Dynamics of localized waves: Pulsed microwave transmissions in quasi-one-dimensional media

We have measured pulsed microwave transmission through quasi-one-dimensional (quasi-1D) samples with lengths up to three times the localization length as determined from measurements of the variance of intensity fluctuations. Measurements are analyzed using four complementary approaches, each appropriate in a specific time range: (i) diffusion theory; (ii) self-consistent localization theory (SCLT) with a renormalized diffusion coefficient in space and frequency, D(z,Omega); (iii) a dynamic single parameter scaling (DSPS) model, which reflects the decay of localized modes which do not overlap in space and frequency; and (iv) simulations of 1D random media. For times up to twice the diffusion time tau(D), diffusion theory gives an excellent fit to the data. For times up to 4 tau(D), the slowing of the decay rate of transmission is in accord with SCLT. For longer times, transmission decays more slowly than the predictions of the SCLT, indicating the inability of this modified diffusion theory to capture the decay of long-lived localized states. Beyond the Heisenberg time, the decay rate approaches the predictions of the DSPS model, reflecting the increasing proportion of wave energy in long-lived localized states. The decay rates obtained from 1D simulations are then in good agreement with measurements in quasi-1D samples and coincide with decay rates given by the DSPS model.