Focal depth localization for highly focused transducers in isotropic materials

Focusing equations aim to define the point in a solid at which a transducer beam will reach a minimum cross section. The most commonly used focusing equation relies on a small angle assumption that inherently excludes sharply focused transducers with significant curvature. In this article, a revised focusing equation is proposed for normal and oblique incidence through a fluid-solid interface. The closed-form equation is derived using ray tracing approaches similar to the conventional expression but circumvents the paraxial approximation, extending the applicability to sharply focused probes. Both conventional and modified focusing equations are compared through normal and oblique incidence ray diagrams, and the proximity to the computationally derived geometric focus is explored. The proposed modification to the focusing equation generally results in a closer approximation to the geometric focus, a smaller beam cross section, and a greater time convergence when compared to the conventional focusing equation.