High-frequency dynamics of wave localization

We study the effect of localization on the propagation of a pulse through a multimode disordered waveguide. The correlator $〈u\left({\omega }_1{)u}^{*}\right({\omega }_2)〉$ of the transmitted wave amplitude u at two frequencies differing by $\delta \omega$ has for large $\delta \omega$ the stretched exponential tail $\propto \exp \left(-\sqrt{{\tau }_D\delta \omega /2\mathrm{}}\right).\mathrm{}$ The time constant ${\tau }_D{=L}^2/D$ is given by the diffusion coefficient D, even if the length L of the waveguide is much greater than the localization length $\xi .$ Localization has the effect of multiplying the correlator by a frequency-independent factor $\exp (-L/2\mathrm{}\xi ),$ which disappears upon breaking time-reversal symmetry.