Signature of wave localization in the time dependence of a reflected pulse

The average power spectrum of a pulse reflected by a disordered medium embe dded in an N-mode waveguide decays in time with a power law t(-p) We show t hat the exponent p increases from 3/2 to 2 after N-2 scattering times, due to the onset of localization. We compare two methods to arrive at this resu lt. The first method involves the analytic continuation to an imaginary abs orption rate of a static scattering problem. The second method involves the solution of a Fokker-Planck equation for the frequency dependent reflectio n matrix, by means of a mapping onto a problem in non-Hermitian quantum mec hanics.