Bridging the numerical-physical gap in acoustic holography via end-to-end differentiable structural optimization

Abstract

Acoustic holography provides a practical means of flexibly controlling acoustic wavefronts. However, high-fidelity shaping of acoustic fields remains constrained by the numerical-physical gap inherent in conventional phase-only designs. These approaches realize a two-dimensional phase-delay profile as a three-dimensional thickness-varying lens, while neglecting wave-matter interactions arising from the lens structure. Here, we introduce an end-to-end, physics-aware differentiable structural optimization framework that directly incorporates three-dimensional lens geometries into the acoustic simulation and optimization loop. Using a novel differentiable relaxation, termed Differentiable Hologram Lens Approximation (DHLA), the lens geometry is treated as a differentiable design variable, ensuring intrinsic consistency between numerical design and physical realization. The resulting Thickness-Only Acoustic Holograms (TOAHs) significantly outperform state-of-the-art phase-only acoustic holograms (POAHs) in field reconstruction fidelity and precision under complex conditions. We further demonstrate the application of the framework to spatially selective neuromodulation in a neuropathic pain mouse model, highlighting its potential for non-invasive transcranial neuromodulation. In summary, by reconciling numerical design with physical realization, this work establishes a robust strategy for high-fidelity acoustic wavefront shaping in complex environments.

Figures (16)

• Figure 1: Conventional phase-only versus geometry-centric thickness-only acoustic hologram optimization. (A, D) Pipelines of the conventional phase-only and the proposed thickness-only acoustic hologram. (B, E) Optimized holograms and simulated acoustic fields in the phase-only and thickness-only hologram frameworks, respectively. (C, F) Implemented holograms and measured acoustic fields in physical domain where the holograms are fabricated as lenses.
• Figure 2: Differentiable hologram lens approximation and end-to-end optimization of acoustic hologram geometry. (A) End-to-end differentiable optimization pipeline for TOAH design. A continuous lens parameter map $\theta$ is mapped to a three-dimensional lens geometry via the DHLA, followed by acoustic wave simulation and loss calculation. The histogram illustrates the resulting quasi-binary distribution of the optimized lens. (B) Optimization trajectory of the proposed framework. The loss function decreases over iterations as the hologram geometry evolves to better reproduce the target acoustic field. Representative snapshots of the lens geometry and corresponding simulated acoustic fields illustrate the progressive improvement in reconstruction during optimization.
• Figure 3: Transcranial ultrasound neuromodulation setup using the proposed TOAH framework. (A) Target regions for multi-focal neuromodulation in the mouse brain, including the ventral posterior nuclei (VP) and periaqueductal gray (PAG). (B) Numerical simulation setup for transcranial acoustic field synthesis through a rodent skull model. Red circles indicate the target focal points. (C) Exploded view of the holographic ultrasound module consisting of a transmit aperture, the hologram lens, and an acoustic coupling cone. (D) Experimental setup for transcranial stimulation using a stereotaxic transducer holder and a custom ultrasound transducer aligned with the rodent model. (E) Structural design of the transducer assembly showing the PZT-4 disk, matching layer (alumina–epoxy), air backing layer, and the hologram lens fabricated via 3D printing.
• Figure 4: Consistency between optimization and fabrication domains. (A) Reconstructed three-dimensional acoustic pressure fields in both domains for single-, dual-, and tri-focal configurations. (B) Quantitative performance metrics: (a) cross-domain field accuracy, (b) peak focal pressure, (c) energy leakage ratio, (d) focal intensity uniformity (red dashed line indicates the threshold for insufficient uniformity), (e,f) axial and lateral full width at half maximum (FWHM), and (g) -6 dB focal volume.
• Figure 5: Ex vivo validation of transcranial acoustic field reconstruction. Representative experimentally measured acoustic intensity fields for single-, dual-, and tri-focal configurations using Diff-PAT and the proposed TOAH. Left panels show the measured three-dimensional intensity distributions, and right panels show the corresponding –6 dB thresholded fields highlighting the reconstructed focal regions.