Analysis of phonon echoes in simple insulators by means of Mathieu's equation

The equation of motion for acoustic waves, including the nonlinear effects of a strain-dependent dielectric constant in a simple insulator, is derived and analyzed for propagation in one dimension. The time behavior of these acoustic waves is found to be governed by Mathieu's equation when the material is immersed in a homogeneous time-varying electric field. This behavior is shown to produce time-reversed phonon echoes when the electric field is pulsed. The amplitudes of the time-reversed components are found in terms of the amplitude and pulse width of the electric field, and the ratio of the frequencies of the time-varying electric field and the acoustic wave.