Narrow-band performance of phase-conjugate arrays in dynamic random media

Abstract

The theoretical narrow‐band performance of acoustic phase‐conjugate arrays in the presence of static and dynamic random media is presented. For a static random medium, analytical formulas are derived for the mean focus field of a Gaussian‐shaded volumetric phase‐conjugate array. The results suggest that random refraction allows phase‐conjugate arrays to ‘‘super focus,’’ that is, produce a focal region smaller than the free‐space diffraction limit. More specifically, vertical phase‐conjugate arrays are predicted to have horizontal directivity. In a dynamic random medium, phase‐conjugate array performance is degraded by changes in the medium that occur between the time that the acoustic signal is launched from its source and the time that the array’s transmission is received back at the source location (the round‐trip time delay). Formal results are obtained for an intrinsic signal‐to‐noise ratio for the signal sent from the phase‐conjugate array based on a combination of multiple scattering from static random refraction and single scattering from dynamic random refraction in the acoustic medium. These formal results are reduced to analytical formulas for a random medium characterized by the statistics of oceanic internal waves. The intrinsic signal‐to‐noise ratio is found to be proportional to: the inverse square of the round‐trip time delay, the inverse square of the acoustic frequency, the inverse first power of the array‐source range, and a simple function that combines the size of the array and the parameters of the random medium. For a typical deep‐water oceanic medium at acoustic frequencies near 10 kHz, phase‐conjugate array performance may be unaffected for round‐trip time delays as long as a minute.