Experimental Observation of a Fundamental Length Scale of Waves in Random Media

Waves propagating through a weakly scattering random medium show a pronounced branching of the flow accompanied by the formation of freak waves, i.e., extremely intense waves. Theory predicts that this strong fluctuation regime is accompanied by its own fundamental length scale of transport in random media, parametrically different from the mean free path or the localization length. We show numerically how the scintillation index can be used to assess the scaling behavior of the branching length. We report the experimental observation of this scaling using microwave transport experiments in quasi-two-dimensional resonators with randomly distributed weak scatterers. Remarkably, the scaling range extends much further than expected from random caustics statistics.

Figure 1

Experimental branched flow at 25 GHz for (a) configuration 1 (caps type 1), (b) configuration 2 (caps type 2), and (c) configuration 3 (caps type 2). The contours of the scatterers are indicated by circles, and a color code from white (low intensities) to blue (high intensities) is used. The source antenna position $r_a=(0,0)\mathrm{mm}$ is marked by a filled circle.

Figure 2

Scaling of the branching length with the strength of the random potential $ϵ$. (a) Scintillation index ${\sigma }_I^2(x)$ as a function of the distance from the source for different values of $ϵ$. (b) Peak positions of the scintillation curves obtained from $s_I^2(x)$ and ${\sigma }_I^2(x)$ (inset). Both curves show a scaling of ${ϵ}^{-2/3}$ (red dashed lines). The black dotted lines indicate the standard deviation of the individual peak positions around their mean value. The scaling is confirmed in panel (c), where the curves from the left panel are shown with a rescaled $x$ axis, on which all peaks occur at approximately the same distance. The peaks of the two curves for the strongest potential, i.e., the two leftmost curves of panel (a), start to decrease in amplitude with growing $ϵ$, which we attribute to the onset of a significant amount of backscattering.

Figure 3

Autocorrelation function of potential 1 (blue dashed line), 2 (red solid line), and 3 (orange solid line) with corresponding Gaussian fits (orange dotted lines). For the fits all values above 0.5 (horizontal, black dotted line) were considered. The horizontal dotted line at 0 emphasizes the undershoot of the functions. The inset shows a photo of a configuration with the caps of type I.

Figure 4

Scintillation index $s_I^2(r)$ for configuration 1 at 30 GHz. The dotted vertical line indicates the extracted maximal position $r_s^{\mathrm{peak}}$.

Figure 5

Rescaled maxima positions of the scintillation index $r_s^{\mathrm{peak}}$ for different frequencies and configurations. The expected scaling behavior of $-2/3$ is indicated by the dashed red line and the standard deviation of the individual peak positions of the simulations [see Fig. 2b] are marked by the dotted black lines.