Parametric acoustic array lensed by a gradient-index phononic crystal

This work presents a theoretical study of a parametric transmitter employing a small ultrasonic transducer and an acoustic lens for the collimation of the non-directional primary ultrasonic waves into a highly-directional beam. The acoustic lens is represented by a gradient-index phononic crystal (GRIN PC) composed of an array of toroidal scatterers. Parameters of the GRIN PC lens are determined employing an optimization procedure that maximizes the minimum value of the primary-wave amplitude over a wide frequency range at a distant point in front of the transducer-lens system. The Westervelt equation is used as a wave equation taking into account diffraction, nonlinearity, and thermoviscous attenuation. The wave equation is solved numerically in the quasi-linear approximation in the frequency domain employing the finite element method. The numerical results show that employing a simple GRIN PC lens, a highly-directional low-frequency beam can be parametrically radiated from a small ultrasonic transducer.