Spectral Measurement

The different powders limit our choice of strong scattering samples askl^∗is fixed for each powder within a small deviation due to different filling fractions. An el-egant way to tunekl^∗to a certain extent is by changing the wavelength and thus kl^∗. This way we can examine one sample and changekl^∗without changing the

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Python LabVIEW

Figure 6.5: Spectral measured inverse turbidities as determined from the width of the coherent backscattering cone are shown. (left) measurements for R104 are shown and (right) for R700. Red points are evaluations with the LabVIEW code and black ones with the new Python program. In both cases a clear trend of increasingkl^∗with increasing wavelength can be observed.

sample, giving us the great possibility to characterise the ‘transition to localisa-tion’ in a complete different way as explained before. Therewith we can rule out effects of sample quality. Here we investigate R104 and R700 spectrally.

We do not know a priori the wavelength dependence ofl^∗, and thus measured the dependency of the coherent backscattering cone on the wavelength. The es-timated spectral values ofkl^∗are shown in fig. 6.5. We can see, within the error bars, a roughly linear increase withλ. We thus extrapolate/ interpolatekl^∗’s for wavelengths where we could not measure the backscattering cone. For R104 and R700 the obtained trend is comparable. This measurement shows us that de-creasingλwill increase the scattering strength within the range we measure. A linear fit to the LabVIEW fitted data gives the following dependency:

kl_R700^∗ = −5.7912+0.0144·λ (6.1) kl_R104^∗ =0.2114+0.0065·λ (6.2)

Time of flights of different incident wavelengths for a R104 sample of the size L = 1.02 mm are shown in fig. 6.6(left). The same for a R700 sample withL= 0.98 mm can be found in6.6(right). For both samples the long time tail deviates more for shorter wavelengths. The scattering strength of R104 is lower than for R700 which is why the deviation is not as strong. At the longest wavelengths the decrease is almost linear, approaching the ‘mobility edge’. The very similar long time behaviour of these two samples, despite their differentkl^∗, suggests that by approaching the ‘mobility edge’ the wave transport does not suddenly change, it is rather a smooth transition. Additionally we can observe for both samples

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Determination of the ‘Transition to Localisation’

Figure 6.6: Time of flights for R104 (left) and R700 (right) for different incident wavelengths are shown. Legend in nm. The deviation at long times from a linear decrease is bigger for R700 (lowerkl^∗). For largerλwe observe a convergence to diffusion. For both samples diffusion gets slower with decreasing wavelength. (right) Same figure as in [25].

a shift ofτmax to longer times, indicating that the transport is slower at lower wavelengths. The diffusion constants obtained from a classical fit to the data are shown in fig. 6.7(right). ThatD increases for bigger wavelengths could be seen as an indication of approaching the diffusive regime.

By evaluating the difference between the measured data and a diffusion fit, we can get a quantification of the excess of photons at long times. This way we can show more clearly the onset of ‘localisation’. The excess of photonsΞis defined as follows:

The integration limits were chosen to start at a time (2.0τmax) where ‘localisation’

signs are expected at the earliest, and to stop before the signal vanishes in noise (3.5τmax). The result for both samples is shown in fig. 6.7(left). We can see that the excess for R700 rapidly decreases untilkl^∗≈3.2. In contrast for R104Ξ is much lower and decreases almost instantly to one. Here we observe a discrep-ancy between both samples. The excess for R700 is already unity atkl^∗≈3.2, whereas for R104 we can still see a small excess atkl^∗≈3.85. This significant dis-crepancy is present in all the data presented here. The most probable reason lies in the estimation ofkl^∗from the coherent backscattering cone. Since the qual-ity of the cone depends strongly on the adjustment and the calibration, this is a possible source of error, hampering a fit of the data. Additionally the estimation of the effective refractive index could be imprecise, leading to a differentkl^∗as it should be. We can see in fig. 6.5that the new data evaluation leads to values

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R104 R700

Figure 6.7: (left) The excessΞis shown for R104 and R700. For R700 it decreases rapidly to unity, whereas for R104 even the biggest excess is close to unity. Same data for R700 as in [25] (re-evaluated). (right) Depend-ency of the classical diffusion constant D_B, as obtained from a dif-fusive fit, is shown as a function of the incoming wavelength. For both samplesDis increases withλ. Same data for R700 as in [25] (re-evaluated).

ofkl^∗that are closer together, however the range ofkl^∗still does not overlap for both samples either. Thus we do not use the new data evaluation to not lose the comparability with experiments performed before.

The corresponding transmission profiles can be found in fig. 6.8for R104 (left) and R700 (right). Again the two samples look comparable. For the shortest wave-length the profile for R104 shows a peak, indicating that the sample is roughly as long as the ‘localisation’ length, whereas for R700 a nice plateau can be ob-served. For both samples, the longest wavelength shows ‘sub-diffusive’ beha-viour. The transition from ‘localisation’ to ‘sub-diffusion’ can be determined for R104 at 600 nm and for R700 at 640 nm. These observations are again in agree-ment with the different scattering strengths of both powders. At this point the transmission profiles are ‘better’ than the time of flights, as a distinction between

‘localisation’ and ‘sub-diffusion’ can easily be made. The decrease of diffusion with decreasing wavelength can also be observed via the slope of the linear part of the curves and is comparable to the time of flight measurements.

By varying the incident wavelength we are able to see a crossover from ‘localisa-tion’ to ‘sub-diffusion’. While it is rather hard to locate with the time of flights, the transmission profiles are giving enough information. This gives us the possibility to determine the transition from ‘localisation’ to diffusion in combination with the size dependent measurements, which we will do in the next section.

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Determination of the ‘Transition to Localisation’

Figure 6.8: Transmission profile measurements at different incident wavelengths are shown for for R104 (left) and R700 (right) are presented. Legend in nm. For R700 we can clearly see plateaux for the shortest wavelengths, whereas R104 only shows peaks (L≈ξ), being in accord with the dif-ferent scattering strengthskl^∗. A crossover to ‘sub-diffusion’ can be seen for R104 at 600 nm and for R700 at 640 nm. (right) Same figure as in [24].