Superresolution and duality for time-reversal of waves in random media

We analyze the time reversal of waves in a turbulent medium using the parabolic Markovian model. We prove that for waves in a fractal medium with a sufficiently small Fresnel number the time reversal resolution can be a nonlinear (between linear and quadratic) function of the wavelength and independent of the aperture. We establish the duality between the forward propagation and time reversal. The duality holds true for any media and has two aspects: First there is an uncertainty inequality between the turbulence-induced wave spread and time-reversal resolution. The inequality becomes an equality when the wave structure function is Gaussian. Second, the turbulence-induced resolution in time reversal is identical to the turbulence-induced coherence length. As a consequence, the turbulence-induced aperture can be estimated by 2π times the forward spread, independent of the original aperture.