Weak Localization of Light in Superdiffusive Random Systems

Lévy flights constitute a broad class of random walks that occur in many fields of research, from biology to economy and geophysics. The recent advent of Lévy glasses allows us to study Lévy flights—and the resultant superdiffusion—using light waves. This raises several questions about the influence of interference on superdiffusive transport. Superdiffusive structures have the extraordinary property that all points are connected via direct jumps, which is expected to have a strong impact on interference effects such as weak and strong localization. Here we report on the experimental observation of weak localization in Lévy glasses and compare our results with a recently developed theory for multiple scattering in superdiffusive media. Experimental results are in good agreement with theory and allow us to unveil the light propagation inside a finite-size superdiffusive system.

Figure 1

(a) Electron micrograph of the interior of a Lévy glass. (b) Sketch representing the optical mechanism lying behind the coherent backscattering cone in a Lévy glass. $L=300\mu \mathrm{m}$ is the thickness of the sample and $ø$ the diameter of the spheres. In both images the scale invariance of the system is evident.

Figure 2

(a)–(c) Measured coherent backscattering cone from diffusive samples and Lévy glasses with $\alpha =1.5$ and $\alpha =1$, respectively. In red, fit to the experimental data according to standard diffusion theory. Insets: Residuals of the fits.

Figure 3

Comparison between the calculated superdiffusive cones obtained with the fractional derivative approach and the measured Lévy cones (blue for $\alpha =1$, red for $\alpha =1.5$).

Figure 4

(a) Amplitude of the Fourier transform (FT) of the measured and calculated coherent backscattering (CB) for small ${\rho }_y$. (b),(c) Amplitude of the Fourier transform of the calculated CB and intensity distribution $f(x={\ell }_{\alpha },x_0={\ell }_{\alpha },{\rho }_y)$ for large ${\rho }_y$.