influence on the measured widths around the maximal intensity as can be seen in fig. 4.12. The cause for this is not yet fully understood, but one problem is, that the voltage of the ring-shaped cathode causing the acceleration of the photo-electrons has to be switched fast, but in a manner to not give rise to bursts in the photoactive material. The opening behaviour is therefore like the one of an iris diaphragm being opened over a certain time scale with probably some boundary effects due to charging. The slight underswing (see arrow in fig. 4.12) depends on the used mode but will be visible for any data shown in the results section 5.2.
4.2 Coherent Backscattering Cone Setup
As has been described in section 2.5, the coherent backscattering cone gives a measure for kl∗ independently of ToF measurements. The setup of our CBS measurement device is described in detail in [GSF+07] and [Fie10] from where the following section is mainly adopted.
Figure 4.13 shows a sketch of the setup. A laser beam is illuminating the sample from above. 256 photo diodes are arranged in a half-circle around the sample. The photo diodes collect the incident light and can be read out simultaneously, providing a “one-shot” measurement of the whole cone.
The circular polarizer foils (M3, J53-333) in the laser beam and in front of the photo diodes block singly scattered light from the photodiodes with
sample lens laser
circular polariser circular polarizer
photodiodes telescope
photo diode neutral density
filters
focal point
Figure 4.13: Coherent Backscattering Cone Setup. Sketch adapted from [Fie10]
an efficiency of 97 %, since single scattering flips the helicity of circularly polarized light.
The cone is, even for very low values of kl∗, rather small, therefore the position of the photo diodes in the arc are positioned with different distances:
angles θ < 9,75◦ are detected via photo diode arrays containing 16 diodes (Hamamatsu, S5668), thus providing an angular resolution of 0,15°. Single photo diodes (Hamamatsu, S4011) provide the rest of data points, positioned with angular resolution of 0,7° for 9,75◦ < θ < 19,55◦, ≈ 1◦ for 19,55◦ <
θ < 60◦, and ≈ 3◦ for 60◦ < θ < 85◦. The cone tip cannot be resolved, since the laser beam has to pass through the arc.
For a better linear power calibration, the power meter used with earlier experiments ([Fie10]) is replaced by a photo diode, which is protected with a band pass filter from surrounding light and proves to be more exact in linear response.
To measure a cone, one first uses a Teflon block and measures the response of the photo diodes for different incident powers, while tracking the power.
The Teflon block has a coherent backscattering cone of≈0,03◦, i.e. smaller than the detection accuracy of the setup, so it can be used as a source for incoherent background. Then the actual sample is placed into the setup and the cone is recorded, while tracking the incident power to the sample. During all measurements the samples are rotated to avoid static speckle patterns on the arc of photo diodes.
4.2.1 Data Analysis
Using the Teflon data, a response function correlating incident power and count rate is calculated for each photo diode. With these, the raw data of the samples’ cones are corrected, including the albedo properties of the different samples, as described in [Fie10]. Additionally, the cosine background caused by Lambert’s Law is removed by division, resulting in the cone as shown in fig. 4.14 (black dots). On this, a theoretical cone shape can be fitted.
As has been shown in sec. 2.5, the FWHM of the CBS cone is directly connected tokl∗ by the expression
FWHM−1 =
1 + 1 +R 1−R
2
3kl∗ (4.4)
including internal reflection, whereRis the reflectivity as defined in [ZPW91].
R is connected to the effective refractive index neff of the sample which is calculated according to sec. 3.1.4. For typical values of neff≈1,5−1,8 one gets R≈0,5−0,75.
4.2 Coherent Backscattering Cone Setup Setups
- 5 0 0 5 0
0 , 0 0 , 3 0 , 6
0 , 9 e x p . d a t a
f i t
enhancement
a n g l e ( d e g )
Figure 4.14: Cone shape after including albedo effects and correcting for Lambert’s Law.
Chapter 5 Results
As has been discussed in the theory chapter 2, non-linear effects within a mul-tiple scattering sample can lead to a spectral broadening of the incident laser light and can increase localization effects. Putting these findings together with the intuitive picture of Anderson Localization corresponding to closed loops, one can argue, that photons traveling in an interference loop can, due to the increased energy density in that loop, experience a relatively higher, and thus significant amount of linear effects. This holds, even if the non-linearity of the medium in which they travel is small but non-vanishing. In other words: the longer the traveling time of photons within such a medium, the more pronounced should the shift to lower or higher wave-lengths be, if they experience loops. Given that TiO2 particles have been used in different experiments, see e.g. [ZLMM97], [IKNR08], [ABG+11] to increase or show non-linear effects in multiple scattering media, one expects such effects to appear in TiO2 samples showing signs of Anderson Localization. In partic-ular, the non-exponential decay measured in earlier experiments [SGAM06]
should consist of photons which have spent a considerable path on closed loops, i.e. have experienced significant non-linearities. Therefore, the pho-tons at late times should show increased red- and green-shifts with respect to the reference laser wavelength.
5.1 Spectrally Resolved 1D Transmission
Power Dependent Non-Linearity
In order to see if we indeed could find any evidence of the idea outlined above, we first took a R700 sample, which is the sample with the lowest kl∗ available and tested for power-dependent non-linear responses. We measured the response to the incident power with the Hamamatsu PM and a counter device, using the dye laser system either in pulsed mode (and calculating back to cw) or in cw mode. In order to vary the incident power and keep the PM in the counting mode proportional to the incident power we use appropriate neutral density gray filters in front and behind the sample and correct for
1 0- 2 1 0- 1 1 00 1 01
Figure 5.1: Measurement of power-dependent non-linearity of an R700 sam-ple (m = 481 mg, L= 1,25 mm). Static transmission over almost 4 decades of incident power does not show any non-linearity in the left-hand side graph (statistical measurement errors are smaller than symbol size). However, on the right-hand side, the ToF measurements clearly exhibit a power depen-dence in the intensity of later arriving photons.
their OD. As can be seen in the left-hand graph of fig. 5.1, this yields no power dependence over almost 4 decades of incident power within the errors, as can be concluded by the slope of the fit in the double logarithmic plot given. In contrast, the dynamic time of flight measurement of the very same sample shows an increase of photons with long traveling times through the sample with increase of incident power in the right-hand side of fig. 5.1. From the ToF it can be seen, that at t = 13 ns the number of photons is roughly doubled for an increase of factor 1000 in incident power. Translated into the static transmission measurement, we would need to resolve the difference of the integrated curves, which is not possible since the integrated intensity of the main peak is so high, that the difference of the curves will be within the noise level of the measurement. We therefore can conclude that the non-linearity in power is at the order of 10−5 or less.
As has been shown by M. Stoerzer in [SGAM06], the non-exponential decay is kl∗-dependent and he extracts the critical kl∗ for the transition to localization atkl∗ ≈4,21. We therefore took samples available with different kl∗and measured their non-linearity using the ToF measurements. The result is shown in fig. 5.2, in which all curves are normalized to their corresponding τmax which roughly eliminates the sample propertiesLandD0 (hidden inτD)
1This is in general agreement with the Ioffe-Regel criterion [IR60]kl∗≈1. The differ-ence in value might be accounted to the fact, that, as has been stated by [FC12], non-linear effects facilitate Anderson Localization.
5.1 Spectrally Resolved 1D Transmission Results sample L [mm] D0 [m2/s] τabs[ns] τmax[ns] kl∗
R700 0,81 11,5 0,85 2,68 2,7
R902 0,95 11,0 1,17 3,56 3,6
R104 0,92 17,5 1,02 2,46 3,9
R104AA 0,94 19,3 1,00 2,42 5,7
Table 5.1: Sample properties of the samples used for the graphs shown here, all properties are extracted using the BPF590.
as has been discussed in sec. 2.1.2, equation (2.16) for better comparability.
All the samples show a power-dependent change in intensity for long times. Additionally, the difference between the measurements of OD3 behind the sample to the measurements of OD0, i.e. no grey filter behind the sample seem to be roughly comparable at a first glance.
To quantitatively check this, we integrate over the ratio between the curves (which is, in a semilog plot, equivalent to the difference of the cor-responding two curves) in an interval between 3,0 and 3,7 τmax, the interval being marked in all plots:
ΞOD 0(x) := The integration boundaries are chosen as follows: starting from 3,0 τmax, the difference between all the curves are clearly visible. For the upper bound-ary one has to take into account, that effects of the afterpulse set in at 13,2 ns.
As R902, being the thickest sample with lowD0 for the curves displayed, has the biggestτmax as can be seen in table 5.1, this value sets the upper limit of the upper boundary to 13,2 ns/3,56 ns≈3,71.
On the one hand, the integral2 Ξ allows for integration over the noise and therefore reduces the statistical error, on the other hand it takes only into account the region, where the differences are clearly observable, and is far enough away from any after-pulsing for all samples therefore giving a solid measure of the difference of the curves. Furthermore, the ratio between the intensities eliminates the dependence of τabs so that these integrals are independent of the sample-specific propertiesD0,τabs andLwithin the error bars.
Fig. 5.3 exhibits in units of OD (which translates to a log10 power scale), that R700 and R902 show almost the same non-linearity in power, whereas
2The nomenclature in this work will be as follows: the subindex of Ξ denotes the filter, to which the reference of the integral is taken to, whereas the argument characterizes the filter being compared to the reference
0 1 2 3 4 5
Figure 5.2: Power dependent Time-of-Flight measurements for the different samples as indicated in the graph. The OD indicated refers to the filter be-tween sample and detector, which means, e.g. OD3 translates to an increase of incident power of factor 1000 compared to the OD 0 (without filter) curve.
One can see, that all samples show an increase of long-time photons with increasing incident power. Vertical lines indicate the integration boundaries for Ξ as explained in the text. Error bars are generally shown in fig. 4.11 and omitted for better visibility.
5.1 Spectrally Resolved 1D Transmission Results
0 1 2 3
0 , 8 1 , 0 1 , 2 1 , 4 1 , 6 1 , 8 2 , 0 2 , 2 2 , 4
ΞOD0 (OD x)
O D
R 1 0 4 A A R 1 0 4 R 9 0 2 R 7 0 0
Figure 5.3: Power dependence (in units of optical density put between sample and detector) of intensity for long photon paths, ΞOD0 as defined in the text.
R700 and R902 behave almost identically, whereas R104 and the mixture of R104 and Aldrich Anatase show similar less power dependence. Error bars are statistical errors for the integral resulting from the ToF errors.
R104 and the mixture R104AA show less, but similar non-linear power de-pendence. From this we can infer that there is no clear dependence from optical non-linearity and kl∗ for these samples.
Spectrally Resolved Time of Flight
If we now analyse the Time of Flight of the very same samples spectrally resolved with the band pass filters at the same incident wavelength of λ = 590 nm, we get the results shown in fig. 5.4, again with the time axis nor-malized to τmax. One can clearly see, that the deviation from pure diffusion increases with decreasing kl∗ as has been already shown by M. Stoerzer in [SGAM06]. Furthermore, the spectral analysis shows that the BPF590 let-ting pass the incident wavelength indicates pure classical diffusion for all samples. The deviation from pure classical diffusion seems to increase with the distance of the passing wavelength window of the used filter from the incident wavelength, giving rise to the interpretation, that the photons caus-ing the deviation from classical diffusion must be shifted in energy, i.e. on long paths in the sample the wavelength of the photons either increases or decreases. The fact, that we find wavelength shifts to lower values than the incident light excludes fluorescence as a possible explanation for the increase in photons having spent long time in such a sample. Also, the bandwith of the incident laser pulse, which amounts to roughly 3 nm (see sec. 4.1.1)
can-BPF virtual OD @ λ= 590 nm
560 2,5
570 3,3
580 1,8
590 0,4
600 0,6
610 1,2
Table 5.2: OD of the different band pass filters at incident wavelength of λ = 590 nm. Note, that the BPF570 is optically thicker than expected, regarding the behaviour of the other filters.
not solely explain the wavelength shift, which is prominent for the filtering window roughly 20 nm away from the incident wavelength.
As we have seen in the power dependent measurements that R700 and R902 show the same amount of non-linearity, they here clearly exhibit a different deviation from classical diffusion. In order to better characterize the deviation, we again use an appropriate integral over the ratio of each curve in the interval 3,0τmax to 3,7τmax to the reference curve (set to be the curve for BPF590, which is the passing filter for the incident wavelength):
ΞBPF590(x) :=
Z 3,7τmax
3,0τmax
1 0,7τmax
Ix(t)
IBPF590(t)dt (5.2)
In order to make this quantity comparable for all samples, we need to take into account the different non-linear behaviour of the pairs R700 / R902 and R104/R104AA found before and correct ΞBPF590(as a quantity characterizing the deviation from classical diffusion) for purely non-linear power dependent effects. To do so, we analyzed, from the knowledge of the grey filters put in the beam beam before and after the sample and the resulting count rate, which virtual OD is caused by the band pass filters behind the sample with respect to the measurement without filter (i.e. OD0) between sample and detector. This is more accurate than the spectral measurements, since, as discussed in section 4.1.3, with the spectrometer used, the optical density cannot be characterized better than roughly to OD 2,5 - 3,0. The extracted values are given in table 5.2.
Since the BPF590 causes a kl∗-dependent shape change, which will be present in ΞBPF590 whereas the pure non-linear effect seems to be the same for the pairs R104/R104AA and R902/R700 and therefore not (strongly)kl∗
5.1 Spectrally Resolved 1D Transmission Results
Figure 5.4: ToF of the same samples used in fig. 5.2 using the band pass filters indicated, i.e. spectrally resolved. Error bars are generally shown in fig. 4.11 and omitted for better visibility.
dependent, and we want to correct for the latter, we take the quantity3 ΞBPF590(OD x)
ΞBPF590(OD 0) (5.3)
as correction factor. The correction factors for the virtual OD caused by the band pass filters can then be interpolated and the integrals ΞBPF590(BPF x) can be corrected by these factors.
The result is shown in fig. 5.5. One can see, that the BPF570 and the BPF610 cause a maximal deviation from classical diffusion, well outside the error bars for R700 with lowkl∗.
3This quantity is, although comparable, not equal to the earlier used quantity ΞOD0(ODX), simply since addition and division are not interchangeable.
5 6 0 5 7 0 5 8 0 5 9 0 6 0 0 6 1 0 0 , 6
0 , 8 1 , 0 1 , 2 1 , 4 1 , 6 1 , 8 2 , 0 2 , 2 2 , 4 2 , 6 2 , 8
ΞBPF590(x) corrected for powerdependence
B P F
R 1 0 4 A A - k l * = 5 , 7 ( 3 ) R 1 0 4 - k l * = 3 , 9 ( 2 ) R 9 0 2 - k l * = 3 , 6 ( 3 ) R 7 0 0 - k l * = 2 , 7 ( 3 )
Figure 5.5: Deviation from classical diffusion of the different samples for the different band pass filters. Data is corrected for possible pure power-dependent effects as explained in the text. Error bars include the statistical errors of the integrals Ξ as well as the uncertainty in the measurement of the virtual OD caused by the band pass filters. Error bars are statistical errors of the integral including error of the correction factor.
Dependence from kl
∗for R700
To check, whether this behaviour is just a coincidence, another R700 sample was created and measured spectrally resolved with incident laser wavelengths of λLaser = 575 nm and 610 nm. The results, again normalized to τmax, are shown in fig. 5.6. As can be seen, using the band pass filter passing the in-cident wavelength, the curves again suggest diffusion, whereas moving away from the incident wavelength, the non-exponential decays increase, as we al-ready have seen in the detailed analysis above. The missing filters for the 610 nm-spectrum result from quite low incident power at this wavelength, so that the count rate was too low to measure with other than the filters shown. The wavelength dependence of kl∗ was determined with a coher-ent backscattering cone measuremcoher-ent at roughly the same wavelengths as the incident ones for the spectra, with simulated refractive indices including the appropriate wavelength. One can clearly see, that the deviations from classical diffusion increase significantly with decreasing kl∗ using the same sample. It has been checked for the λ = 575 nm spectrum with an OD2
5.1 Spectrally Resolved 1D Transmission Results
Figure 5.6: Spectrally resolved ToF for R700 at laser wavelength of 575 nm (left-hand side) and 610 nm (right-hand side). Data not corrected for power dependence, since power dependence does not change with wavelength within error bars (see text).
filter (for checking OD3, the incident laser power was too low), that the pure non-linear response characterized with ΞOD0 is the same within the errors as for λ = 590 nm, therefore not being able to be the cause for the increased split.
Measuring the ToF for the same R700 sample at different wavelengths, we can clearly see in the left-hand graph in fig. 5.7, that the non-exponential long-time behaviour increases with decreasing wavelength, corresponding to a decreasing value of kl∗. In other words: we can see a change from classi-cal to non-classiclassi-cal behaviour of the same sample simply by decreasing the wavelength and hence kl∗. Note also that the maximum of the curve shifts towards longer times as the time axis in this case is not scaled to τmax, in-dicating a slowing down of diffusion. Extracting D0 and τabs from the ToF curves with band pass filters passing the incident wavelength for different wavelengths, shown on the right-hand side of fig. 5.7, one can see, that D0
indeed, as well as τabs decreases with decreasing wavelength. The reduction of τabs is expected, since one slowly approaches the band gap of titania and therefore absorption increases.
Although absorption increases, we still can see an increasing portion of long path lengths for decreasing kl∗. As we, in this case, have measured the very same sample with constant intrinsic power-dependent non-linearity and see a significant change withkl∗ of the deviations, we conclude that the long path length photons, being shifted in wavelength by the material-intrinsic non-linearity, can be interpreted as resulting from path-reversal interference loops where the energy density is increased. When kl∗ becomes lower by a
0 5 1 0 1 5
Figure 5.7: Left-hand side: ToF curves from the same R700 sample with dif-ferent incident wavelengths without filter behind the sample: one can clearly see, that with decreasing wavelength, the non-exponentiality of the long-time behaviour increases as well as the diffusion coefficient. The inset shows the dependence ofkl∗ from the wavelength of R700 for the wavelengths available in the coherent backscattering setup. Right-hand side: parameters for this R700 sample for differentkl∗ (determined via thekl∗-wavelength dependence data of the inset of the left-hand side figure by using a linear fit), extracted
Figure 5.7: Left-hand side: ToF curves from the same R700 sample with dif-ferent incident wavelengths without filter behind the sample: one can clearly see, that with decreasing wavelength, the non-exponentiality of the long-time behaviour increases as well as the diffusion coefficient. The inset shows the dependence ofkl∗ from the wavelength of R700 for the wavelengths available in the coherent backscattering setup. Right-hand side: parameters for this R700 sample for differentkl∗ (determined via thekl∗-wavelength dependence data of the inset of the left-hand side figure by using a linear fit), extracted