Intensity of Waves Inside a Strongly Disordered Medium

Anderson localization does not lead to an exponential decay of intensity of an incident wave with the depth inside a strongly disordered three-dimensional medium. Instead, the average intensity is roughly constant in the first half of a disordered slab, sharply drops in a narrow region in the middle of the sample, and then remains low in the second half of the sample. A universal, scale-free spatial distribution of average intensity is found at mobility edges where the intensity exhibits strong sample-to-sample fluctuations. Our numerical simulations allow us to discriminate between two competing local diffusion theories of Anderson localization and to pinpoint a deficiency of the self-consistent theory.

Figure 1

Spatial distributions of the average wave intensity inside slabs of disordered medium of different thicknesses $k_{0}L=8–14$. Symbols correspond to the point-scatterer model (1) with $\omega$ in between the two mobility edges ${\omega }_c^{\mathrm{I}}={\omega }_0+0.256{\mathrm{\Gamma }}_0$ and ${\omega }_c^{\mathrm{II}}={\omega }_0+0.935{\mathrm{\Gamma }}_0$ for $\rho /k_0^3=0.15$ [30], $k_{0}R=20$, and $k_0{R}_1=10$. Almost coinciding dashed and solid lines show fits of SC (7) and SUSY (8) theories to the point-scatterer data for $k_{0}z>2$ with the localization length $\xi$ as a free parameter. The upper inset shows the best-fit values of $\xi$ following from SC (black squares) and SUSY (red circles) theories. Dashed lines show average values of $\xi$. The lower inset shows the average transmission through the slab as a function of slab thickness.

Figure 2

Same as Fig. 1 but at the low- (a) and high-frequency (b) mobility edges $\omega ={\omega }_c^{\mathrm{I}}$ and $\omega ={\omega }_c^{\mathrm{II}}$, respectively, and for $k_{0}R=25$. Dashed and solid lines show, respectively, SC and SUSY theory fits to numerical results with the decay length $\eta$ as a free fit parameter. The fits were performed for $k_{0}z>2$ (a) or $k_{0}z>3$ (b). The upper insets show the best-fit values of $\eta$ for SC (black squares) and SUSY (red circles) models, with average values of $\eta$ represented by dashed lines. The lower insets show the average transmission through the slab as a function of slab thickness.

Figure 3

(a) Relative fluctuations of intensity at the two mobility edges for a slab of thickness $k_{0}L=12$ and all other parameters as in Fig. 2. (b) Gray scale plot of IPR of quasimodes for a representative random configuration of scatterers. Each point corresponds to a quasimode of frequency $\omega ={\omega }_0-({\mathrm{\Gamma }}_0/2)\mathrm{Re}\mathrm{\Lambda }$ and decay rate $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\mathrm{Im}\mathrm{\Lambda }$, where $\mathrm{\Lambda }$ is an eigenvalue of $\widehat{G}({\omega }_0)$. The gray scale of the point reflects the IPR of the corresponding eigenvector. Dashed lines show the two mobility edges.