Probing two-dimensional Anderson localization without statistics

We investigate the possibility of using the independence of the transmitted speckle pattern on the illumination condition as a signature of Anderson localization in a single configuration of a two-dimensional and open disordered medium. The analysis is based on exact numerical simulations of multiple light scattering. We introduce a similarity function that we propose as a reliable observable to probe Anderson localization without requiring any statistical averaging over an ensemble.

Figure 1

Sketch of the disordered medium. The incident Gaussian beam is focused on the input interface with an angle $\theta$. ${\mathbf{r}}_0$ is a point inside the medium where the LDOS $\rho ({\mathbf{r}}_0,\omega )$ is computed. ${\mathbf{r}}_s$ is a point on the observation plane where the transmitted intensity $I({\mathbf{r}}_s,\omega )$ is computed. The red solid line is an example of speckle pattern calculated in the observation plane.

Figure 2

Normalized transmitted intensity spectrum $I({\mathbf{r}}_s,\omega )/I_0$ for $\theta =0$ (solid blue line) and normalized LDOS spectrum $\rho ({\mathbf{r}}_0,\omega )/{\rho }_0\left(\omega \right)$ (dashed red line) at a point placed at the center of the disordered region. (a) Localized regime. The spectrum is centered at frequency ${\omega }_c^l$ for which $\xi /L\sim 0.09$. (b) Diffusive regime. The spectrum is centered at frequency ${\omega }_c^d$ for which $\xi /L=40$.

Figure 3

Normalized transmitted intensity $I(x,\theta ,\omega )/I_0$ versus the normalized transverse direction $x/\lambda$ in the localized regime [$\omega ={\omega }_c^l$, (a)] and in the diffusive regime [$\omega ={\omega }_c^d$, (b)] at normal incidence (solid blue line) and for an incidence angle $\theta ={30}^{\circ }$ (dashed red line).

Figure 4

Similarity function $C(\theta ,\omega )$ versus the incidence angle $\theta$ in the localized regime ($\omega ={\omega }_c^l$, dashed red line) and in the diffusive regime ($\omega ={\omega }_c^d$, solid blue line).