for including non-linear effects. However, there exist some theoretical approaches in literature, which try to include non-linear behaviour of the samples, where we will shortly discuss two models.

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2.6.1 Direct Solution of Maxwell Equations for Non- linear Effects

Conti et al. [CAR07] are describing the laser pulse propagation through a sample of 1000 TiO2 spheres solving Maxwell equations. They report spectral broadening of the light traveling through the sample, when they set the TiO2 non-linear term χ2 very small, but non-vanishing, if the intensity of the light reaches a certain threshold. Note however, that this finding cannot be taken directly for explaining the experimental findings in this work, since their samples are orders of magnitude smaller than our real samples, their pulse is much shorter and they can only simulate to time frames our experiment cannot even resolve properly. Still, it gives rise to the qualitative assumption, that, for sufficiently high energy densities, spectral broadening could be observed.

2.6.2 Non-linear Schroedinger Equation

V. Folli et al. [FC12], use a non-linear non-local Schroedinger equation of the form

i∂

∂tψ + ∂²

∂²xψ =V(ψ)−sψ Z +∞

−∞

χ(x⁰−x)|ψ(x⁰)|²dx⁰ (2.42) whereV(ψ) is the random scattering potential,χis the normalized (R

χ dx= 1) non-linear response function of the medium and s is a switch for focusing (s= 1) and defocussing (s=−1) non-linearity. This ansatz can be used for non-linear optics, because it includes wave-character (2nd derivative in space) as well as time evolution, including the scattering potentialV. The integral can be seen as a function memorizing, which kind of interaction, characterized by χ, the wave ψ has seen. Since ψ is connected to the physical quantity E, one can see, that the integral includes non-linear effects proportional to intensity timesE-field. Using this model, Folli et al. state: ”In the focusing case, the localization length decreases [...]” [FC12]. Schwartz et al. showed 2007 in [SBFS07] indeed that non-linearity increases localization in optical 2D lattices - although they watch it from the third dimension. This is known as “transverse localization” with the idea, that in the third dimension ∆k is so small, that the Ioffe-Regel criterion is fulfilled. Lahini et al. [LAP⁺08]

found in 2008 similar decrease for the localization length with non-linearities in 1D optical waveguides.

In chapter 5, we will present an experimental study of a spectrally resolved optical time of flight experiment in 3D samples, which shows, that photons

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2.6 Non-linear Effects in Random Media Theoretical Background having traveled on long paths will have changed their wavelength away from the impinging wavelength.

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Chapter 3 Samples

We have seen, that the turbidity of a multiple scattering medium must be high, or in other words, kl^∗ must be low, for localization effects to possibly show up. This means, that the scattering ability of the scatterers in such a medium must be high. For optical samples, the scattering cross-section is mainly determined by the refractive index difference between scatterer and its surrounding, therefore we need a high refractive index and, at best, vacuum or air as surrounding medium.¹ Furthermore, the absorption has to be as low as possible so that the photons can travel long enough in the sample to exhibit the looping effect. As we have also seen in the discussion of single scattering effects, efficient scattering is best possible for Mie-like scatterers, so the size of the scatterers should be in the order of the used wavelength.

We use titanium dioxide, TiO2, known also as “titania”, as material for the scatterers in this work, which fulfills all the criteria mentioned above. Air is used as ambient material for the scatterers. Titania exists in three structures:

1. Anatase, with refractive index n = 2,5 and density % = 3,89^g/cm³. This conformation irreversibly changes slowly to the rutile conformation (above 700fast)

2. Brookite, which has no industrial importance and is present in mineral form

3. Rutile, with refractive indexn= 2,73 and density%= 4,26^g/^cm³, used as pigment e.g. in wall color, paper, drugs, sunscreen, food (known there as E 171)

Due to the quite high band gap of titania with Ebg = 3,05 eV (corresponding to λ = hc/E ≈ 400 nm), the absorption starts to increase in the blue spectrum, but is very low for the wavelengths available with our laser systems (550 nm to 665 nm). This results in a very high albedo of > 99 %.

Together with the high mean refractive index for the rutile conformation,

1Of course, one could think of even more fancy materials with n <1, but those are extremely sensitive to the frequency of the wave.

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its chemical stability, and the well-known industrial processes for synthesis of those particles on the 100 nm-scale with different sizes and qualities, such particles are very easy to obtain in large quantities. The particles synthesized industrially show absorption times in the order of 1 ns, translating to a microscopic absorption length la ≈20 cm.

There are, in fact, only a few other materials showing a high refractive index for optical wavelengths. Wiersma et al. [WBLR97] have used porous GaAs samples with n = 3,48, scattering infrared light at 1064 nm. V. d.

Beek [vdBBJ⁺12] measured, that these samples show a kl^∗ ≈ 5−6 and an absorption length of ≈ 6−8µm. Unfortunately, the band gap is too low (Ebg = 1,43 eV,λ ≈870 nm) for our old laser system.

Schuurmans [SMVL99] used porous samples of GaP (refractive index of

≈ 3,3 and indirect bandgap of Ebg = 2,24 eV, (λ = 550 nm)) [FSV⁺99] for studies of localization and reached kl^∗ ≈3,2 at λ = 685 nm. However, this material shows a rather short absorption length ofLa≤80µm, which is still too small for sufficiently measuring localization effects with Time-of-Flight experiments.

3.1 Titania Powders

3.1.1 Commercially Available Particles

Since titania particles are widely used for industrial products e.g. in white paint, toothpaste or food, many different kinds of rutile titania particles are available. We thank DuPont for the different samples they sent us for free. Additionally, we used Aldrich anatase (248576 Aldrich, -325 mesh) as a sample with lower index than rutile in order to be able to changekl^∗ while keepingτabs roughly at the same value. The main properties of the samples used in this work are given in table 3.1.

The listed size and polydispersity of the samples was extracted by taking SEM pictures of the samples, creating a histogram of roughly 100 particles.

The particles are identified by vision and measured manually. Fitting a Gaussian in the form

y =y₀+ A wp_π

2

e⁻²(^x−xcw )² (3.1) in which the parameterwcorresponds roughly to the FWHM of the Gaussian as shown in fig 3.1, the polydispersity is calculated by w/xc.

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3.1 Titania Powders Samples

Diameter Polydispersity kl^∗^a

R700 245 nm 45 %^b 2,8 [St¨o06]

R902 279 nm 38 % 3,4 [St¨o06]

R104 233 nm 25 % 3,7

Aldrich Anatase 170 nm 47 % 6,4

a @ 590 nm, average over many samples

bTypo in [St¨o06]

Table 3.1: Properties of the commercial samples used. Diameter and polydispersity were determined via electron micro-graph pictures of the particles.

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

048

1 2 1 6 2 0 2 4 2 8 3 2

d i a m e t e r ( n m )

# of particles

A l d r i c h A n a t a s e xc = 1 7 0 n m F W H M = 7 9 , 3 8 n m

Figure 3.1: On the left-hand side a SEM picture of an Aldrich anatase sample, on the right-hand side the histogram of a set of 120 particles and the corresponding Gaussian fit.

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3.1.2 Custom Made Particles

Additionally to the commercially available samples, S. Shafaei and S. Kam- merer continued the work of J. Widoniak and S. Eiden [EAWM04] to improve the synthesis of custom made samples with the aim to improve the cleanli- ness while at the same time controlling the size and the polydispersity of the particles.

The synthesis reaction used is a hydrolysis of titanium tetraalkoxide in an ethanol-salt solution as indicated in the following reactions:

Ti(OR)₄ + 4H2O −→ Ti(OH)₄+ 4ROH (3.2) Ti(OH)₄ −→ TiO2 xH2O + (2−x)H2O (3.3) The colloidal particles synthesized in this procedure are in an amorphous state of TiO2. For getting the necessary refractive index for highly turbid samples, as required for low kl^∗, the conformation has to be changed to rutile. To do so, one has to heat up the particles to roughly 1000°C for some minutes. One problem arising here is, that any residual of organic solvent burns to carbon during that procedure, therefore massively decreasing the absorption time, so that many batches of particles show a gray tone already visible by eye. S. Kammerer was able to reduce the residual organic solvents by different cleaning steps, including ozone treatment and heating in inert atmosphere. However, the particles still exhibited low absorption times, although, as chemical analysis showed, there was no carbon found within the analysis errors. The reason for this is still not clear but a possible explanation is that using this procedure the resulting nano-particles consist of many crystals, whose boundaries show defects and therefore sites where absorption can take place. This hypothesis, however, cannot be verified easily, since the available x-ray apparatus lacks a sufficiently high resolution of the line width of the Bragg peaks to estimate the size of the single crystallites.

3.1.3 Sample Holder

The powders discussed above need to be compressed and put into a solid framework to get sufficiently high filling fractions without any optical short- cuts, and for moving them to the detectors for the Time-of-Flight (ToF) measurements and to the Coherent Backscattering Cone (CBS) setup without destroying or changing them.

Fig. 3.2 shows the sample holder used, consisting of a hollow cylinder (inner diameter: 1,5 cm), a stamp with glass plate and a cover glass. The samples are created by closing the cylinder with the cover glass plate, then

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3.1 Titania Powders Samples

Figure 3.2: Parts of the sample holder (from left to right): hollow cylinder, stamp with glass (entrance window of laser light), cover glass.

putting a certain mass of powder controlled with a precision scale (precision:

1 mg) into it. Afterwards the powder is compressed with the stamp and fixed by fastening the screws by hand. Then the thicknessLof the sample is measured and the volume fraction is calculated, using the known densities of TiO2 in rutile %rutile = 4,26^g/^cm³ or anatase %rutile= 3,89^g/^cm³, respectively.

For measuring the CBS cone one needs to remove the frontal glass plate, which is done by first using additional four screws indicated in fig. 3.3 to keep the stamp at its position to avoid that the sample would be accidentally pressed out of the cylinder and destroyed, then removing the cover glass plate.

3.1.4 Effective Refractive Index

As we have discussed in sec. 2.5, correcting the cone for internal reflections needs the refractive indexn_eff of the sample. This index cannot be measured directly due to the inherent index differences within our heterogeneous samples, but has to be estimated or calculated. The easiest, but very crude way, would be to linearly interpolate from the volume fractionf and the knowledge of the index of the materialnsc, the effective index toneff =f nsc+(1−f)nair. A better estimate would be to use Garnett’s theory [Gar04], which is valid for small particles (i.e. more like Rayleigh-scatterers) or a small mismatch of refractive indices between scatterer and ambience. For comparison, the formula for the dielectric constantdescribing spherical scatterers is [St¨o06]:

eff = (1−f)amb+f β

1−f+f β (3.4)

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Figure 3.3: Assembled sample holder (from left to right): complete sample holder with sample for ToF measurements, sample holder without cover glass for CBS measurements, backside with screws for compressing (screws not labeled) and fixing the stamp (red screw heads).

with β being a constant including geometrical effects of the scatterer, which reads for spherical scatterers as follows:

β = 3_air sc+ 2amb

(3.5) Garnett theory generally fails to describe a densely packed system of Mie scatteres with fairly high index contrast, as present in our samples. There- fore we use the so-called Energy Coherent Potential Approximation (ECPA), developed by Busch and Soukolis [BS96, SDE94].

The idea of ECPA is to pick out one spherical scatterer from the multiple scattering medium and put it into a homogeneous medium with a fixed dielectric constant (representing the averaged medium) and assume energy conservation, i.e. the energy density of a volume with a scatterer inside must be equal to the energy density without a scatterer in the same volume. The first one is rather complicated and involves calculating Mie coefficients, the latter one can be calculated rather easily. Then one varies the effective dielectric constant coupled to neff of the sample and minimizes the difference of both energy densities.

We calculate the index of a given sample using ECPA, knowing the volume fraction and size distribution of the particles, in the following way:² The

2It should, however, be mentioned, that this whole procedure cannot be more than an estimation of an effective index, since it relies on the approximation of a smooth surface between sample and ambience, which is clearly not the case for any real sample.

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3.2 Sponge-like Solid Structures Samples

0 , 3 0 , 4 0 , 5 0 , 6

1 , 3 1 , 4 1 , 5 1 , 6 1 , 7 1 , 8 1 , 9 2 , 0

refractive index

v o l u m e f r a c t i o n R 7 0 0

R 1 0 4 A n a t a s e R 9 0 2

Figure 3.4: Calculated (ECPA) refractive indices of the used TiO₂ powders for the volume fractions of the measured samples in the experiments. Error of the calculation code is roughly symbol size. For comparison, the dashed line shows the linear approximation for n_eff, the dash-dotted line shows Garnett theory for rutile scatterers.

Gaussian distribution (as characterized according to sec. 3.1.1) of the given sample’s particle diameters is taken and about 10 diameters are simulated with the ECPA software described in [Twe02] based on [BS96] for the volume fraction of the sample, which we can calculate from the geometry of the sampler holder, the properties of TiO2 and the measured thickness L of the sample. Then the indices are weighted with their distribution and summed up to get neff for the given sample.³ The results for the samples used in this work can be see in fig. 3.4 compared to the two methods mentioned above.

Note that, although the anatase particles have a lower index than the other samples being rutile, the refractive index can be higher for the same volume fractions, rooted in the different size distribution. The results of ECPA are within the naive expectations of the linear extrapolation and Garnett theory.

3.2 Sponge-like Solid Structures

The group of K. Fujita at Kyoto University provided us with sponge-like solid titania gels like the one shown in fig. 3.5 for characterization with the Time-of-Flight setup in order to check, if such structures could show stronger

3The code has been tested for robustness by shifting the calculated diameters while keeping the distribution constant and calculating the resulting indices. From these, one can estimate an error of<1 % for the results of the simulation code

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scattering properties than our powders.

Figure 3.5: Solid sponge-like titania structures from the group of K. Fujita, picture from [Fuj07].

They fabricated these structures with different filling fractions of air bub- bles from 33 to 46 % in a matrix of rutile TiO2 and calculated the refractive indices using Garnett-theory [Gar04] to yield neff ≈ 1,8−2,1 [Fuj07].

However, Time-of-Flight measurements showed, that the absorption in these samples was, similarly to our custom-made samples, too high for being able to observe localization effects.

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Chapter 4 Setups

In the theory chapter, we have seen, that we need to have time-resolved measurement techniques in order to clearly distinguish between localization and absorption effects. In the following sections, the different setups used for data acquisition of this work are introduced.

4.1 Time-of-Flight Setups

Most data was collected with the technique of measuring the time-of-flight distribution of single photons through highly scattering, thick samples. The flight time directly translates to the path length distribution of the diffusing photons by the factor c/neff. In order to realize an optical Time-of-Flight experiment one basically needs the following ingredients:

a pulsed light source - in our case a laser system of sufficiently high intensity and monochromaticity

a start signal, coming from the incident laser pulse (i.e. the information:

pulse has hit the sample)

a stop signal, coming from a detected photon (i.e.: photon has passed through the sample and exited)

a system measuring the time between start and stop signal with suffi- cient accuracy

The experiment is then repeated many times and a histogram of the occurring time differences is taken. Fig. 4.1 schematically shows the basic setup. In the experimental realization, a Becker & Hickl SPC 140 card is used as time measurement system together with a Becker & Hickl DCC 100 card for controlling the photon detector. Technically, a reversed Time-of-Flight is taken, i.e. the measurement only starts, when a diffused photon has been detected, whereas the stop signal is given by a photo diode being illuminated by a small fraction of the laser beam. The time axis is then reversed. This does not change the result, but increases the efficiency of the setup, since

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beam splitter

sample pulsed

light source (laser)

photo diode (t )1

PC (histogram of - t ) 1t 2

detector (t )2

delay line (t > t )₁ ₂

Figure 4.1: Schematic setup for an optical single photon time-of-flight measurement. The timest1 andt2 indicate a reversed Time-of-Flight i.e. the detector signal starts the time measurement, the reference signal of the photo diode stops it, thus optimizing the setup for each photon detected to be recorded.

each photon being detected behind the sample will add to the distribution, whereas in a standard Time-of-Flight setup where the start signal is given by the laser pulse, many start signals would not have a corresponding stop signal from a photon having traveled through the sample due to the chosen count rate [Bec06]. More details will be explained in sec. 4.1.2.

4.1.1 Light Sources

Two laser systems were used as light sources. The older one is a synchronously pumped ps-system, which has been used by M. Stoerzer in his thesis [St¨o06] and S. Fiebig in her thesis for acquiring the results published amongst others in [ASB⁺07], [ASM06], [FAB⁺08], [SGAM06]. The newer one is a synchronously pumped fs-system.

ps-laser-system

This laser system, being capable of creating pulse durations at the order of 20 ps at a repetion rate of 76 MHz, consists of a customized actively mode- locked Ar-Ion Laser Innova I400 (Coherent) operating at the 514 nm emission line of Argon, which synchronously pumps a dye laser using a Rhodamine 6G methanol-ethyleneglycol-solution in a jet as active material. Active mode locking of the Ar-ion laser is done by using an acousto-optic modulator (AOM) crystal as wavelength selection prism in its cavity. For high qual- ity and short pulse durations, the pulses cycle 30-40 times in the dye cavity, before they are extracted via Bragg reflection of an intra-cavity AOM, which

Anderson Localization of Light in the Presence of Non-linear Effects