Ultrasonic four-wave mixing mediated by an aqueous suspension of microspheres: Theoretical steady-state properties

Abstract

A novel mechanism is analyzed for the interaction of sound with sound that relies on the slow response of the acoustic medium to the radiation pressure of a pump wave. The coupling mechanism is analogous to optical four-wave mixing in photorefractive materials. Counter propagating ultrasonic pump waves establish a standing wave in a dilute suspension of microspheres. In response, the spheres are attracted to pressure nodes and form spatially periodic bands. The Bragg reflection of a higher frequency probe wave from the concentrated layers of microspheres generates a fourth wave. Its amplitude is calculated by two methods that are found to give mutually consistent results for the dependence of the reflectivity on the pump wave amplitude. Both methods use a Born approximation that neglects the depletion of the probe wave propagating through the layered region. One of the calculations uses a symmetric Epstein layer to approximate the local refractive index variations. The other method is based on a Fourier decomposition of the spatial acoustic refractive index variations for layers of particles given by a Boltzmann distribution in a periodic energy well. The Bragg reflectivity is found to saturate with increasing pump pressure when the energy well becomes deep in comparison to the thermal energy kBT.