Differentiable topology optimization for high-fidelity acoustic holography beyond the thin-element approximation

Abstract

Acoustic holography in the megahertz range can impact applications from manufacturing and non-destructive testing to transcranial ultrasound. Designing lens topologies for complex holograms in the sub-megahertz to low-megahertz regime most relevant to transcranial therapy is difficult because wave propagation through an acoustically thick lens cannot be ignored. Here we show that the heterogeneous angular spectrum approach (HASA), which admits in-plane varying speed-of-sound maps and supports rapid differentiable optimization with the Adam optimizer, can generate three-dimensional lens topologies directly. With it we quantify the failure of the thin-element approximation (TEA) at these frequencies, where lenses enter an acoustically thick regime (thickness L much greater than the wavelength) governed by refractive walk-off, and we introduce a dimensionless refractive walk-off number Nw that defines a frequency-dependent operating regime. Topology optimization outperformed TEA-based design in all 45 configurations tested, with the largest structural gains at or below 1 MHz; for high-numerical-aperture designs the fidelity at 1 MHz matched that at 2 MHz, while structural similarity was highest at 0.5 MHz. The sub-megahertz to low-megahertz window is therefore a favorable operating regime for high-fidelity transcranial holography rather than one to be avoided. Free-field hydrophone measurements of 3D-printed lenses reproduced the designed patterns, and because the framework also accounts for aberration in heterogeneous media, patient-specific lenses corrected skull-induced aberration in simulation and ex vivo experiment, generalized across skull segments, and shaped volumetric beams through the human skull. Together these results provide a framework for designing simple, economical, and high-performance ultrasound systems.