Dynamic effect of phase conjugation on wave localization

We investigate what would happen to the time dependence of a pulse reflected by a disordered single-mode waveguide if it is closed at one end, not by an ordinary mirror, but by a phase-conjugating mirror. We find that the waveguide acts like a virtual cavity with resonance frequency equal to the working frequency ${\omega }_0$ of the phase-conjugating mirror. The decay in time of the average power spectrum of the reflected pulse is delayed for frequencies near ${\omega }_0.$ In the presence of localization the resonance width is ${\tau }_s^{-1}\exp (-L/l),$ with L the length of the waveguide, l the mean free path, and ${\tau }_s$ the scattering time. Inside this frequency range the decay of the average power spectrum is delayed up to times $t\simeq {\tau }_s\exp \mathrm{}(L/l).$