Dynamics of Anderson Localization in Open 3D Media

We develop a self-consistent theoretical approach to the dynamics of Anderson localization in open three-dimensional (3D) disordered media. The approach allows us to study time-dependent transmission and reflection, and the distribution of decay rates of quasimodes of 3D disordered slabs near the Anderson mobility edge.

Figure 1

Time-dependent diffusion coefficient for a disordered slab of thickness $L=100\ell$ in the regime of weak disorder (the product of wave number $k$ and mean free path $\ell$ exceeds unity). Dashed lines show the result of the perturbational calculation (4). Inset: the first peak of the leakage function $P_T(\alpha )$. We find $P_T(\alpha )\sim \delta (\alpha -1/t_D)$ and $D(t)/D_B\equiv 1$ in the limit $k\ell \to \infty$.

Figure 2

Time-dependent transmission coefficient $T(t)$ of a 3D disordered slab of thickness $L$ in the localized regime. For each value of $k\ell$, curves corresponding to $L/\xi =10$, 12, and 14 are shown ($\xi$ is the localization length). For clarity, the curves corresponding to $k\ell =0.7$ and 0.6 are shifted upwards by 3 and 6 units, respectively. The $t^{-(1+s)}$ slope (with $s\approx 0.85$, dashed line) obeyed by all curves crosses over to exponential decay for times $t>1/{\alpha }^{*}$. Lower inset: leakage threshold ${\alpha }^{*}$ vs slab thickness $L$ at $k\ell =0.6$ (+), 0.7 (circles), and 0.8 (triangles). The dashed line is ${\alpha }^{*}=(D_B/{\xi }^2)\exp (-L/\xi )$. Upper inset: leakage functions $P_T(\alpha )$ behave as ${\alpha }^s$ (dashed line), except for very small $\alpha <{\alpha }^{*}$ where they vanish rigorously.

Figure 3

Leakage function $P_R(\alpha )$ in reflection from a 3D disordered half-space in the localized regime. Dashed lines are linear fits to the initial parts of the curves. Arrows show ${\alpha }_0$ up to which the linear dependence of $P_R(\alpha )$ on $\alpha$ is expected. Inset: ${\alpha }_0$ as a function of $k\ell$. Dashed lines show approximate analytic results ${\alpha }_0=(D_B/4{\xi }^2)[1-6(k\ell {)}^4]$ for $k\ell \downarrow 0$ and ${\alpha }_0=(8D_B/27{\xi }^2)(1-k\ell )$ for $k\ell \uparrow 1$.

Figure 4

Distribution of leakage rates $P(\Gamma )$ for a slab of thickness $L$ in the localized regime (localization length $\xi$). For clarity, only the results for $k\ell =0.8$ are presented. For other $k\ell <1$ the curves are identical, except that the region of diffuse behavior $D_B/3{\xi }^2<\Gamma <D_B/{\ell }^2$ shrinks when $k\ell$ decreases and $\xi$ approaches $\ell$.