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 3 / 2 to 2 after N2 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.