Two-dimensional axisymmetric numerical simulation of supercritical phase conjugation of ultrasound in active solid media

In the present study a two-dimensional axisymmetric numerical model is developed for supercritical parametric phase conjugation of ultrasound in a solid active element of cylindrical shape and finite length. The pseudospectral time domain algorithm (PSTD) is used owing to its efficiency to model large-scale problems. PSTD solves elastic wave equation in time-dependent heterogeneous isotropic and axisymmetric anisotropic solids using FFTs for high order approximation of the spatial differential operator on staggered grid, and a fourth-order Adams–Bashforth time integrator. In order to truncate the computational domain absorbing boundary conditions are introduced with complex frequency shifted perfectly matched layers. This procedure is highly effective at absorbing signals of long time-signature. The free surface of the active ceramic rod is introduced through the method of images. A systematic study of the influence of lateral limitations of the active medium on parametric wave phase conjugation of sound is made. It is shown that retro-focusing of the incident pulse takes place even in the case of mode conversions 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 retro-focusing of ultrasound observed in experiments.