Time-Reversal Aperture Enhancement

Time-reversal refocusing for waves propagating in inhomogeneous media have re- cently been observed and studied experimentally in various contexts (ultrasound, underwater acous- tics, ... ); see, for instance, (M. Fink, Scientific American, November (1999), pp. 63-97). Important potential applications have been proposed in various fields, for instance in imaging or communica- tion. However, the full mathematical analysis, meaning both modeling of the physical problem and derivation of the time-reversal effect, is a deep and complex problem. Two cases that have been considered in depth recently correspond to one-dimensional media and the parabolic approximation regime where the backscattering is negligible. In this paper we give a complete analysis of time- reversal of waves emanating from a point source and propagating in a randomly layered medium. The wave transmitted through the random medium is recorded on a small time-reversal mirror and sent back into the medium, time-reversed. Our analysis enables us to contrast the refocusing proper- ties of a homogeneous medium and a random medium. We show that random medium fluctuations actually enhance the spatial refocusing around the initial source position. We consider a regime where the correlation length of the medium is much smaller than the pulse width, which itself is much smaller than the distance of propagation. We derive asymptotic formulas for the refocused pulse which we interpret in terms of an enhanced effective aperture. This interpretation is, in fact, comparable to the superresolution effect obtained in the other extreme regime corresponding to the parabolic approximation. However, as we discuss, the mechanism that generates the superresolution is very different in these two extreme situations.