2D numerical simulation of supercritical phase conjugation of ultrasound in active solid

In the present study a 2D axisymmetric numerical model is developed for a cylindrical magnetostrictive ceramic active element of a finite length emitting phase conjugate wave into a liquid. The pseudo-spectral time domain algorithm (PSTD) is used owing to its efficiency to model large-scale problems. PSTD solves the elastic wave equation in time dependent heterogeneous isotropic and axisymmetric anisotropic solids using FFT for high order approximation of the spatial differential operator on staggered grid, and a 4^th order Adams-Bashforth time integrator. The absorbing boundary condition, to truncate the computational domain, is introduced with complex frequency shifted perfectly matched layers (CFS-PML), found to be highly effective at absorbing evanescent waves and signals of long time-signature. A systematic study of the influence of lateral limitation of the active medium on parametric WPC of sound is made. It is shown, that retrofocusing of the incident pulse takes place even in the case of mode conversion inside the active zone. Nevertheless, amplitude and form of the obtained conjugate pulse depend on the simulated configuration. Numerical simulation correctly describes the parametric amplification and retrofocusing of ultrasound observed in experiments, when the conjugator is loaded at one end by water, and is free on the opposite one.