Nonergodic extended phase for waves in three dimensions

Abstract

Wave transport in complex media is determined by the nature of quasimodes at the microscopic level. In three dimensional disordered media, waves generally undergo a phase transition from diffusion to Anderson localization, characterized by exponentially localized modes. A remarkable exception are electromagnetic waves, whose vector-like nature prevents Anderson localization to occur. Here we demonstrate that both scalar and vector (electromagnetic) waves exhibit a non-ergodic extended phase characterized by fractal quasimodes, for a broad range of disorder strengths. While electromagnetic waves remain in the non-ergodic extended phase at high disorder strength, scalar waves eventually enter a localized regime. These results pave the way for the engineering of anomalous wave transport phenomena in disordered media without spatial correlations.

Figures (4)

• Figure 1: Phase diagram as a function of the disorder strength for the 3D Anderson model, the Rosenzweig-Porter model, and the point-dipole model for scalar and electromagnetic waves with a characteristic, schematic representation of a typical eigenstate of each phase. Vertical arrows indicate the occurence of the Ergodic and Anderson transitions in each model.
• Figure 2: Mean level spacing ratio $\langle r \rangle$ as a function of the scatterer density $\rho\lambda^3$ for (a) scalar and (b) electromagnetic waves. Horizontal dashed (dash-dotted) lines correspond to the predictions for the extended (localized) regime. Darker colors correspond to a larger number of scatterers, whose range is $N \in [2000,12000]$ in (a) and $N \in [2000,6000]$ in (b). The upper horizontal axes show the Ioffe-Regel parameter $k\ell$ within the independent scattering approximation at resonance. The vertical arrow in (a) shows the location of the Anderson transition.
• Figure 3: (a),(b) Scaling of the inverse participation ratio (IPR) with system size $N$ for scalar and electromagnetic waves, respectively. Symbols correspond to $\rho\lambda^3 = 0.1,4,8,10,30$, increasing from bottom to top. Solid black lines are power-law fits. Panels (c) and (d) present the normalized fractal dimension $D_F$ for the whole range of density. Solid lines show the result of the fitting analysis and the shaded region corresponds to standard deviation. Horizontal dashed (dash-dotted) lines are the predictions for the extended (localized) regime. The vertical line in (a) shows the Anderson transition point extracted from Fig. \ref{['fig2']}(a).
• Figure 4: Derivative of the normalized participation entropy $\tilde{S}^{\prime}$ as a function of the scatterer density $\rho\lambda^3$ for (a) scalar and (b) electromagnetic waves. Darker colors correspond to larger system sizes $N$. Panels (c) and (d) show an enlarged view of the curves from (a) and (b) around the crossing points corresponding to the ergodic transition, whose location is indicated by the arrows.