Efficient realization of on-demand functional ultrasonic fields based on prolate spheroidal wave functions from sampling theorem

The past decades have witnessed great efforts in the on-demand ultrasonic field design in which the time reversal technology was widely used in the whole-space acoustic hologram. In practice, the acoustic field of interest is usually bounded in a finite region with flexible distribution. Here, the use of prolate spheroidal wave functions to generate an arbitrary ultrasonic field in a finite region is proposed. The prolate spheroidal functions, which form a complete set of band limited functions and are orthogonal in the infinite and finite regions, can be efficiently reconstructed by the sampling theorem. To display the validation of the proposed method, two types of functional ultrasonic fields are numerically simulated. One type is the ultrasound standing wave field for which six nodes and two nodes are separately realized for two different types of standing waves in the limited range of (-2λ, 2λ). In addition, a composite standing wave field is stimulated with more complicated nodal distributions. The other type is the ultrasound focusing field, where three focal spots with the mainlobe sizes of λ, 0.5λ, and 0.35λ are demonstrated. It is worth noting that the nontrivial side lobes for super-oscillation focusing are designed to be about 3λ away from the central focal spot (the mainlobe size 0.35λ). This work has much significance in the applications of acoustic tweezing, ultrasonic imaging, and treatment.