arxiv: 2604.12595 · v1 · submitted 2026-04-14 · ⚛️ physics.app-ph

Recognition: unknown

Experimental demonstration for precisely tuning the focal length of finite-aperture focused beams and vortex

Shiyu Li , Yicheng Feng , Weiwei Cui , Zhixiong Gong

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:10 UTC · model grok-4.3

classification ⚛️ physics.app-ph

keywords focused ultrasoundfocal length tuningpiezoelectric transducerfinite aperturestationary phaseultrasonic focusinghigh-frequency acoustics

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The pith

The focal length of finite-aperture focused ultrasound beams varies approximately linearly with excitation frequency near the design frequency.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a planar compact piezoelectric ultrasonic transducer fabricated by truncating an ideal spherical wavefront with a plane. This design enables generation of high-frequency focused ultrasound and allows focal tuning by adjusting the driving frequency. Theory derives an approximate linear relation between focal length and frequency near the design value using the stationary-phase condition. Water-tank experiments confirm the linear tuning and match the theoretical predictions, providing a simple approach for dynamic focal control in compact devices.

Core claim

By truncating an ideal spherical wavefront with a plane, a finite-aperture focused beam is produced whose focal length varies approximately linearly with excitation frequency near the design frequency. The finite-range tuning is interpreted using the stationary-phase condition. Both theory and water-tank measurements agree that the focal length changes linearly with frequency, validating the model.

What carries the argument

Truncation of an ideal spherical wavefront by a plane to form a finite-aperture transducer, with focal position determined by the stationary-phase condition.

If this is right

Focal tuning is possible without mechanical adjustments or complex control systems.
The device is compact and planar, allowing easy integration with microscopic platforms.
High-frequency focused ultrasound can be dynamically adjusted for applications like imaging and particle manipulation.
The linear relation holds near the design frequency, enabling precise control within a finite range.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This frequency-based tuning could simplify real-time adjustments in biomedical ultrasound applications.
The approach may extend to generating and tuning acoustic vortex beams as suggested by the title.
Operation away from the design frequency might introduce non-linear effects not captured by the current model.

Load-bearing premise

The stationary-phase condition and the truncated spherical wavefront model accurately describe the finite-aperture acoustic field near the design frequency.

What would settle it

Water-tank measurements near the design frequency that show the focal length does not vary linearly with excitation frequency or deviate from the theoretical prediction.

Figures

Figures reproduced from arXiv: 2604.12595 by Shiyu Li, Weiwei Cui, Yicheng Feng, Zhixiong Gong.

**Figure 1.** Figure 1: FIG. 1. (color online) Comparison of the focused vortex and focused beam generated by planar interdigital transducers. (a) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (color online) Schematic of the experimental setup for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (color online) Simulated and measured results of the focused vortex. (a) Simulated distributions of acoustic field [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (color online) Simulated and measured results of the focused beam. (a) Simulated distributions of acoustic field [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. (color online) Comparison of theoretical, simulated, [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

read the original abstract

High-frequency focused ultrasound is widely used in biomedical applications such as high-resolution imaging, neuromodulation, particle manipulation, and so on. However, dynamic tuning of the focal plane in conventional systems often relies on mechanically adjustable components or array-based control with complex system and high cost. In this work, an optically transparent, planar compact piezoelectric ultrasonic transducer was designed and fabricated by truncating an ideal spherical wavefront with a plane, enabling high-frequency focused ultrasound generation and convenient integration with microscopic platforms. The acoustic field was characterized experimentally at the focal plane under the design frequency and at propagation planes near the design frequency to evaluate the focal tuning. An approximate linear relation between the focal length and driving frequency near the design one is derived theoretically, and the finite-range tuning behavior is interpreted using the stationary-phase condition. Both theory and experiment show that the focal length varies approximately linearly with excitation frequency near the design frequency. Water-tank measurements agree well with the theoretical prediction, confirming the proposed model. This work provides a simple and cost-effective approach for focal tuning in compact high-frequency ultrasound devices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This shows a workable planar piezo transducer that tunes focal length roughly linearly with frequency via wavefront truncation, with water-tank data backing the stationary-phase model.

read the letter

The main thing to know is that truncating an ideal spherical wavefront into a planar piezoelectric transducer gives an approximately linear shift in focal length when you change the drive frequency near the design point, and the experiments in water confirm the prediction without obvious contradictions. The design is compact, optically transparent, and avoids phased arrays or mechanical parts, which is the practical win for high-frequency ultrasound in biomedical settings like imaging or neuromodulation. They characterize the field at the focal plane and nearby planes, and the theory ties the tuning directly to the geometry and stationary-phase condition. That combination of a simple fabrication route and matching data is what makes the result usable. The linearity is labeled approximate, which fits the model assumptions of frequency-independent amplitude over a narrow band, and the abstract reports good agreement with measurements. The soft spots are the usual ones for this kind of work: no error bars or fit metrics are mentioned in the summary, so the tightness of the match will depend on the full methods and figures. The scope stays narrow to this geometry near the design frequency, with no claim of broad generality. This is for people building or using compact focused ultrasound devices who need a low-cost tuning knob without extra hardware. A reader working on transducer design or biomedical acoustics would get direct value from the experimental confirmation and the clear derivation. It is not a field-changer but a solid incremental step with reproducible elements. I would send it for peer review; the theory-experiment loop is grounded enough to justify referee time, even if the authors need to add quantitative agreement details.

Referee Report

1 major / 0 minor

Summary. The manuscript presents the design and fabrication of an optically transparent planar piezoelectric ultrasonic transducer formed by truncating an ideal spherical wavefront with a plane. This produces high-frequency focused ultrasound suitable for integration with microscopic platforms. Using stationary-phase analysis on the truncated-wavefront model, the authors derive an approximate linear relation between focal length and excitation frequency near the design frequency. Water-tank experiments characterize the acoustic field at the focal plane under the design frequency and at nearby propagation planes, with the data reported to agree with the theoretical prediction of linear focal tuning.

Significance. If the central claim holds, the work offers a passive, frequency-based method for focal-length tuning in compact high-frequency ultrasound transducers without mechanical components or phased arrays. This is potentially useful for biomedical applications such as imaging and neuromodulation. The combination of a stationary-phase derivation with experimental checks is a strength, though the experimental validation lacks quantitative rigor.

major comments (1)

[Experimental results] Experimental results section: The manuscript states that water-tank measurements 'agree well' with the theoretical prediction and thereby confirm the model, yet reports no error bars on measured focal positions, no quantitative fit metrics (e.g., slope comparison, R², or residual analysis), and no details on data acquisition, focal-position extraction algorithm, or exclusion criteria. Because the linear-tuning claim rests on this experimental confirmation, the absence of these elements leaves the strength of the validation unclear.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful review and for identifying ways to strengthen the experimental validation. We have revised the manuscript to incorporate additional quantitative details, methodological information, and metrics as outlined below.

read point-by-point responses

Referee: Experimental results section: The manuscript states that water-tank measurements 'agree well' with the theoretical prediction and thereby confirm the model, yet reports no error bars on measured focal positions, no quantitative fit metrics (e.g., slope comparison, R², or residual analysis), and no details on data acquisition, focal-position extraction algorithm, or exclusion criteria. Because the linear-tuning claim rests on this experimental confirmation, the absence of these elements leaves the strength of the validation unclear.

Authors: We agree that the experimental validation would be more robust with explicit quantitative reporting. In the revised manuscript we have added: (i) error bars on all measured focal positions derived from repeated scans at each frequency; (ii) a description of the data-acquisition protocol (hydrophone positioning, sampling rate, and averaging); (iii) the focal-position extraction procedure (parabolic fit to the on-axis intensity profile to locate the peak); and (iv) a linear regression of the experimental focal lengths versus frequency, including the fitted slope, its uncertainty, the R² value, and a direct comparison with the theoretical slope. These additions confirm the linear trend within the stated uncertainty and directly address the concern about the strength of the validation. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The central derivation applies the standard stationary-phase approximation to a truncated spherical wavefront model whose phase profile is fixed by geometry; the resulting approximate linear focal-length vs. frequency relation is obtained directly from that analytic step and is then compared against independent water-tank measurements. No parameter is fitted to the target data and then re-labeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and the model assumptions are stated explicitly without reducing to the experimental outcome. The reported agreement therefore constitutes external confirmation rather than a closed loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the truncated spherical wavefront model and the stationary-phase approximation for finite-aperture beams; no explicit free parameters are introduced beyond the design frequency itself.

axioms (1)

domain assumption The acoustic pressure field of the finite-aperture transducer can be modeled by truncating an ideal spherical wavefront with a plane.
This truncation defines the transducer geometry and underpins both the focal-length derivation and the stationary-phase interpretation.

pith-pipeline@v0.9.0 · 5493 in / 1205 out tokens · 44881 ms · 2026-05-10T14:10:56.981347+00:00 · methodology

discussion (0)

Reference graph

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Following the steps above, the sound field at any target position can be calculated as: pin(x, y, z) = 1 4π2 ZZ k2x+k2z≤k2 S(k x, kz) y=0 ×e ikxx+ikzz+i √ k2−k2x−k2zydkxdkz

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