Cross density of states and mode connectivity: Probing wave localization in complex media

We introduce the mode connectivity as a measure of the number of eigenmodes of a wave equation connecting two points at a given frequency. Based on numerical simulations of scattering of electromagnetic waves in disordered media, we show that the connectivity discriminates between the diffusive and the Anderson localized regimes. For practical measurements, the connectivity is encoded in the second-order coherence function characterizing the intensity emitted by two incoherent classical or quantum dipole sources. The analysis applies to all processes in which spatially localized modes build up, and to all kinds of waves.

Figure 1

(a) Realization of a 2D disordered medium with $N=2292$ subwavelength scatterers. To avoid border effects, fields are calculated within the inner 20 $\mu \mathrm{m}$ by 20 $\mu \mathrm{m}$ square box defined by the solid line. (b)–(d) Calculated LDOS maps. (b) Diffusive regime ($\lambda =400$ nm). (c) Intermediate regime ($\lambda =1000$ nm). (d) Localized regime ($\lambda =1500$ nm).

Figure 2

Left column: maps of the relative connectivity ${\mathcal{C}}_{12}$ versus ${\mathbf{r}}_2$, for a fixed position of ${\mathbf{r}}_1$ (chosen at the center), in a single realization of disorder. Right column: cross sections along the line $y=0$. From top to bottom the system transits from the diffusive to the localized regimes, with the wavelength as a control parameter.

Figure 3

Maps of the second-order coherence function $G_{\mathrm{class}/\mathrm{quant}}^{\left(2\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )$ in a single realization of disorder, with ${\mathbf{r}}_1$ fixed at the center and ${\mathbf{r}}_2$ scanning the medium, with classical sources (left) and quantum sources (right).

Figure 4

Estimate of the averaged mode connectivity profile $\langle {\mathcal{C}}_{12}\rangle \left(r\right)$ over 240 realizations of disorder, as a function of the distance $r$ between the two locations, for several values of the wavelength $\lambda$. The thin solid line at the bottom is $|J_0\left(k_{0}r\right)|$ for $\lambda =400$ nm, which recreates the connectivity of vacuum, for comparison of the behavior at small distances.

Figure 5

Estimate of the averaged second-order coherence for classical sources $\langle G_{\mathrm{class}}^{\left(2\right)}\rangle \left(r\right)$ as a function of the distance $r$ between the two sources, for several values of the wavelength $\lambda$. The thin solid line at the bottom is $1+|J_0\left(k_{0}r\right){|}^2/2$ for $\lambda =400$ nm, which recreates the case of vacuum, for comparison of the behavior at small distances.

Figure 6

Estimate of the averaged second-order coherence for quantum sources $\langle G_{\mathrm{quant}}^{\left(2\right)}\rangle \left(r\right)$ as a function of the distance $r$ between the two sources, for several values of the wavelength $\lambda$. Dashed lines represent estimates of the averaged value of $\langle (1-{\mathcal{F}}_{12}^2)/2\rangle$ and the error bars show the standard deviation when using one realization of disorder only. The thin solid line is $[1+|J_0\left(k_{0}r\right){|}^2]/2$ for $\lambda =400$ nm, which recreates the case of vacuum.