Phase transition of waves in random media
When waves diffuse through a random medium, the mean free path is no longer the only length scale determining the physical situation. Due to interferences, additional effects will appear on the scale of the wavelength, lambda. A striking prediction made by Anderson in 1958 is that in fact diffusion should come to a halt in a medium where the mean free path is comparable with the wavelength. In the context of electronic systems, this has been used as an explanation for the transition from metallic to insulating behaviour with the addition of impurities. We study this phenomenon for the case of diffusing photons in a system made of particles with a high refractive index on the scale smaller than the wavelength of light. Here, localization can be observed without the complications induced by electron-electron, as well as electron-atom interactions, thus allowing a study of pure Anderson localization.
Deviations from classical diffusion
Time-resolved measurements in contrast can specifically look for long paths, where localization should occur and thus test for the hypothesis of a scale dependent diffusion coefficient. We carry out measurements of time-resolved transmission of a picosecond pulse through a multiple-scattering sample. For samples with very high turbidity (characterized by a low value of the mean free path l* - and measured from the width of the coherent backscattering cone), we observe increasing deviations from the behaviour of classical diffusion of a light-pulse [Stoerzer et al. Phys. Rev. Lett. 96, 063904 (2006)]. This indicates that the transport of light is slowed down for samples on very long paths, where the probability of closed loops is higher. These data can be well fitted with a time-varying diffusion coefficient, which is proportional to 1/t. In the diffusion picture, this translates to the statement that the spread of the photon cloud comes to rest at a certain point, which we identify with the localization length. With a measure of l*, the absorption length and the localization length, it is now also possible to gain information from static transmission measurements. While a combination of diffusion and absorption well describes the transmission through a sample with a value of kl*= 25, it fails to describe a sample with kl* = 2.5. When including the localization length however, good agreement with the data is found over 12 decades without any adjustable parameters.
Static tranmission measurements trough a sample with kl* = 2.5. The dashed line gives the theoretical expectation of classical diffusion with absorption, whereas the full line also includes localization effects.
Direct determination of the localization length
Using sub-nanosecond scale imaging, it is possible to determine the spatial distribution of photons leaving the turbid sample as a function of time. This directly allows a determination of a localization length, as diffusife processes lead to an absorption independent linear increase in the width of the exiting photons. A direct image of this is shown in the image below, where the increase of the width at late times can be seen to come to a halt.
The temporal development of the width shows a plateau in localizing samples after an initial linear increase corresponding to diffusion of light at early stages. The transition in three dimensions with turbidity can then be scanned by a change in the wavelength used, which is shown in the figure below for one and the same sample. At the lowest values of kl* = 2, corresponding to the lowest wavelengths, the width saturates, whereas at higher values of kl* = 4.5, corresponding to the highest wavelength, the width increases at all times measures, albeit with a change to non-linear behaviour at late times.
Determination of scaling properties at the transition
The determination of the localization length from time-resolved measurements allows the first determination of the critical exponent of localization, which has been studied with numerical simulations or a one-parameter scaling theory. While the results are in reasonable agreement with the scaling theory, they disagree with the numerical results, which predict a value of 1.5 in contrast to the experimental value close to 0.5. See also our 2006 EPL on the subject under publications.