Scaling properties of delay times in one-dimensional random media

The scaling properties of the inverse moments of Wigner delay times are investigated in finite one-dimensional (1D) random media with one channel attached to the boundary of the sample. We find that they follow a simple scaling law which is independent of the microscopic details of the random potential. Our theoretical considerations are confirmed numerically for systems as diverse as 1D disordered wires and optical lattices to microwave waveguides with correlated scatterers.

Figure 1

(Color online) Scaled inverse delay times [Eq. (1)] for the Anderson model. Various symbols correspond to different disordered potentials $ϵ∊\{0.1,0.5,1,5,10\}$ and $\mid E(k=\sqrt{\pi })\mid <1$. Blue hollow symbols denote delay time data for $q=1$. For comparison, red solid symbols denote $q^{\prime }=2$ information length data, i.e., $l_L^{\left(q^{\prime }\right)}∕L$ versus $\lambda =2l_{\infty }(ϵ,E)∕L$. The dashed line is the result of the best fit from Eqs. (12, 13). Inset: same as in the main figure but now $q=2$ for delay times and $q^{\prime }=3$ for information lengths.

Figure 2

(Color online) Scaled inverse delay times for microwaves propagating in 1D waveguide. The different symbols correspond to energies $\mid E(k=0.5\pi )\mid <1$, $\mid E(k=0.7\pi )\mid >1$ being on both sides of the critical wave vector $k=0.57\pi$. A nice data collapse is observed, indicating that in both cases, the statistical properties of delay times (and thus the structural properties of wave functions) are unaffected by the correlation and correspond to exponentially localized wave functions; albeit the localization length for $k=0.5\pi$ is much larger than for $k=0.7\pi$. This is reflected in the overall scaling parameter $⟨{\tau }_{\infty }^{-1}⟩$. The dashed line is the result of the best fit from Eqs. (12, 13). Inset: the experimental transmission coefficient showing pass and stop bands is displayed by the blue line (left axis). The values for $⟨{\tau }_{\inf }^{-1}⟩$ are shown by the red circles (right axis) (Ref. 33).

Figure 3

Scaled inverse delay time for the disordered optical lattice system, with $ϵ∊\{4.556,0.5\}$ and filling factor $p∊\{0.01,0.025,0.05,0.1,0.9\}$. The nice data collapse confirms the universality of the scaling law [Eqs. (1, 2)]. The dashed line is the result of the best fit from Eqs. (12, 13).

Figure 4

Scaling of Eqs. (1, 2) in the variables from Eq. (12). Inset: same as in the main figure but now for the $q=2$ case.