Signature of Wave Localization in the Time Dependence of a Reflected Pulse

The average power spectrum of a pulse reflected by a disordered medium embedded in an $N$-mode waveguide decays in time with a power law $t^{-p}$. We show that the exponent $p$ increases from $\frac{3}{2}$ to $2$ after $N^2$ scattering times, due to the onset of localization. We compare two methods to arrive at this result. The first method involves the analytic continuation to an imaginary absorption rate of a static scattering problem. The second method involves the solution of a Fokker-Planck equation for the frequency dependent reflection matrix, by means of a mapping onto a problem in non-Hermitian quantum mechanics.