Absence of Anderson Localization of Light in a Random Ensemble of Point Scatterers

As discovered by Philip Anderson in 1958, strong disorder can block propagation of waves and lead to the localization of wavelike excitations in space. Anderson localization of light is particularly exciting in view of its possible applications for random lasing or quantum information processing. We show that, surprisingly, Anderson localization of light cannot be achieved in a random three-dimensional ensemble of point scattering centers that is the simplest and widespread model to study the multiple scattering of waves. Localization is recovered if the vector character of light is neglected. This shows that, at least for point scatterers, the polarization of light plays an important role in the Anderson localization problem.

Figure 1

Density of eigenvalues of the random Green’s matrix. Gray scale density plots of the probability density $p(\mathrm{\Lambda })$ for $\mathrm{\Lambda }$ values corresponding to long-lived states ($\mathrm{Im}\mathrm{\Lambda }<1$). Dashed lines show the border of the eigenvalue domain following from the diffusion theory and the spiral branches along which eigenvalues corresponding to subradiant states are concentrated [24, 25]. Panels (a) and (b) correspond to a low density of atoms at which the majority of eigenvalues are contained within the boundary imposed by the diffusion theory. Panels (c) and (d) correspond to a high density, for which states with very small decay rates $\mathrm{Im}\mathrm{\Lambda }$ appear in the scalar model, but not in the vector one. The smallest $\mathrm{Im}\mathrm{\Lambda }$ of the vector model is even larger than the prediction of the diffusion theory. The inset of panel (d) shows $N$ atoms (black dots) randomly distributed in a sphere of radius $R$.

Figure 2

Inverse participation ratio of eigenvectors. Gray scale density plot of the average IPR as a function of the eigenvalue $\mathrm{\Lambda }$ of the corresponding eigenvector. Dashed lines are the same as in Fig. 1. At low density, subradiant states localized on pairs of closely located scatterers exist in both scalar (a) and vector (b) models. These states have $\mathrm{IPR}\simeq 1/2$. At high density, Anderson-localized states with large IPR appear in the scalar model (c), but not in the vector one (d).

Figure 3

Scaling in scalar and vector models. (a) Thouless number $g$ as a function of the bare Ioffe-Regel parameter $k_0{\mathcal{l}}_0$ for the scalar model at frequency $\omega ={\omega }_0+{\mathrm{\Gamma }}_0/2$ and for different $N$ values. The curves cross at $g\approx k_0{\mathcal{l}}_0\approx 1$ and then again at $k_0{\mathcal{l}}_0\ll 1$ and $g\approx 1$. Localization transitions take place at these points, as confirmed by the analysis of the scaling function $\beta (g)$ that changes sign at $g\approx 1$ (inset). (b) The same for the vector model. Solid lines in the insets are guides for the eye.