Time reversal of parabolic waves and two-frequency Wigner distribution

We consider propagation and time reversal of wave pulses in a random environment.The focus of our analysis is the development of an expression for the two frequencymutual coherence function for the harmonic wave field. This quantity playsa crucial role in the analysis of many wave propagation phenomena and weillustrate by explicitly considering time reversal in the context oftime pulses with a high carrier frequency.In a time-reversal experiment the wave received by an activetransducer or antenna (receiver-emitter) array,is recorded in a finite time window and then re-emitted into the mediumtime reversed, that is, the tails of the recorded signalsare sent first.The re-emitted wave pulse will focus approximately on the originalsource location.We use explicit expressions for the mutual coherencefunctions and their asymptotic approximations in the regime oflong or short propagation distance and a high carrier frequency to analyzethe refocusing of the wave pulse in the time reversal experiment.A novel aspect of our analysis is that we are able to characterizeprecisely the decoherence length in temporal frequency.This allows us to analyze for instance the time reversal experimentwhen the mirror has a finite aperture in time.