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 frequency mutual coherence function for the harmonic wave field. This quantity plays a crucial role in the analysis of many wave propagation phenomena and we illustrate by explicitly considering time reversal in the context of time pulses with a high carrier frequency. In a time-reversal experiment the wave received by an active transducer or antenna (receiver-emitter) array, is recorded in a finite time window and then re-emitted into the medium time reversed, that is, the tails of the recorded signals are sent first. The re-emitted wave pulse will focus approximately on the original source location. We use explicit expressions for the mutual coherence functions and their asymptotic approximations in the regime of long or short propagation distance and a high carrier frequency to analyze the refocusing of the wave pulse in the time reversal experiment. A novel aspect of our analysis is that we are able to characterize precisely the decoherence length in temporal frequency. This allows us to analyze for instance the time reversal experiment when the mirror has a finite aperture in time.