On-chip phased interdigital metamaterials enable versatile manipulation of surface acoustic waves, microfluids, and micro/nano-objects

Domain: physics
Published: 2025-12-08
Editorial outcome: major_revision

Abstract
Surface acoustic waves (SAWs) offer great potential for quantum information processing, optomechanics, acoustofluidics, and acoustic tweezers. However, existing SAW chips lack the ability to control SAWs in a manner similar to current metamaterials, which can achieve versatile subwavelength-resolution manipulation of bulk acoustic waves. This study presents on-chip phased interdigital metamaterials (PIMs) featuring customized interdigital electrodes whose geometries are encoded with deep-subwavelength-resolution phase profiles, enabling versatile transformation of SAWs and manipulation of fluids and micro/nano-objects. Our on-chip PIMs can transform forward SAWs into waves with desired wavefronts and energy patterns, such as SAWs propagating in a specified direction, a SAW jet with energy confined in a wavelength, and twin jets. They also enable “diode-like” SAW transmission, allowing for routing the information carried by SAWs along a forward pathway while blocking backward communication. Additionally, SAWs generated by PIMs exhibit unique energy patterns, allowing for versatile active control of fluid streaming and micro/nano-object distributions.

Introduction
Surface acoustic wave (SAW) technologies, employing “earthquakes” on piezoelectric chips, have shown great potential in various fields such as quantum acoustics1–4, optomechanics5–9, wireless communication10–13, and acoustofluidics14–19. In quantum acoustics, a superconducting qubit can be coupled with SAWs, exhibiting non-linear SAW reflection and relaxation of an excited qubit through SAW emission4. In optomechanics, SAWs can resonantly excite an optomechanical cavity5,6 and precisely trigger single-photon generation by controlling the injection of electrons and holes into quantum dots9. For communication systems, SAW-based filters, duplexers, and multiplexers play critical roles10,11. In recent years, SAWs have increasingly found numerous applications in life sciences20–25, including bioparticle detection with SAW-based biosensors26–28, processing small volumes of bioliquids with SAW-based acoustofluidic devices29–34, and contactless manipulation of tiny bio-samples ranging from nano- to millimeter scales (e.g., zebrafish35,36, cells37–42, exosomes43–45, and nucleic acids44) using SAW-based acoustic tweezers.

Despite advances in SAW technologies, multiple challenges remain. Interdigital transducers (IDTs) with parallel straight electrodes and concentric arc-shaped electrodes have long been predominantly used for generating straight-wavefront and focused SAWs23–26,46,47, respectively. Beyond these traditional designs, IDTs with various unique electrode geometries have been developed to achieve different functionalities—for example, focused IDTs with electrode geometries derived from slowness curves for improved SAW focusing48–51, spiral-shaped IDTs for generating acoustic vortices19,42,52, and angularly slanted IDTs capable of steering SAW direction via frequency tuning53,54. However, a generalized approach for designing IDTs with complex-shaped interdigital electrodes that can generate SAWs with user-defined wavefronts and energy patterns remains lacking, thus limiting the development of customized SAW devices tailored to generate optimal SAW fields for specific application needs. Moreover, few studies have developed metamaterials capable of transforming incident SAWs into waves with targeted propagation directions, wavefront shapes, and energy profiles, and no SAW metamaterial device to date can simultaneously perform such transformations and enable “diode-like” SAW information delivery. Although some studies have explored mechanisms that use components (e.g., thin film transistors, magnetoelastic coupling layers, etc.) placed in the SAW path to modulate transmitted SAWs55–65, integrating these components into large arrays for subwavelength-resolution SAW field transformation remains challenging due to the difficulty of scaling down the complex intermediate modulators to subwavelength dimensions and/or fabricating arrays that incorporate many of them. Recently, Xu et al. introduced sawtooth-shaped metasurfaces bonded to a substrate for SAW deflection66, and Wang et al. demonstrated both SAW focusing and deflection by depositing metasurfaces composed of pillar arrays with varied designs onto a substrate67. However, these approaches require complex fabrication processes due to the addition of extra microstructures, and their functionalities remain limited to SAW deflection and/or focusing. Third, in applied acoustics, a long acoustic jet with energy highly focused within a wavelength shows great potential for acoustic sensing and diagnostics68–70, and acoustic twin beams with energy confined in paired beams enable unique acoustic tweezers for single-object manipulation71–73. However, no studies have realized on-chip SAW jets that could benefit SAW sensor development or explored SAW twin beams that might enable novel SAW tweezers for arranging micro/nano-objects.

To address these gaps, this study presents phased interdigital metamaterials (PIMs) for multi-mode manipulation of on-chip SAWs, microfluids, and micro/nano-objects, such as transforming SAW wavefronts and energy fields, generating unique vortex streaming, and arranging micro/nano-objects into various patterns. Compared to traditional simple-configuration IDTs, the PIMs are composed of complex-shaped electrodes encoded with deep-subwavelength-resolution, position-dependent phase profiles, thus allowing for generating SAWs with tailored wavefronts and energy fields. These PIM offers a unique approach for developing SAW chips that can steer incident SAWs and transform them into waves with diverse wavefronts and energy patterns—without relying on wave interactions with complex-shaped microstructures (such as the micropillar arrays and sawtooth microstructures used in recently developed metasurfaces66,67), which require intricate microfabrication processes to be integrated onto the substrate. Compared to those metasurfaces designed for SAW deflection and/or focusing66,67, our PIM-based chip can achieve more complex SAW field transformations and enable “diode-like” routing of information carried by SAWs. Through validation experiments, our PIM-based chips were successfully demonstrated to transform forward-direction SAWs into waves propagating in desired directions, long single SAW jets, as well as a pair of SAW jets (i.e., twin SAW jets). These chips also successfully routed the information carried by SAWs along the forward pathway while forbidding information delivery in the backward direction. Furthermore, as our PIMs can generate SAWs with unique energy patterns (e.g., pseudo-standing SAWs, single SAW jets, and twin jets), they enable appealing acoustofluidic functions for actively controlling fluids and micro/nano-objects. These functions were validated by arranging microparticles into various patterns, aligning multi-walled carbon nanotubes (MWCNTs), generating counter-chirality streaming vortices, as well as driving the motions of nano-objects.

Results
On-chip phased interdigital metamaterials (PIMs) for manipulating surface acoustic waves (SAWs), fluids, and micro/nano-objects

The PIM is inspired by acoustic holography, which leverages a wave transformation module (e.g., an acoustic lens or metamaterial) containing customized, position-dependent phase information to modulate acoustic waves generated from a transducer (e.g., a piezoelectric plate)74–76, as illustrated in Fig. 1a. The commonly used acoustic holography approach struggles to manipulate high-frequency SAWs propagating along a piezoelectric wafer’s surface due to the challenges in developing and manufacturing arrays of microscale, subwavelength elements capable of inducing desired phase shifts in a range of 0 to 2π rad. To address these challenges, we developed PIMs that integrate the wave transformation and generation modules by encoding the desired position-dependent, subwavelength-resolution phase information into the shapes of the PIMs’ microscale interdigital electrodes, as illustrated in Fig. 1a (right). This approach allows for engineering both the SAW wavefronts and energy distributions by carefully designing the PIM’s interdigital electrode pattern with subwavelength resolution. Based on this mechanism, four example PIMs have been designed to generate SAWs propagating in a desired direction (Fig. 1b), pseudo-standing SAWs (Fig. 1c) with an energy pattern like that of standing SAWs generated by a pair of parallel IDTs, a long SAW jet with most SAW energy focused in a long narrow beam (Fig. 1d), and SAW twin jets with most SAW energy confined in a pair of long narrow beams (Fig. 1e).

PIMs can also be used to develop SAW chips that transform incident SAWs and achieve “diode-like” unidirectional transmission of the information carried by SAWs. As illustrated in Fig. 1f, by integrating four key modules, including a typical straight-electrode IDT, a lossy material (e.g., high-damping paste or polymer), a gain module (e.g., one-way amplification circuit), and a PIM, we have developed a SAW metamaterial chip capable of transforming forward-propagating SAWs into waves with desired wavefronts and energy distributions. For example, when straight-wavefront SAWs incoming from the left side of the chip reach the straight-electrode IDT, an electrical signal is generated from the IDT. This signal is then amplified by a one-way amplification circuit and sent to the PIM on the right side of the lossy material. This allows the incident straight-wavefront SAWs to be transformed into SAWs with desired wavefronts and energy distributions, depending on the PIM’s electrode pattern. In contrast, the information of SAWs incoming from the right side of the metamaterial cannot be transmitted to the left side, as the amplification circuit ensures one-way transmission and amplification of the electrical signal. The lossy material is also critical to achieving “diode-like” unidirectional transmission of SAW information, as it absorbs SAWs arriving from both sides of the piezoelectric substrate and breaks the wave path through the substrate.

In addition to manipulating SAW propagation, PIMs can facilitate the development of acoustic tweezers for manipulating micro/nano-objects, as well as on-chip flow control devices that can generate versatile fluid streaming fields. On one hand, using a pair of PIMs, customized distributions of acoustic potential valleys can be generated to trap tiny objects using acoustic radiation forces and arrange them in various patterns, such as tilted parallel-line-like and two-dimensional (2D) lattice-like arrangements (Fig. 1g, h). With this design approach, infinite customized SAW energy patterns can be generated to construct diverse object patterns by carefully designing the electrode shapes of PIMs through the encoding of desired phase profiles. On the other hand, traveling SAWs from PIMs can generate acoustic streaming in a liquid layer on the SAW substrate, further inducing drag forces on tiny objects to transport them along streamlines. PIMs open a route for generating customized fluid streaming fields, as the PIMs’ electrode configurations can be engineered to generate various traveling SAW fields. For instance, a PIM designed to generate a long narrow SAW beam can produce a jet flow accompanied by multiple pairs of counter-chirality vortices on different sides of the jet flow (Fig. 1i). When customized to generate two long narrow SAW beams, two fluid jets accompanied by more pairs of counter-chirality vortices can be generated (Fig. 1j).

To support the development of PIMs, we established an analytical model (see Supplementary Note 1 and Supplementary Fig. 1) to quickly predict SAW fields generated by PIMs. Moreover, we have performed laser vibrometry-based SAW field visualization experiments to validate the ability of PIMs to generate SAWs with different wavefronts and energy patterns. Furthermore, using our PIMs, we have developed SAW metamaterial chips and experimentally validated their functions of transforming incident SAWs and performing “diode-like” transmission of SAW information. In addition, through a series of acoustofluidic experiments, we have demonstrated the capability of PIMs to generate acoustic streaming and arrange the distributions of micro/nano-objects. The key experimental results, along with the related data analysis and discussion, are given below.

Generation of versatile surface acoustic wave (SAW) fields

The shape of a PIM’s interdigital electrode is critical for generating SAWs with the desired frequency, wavefront shape, and energy pattern. Based on our proposed approach, which encodes the desired subwavelength-resolution spatially varying phase information into the electrode shape, we have developed multiple PIMs (see Supplementary Figs. 2 and 3 for configurations) to generate customized SAWs, such as tilted SAWs propagating in a desired direction and pseudo-standing SAWs. The mathematical descriptions of electrode shapes are provided in Supplementary Note 2. To characterize the PIMs, we acquired their generated SAW displacement and energy fields using a laser vibrometry system (Supplementary Fig. 4) and further analyzed the generated SAW field’s wavenumber spectrum in the kx-ky domain.

To generate SAWs propagating in a direction of eθ = (cos θ, sin θ) (illustrated in Supplementary Fig. 2a), the PIM’s electrode shape should be encoded with a spatial-varying phase profile having a constant gradient ∇ωc(θ) = cSAW(θ), where ω is the angular frequency and cSAW(θ) is a direction-dependent SAW velocity. For example, when using a PIM (illustrated in Fig. 2b) encoded with a gradient phase profile shown in Supplementary Fig. 3b (top), the laser-measured SAW displacement field in Fig. 2e shows tilted wavefronts perpendicular to the SAW propagation direction θ = 80 deg. This is also confirmed by the wavenumber analysis result in Fig. 2h, showing a high-intensity wavenumber spot in 80 deg in the kx-ky domain. In contrast, SAWs generated by a traditional straight-electrode IDT propagate in the 90 deg direction (Fig. 2d) and have a high-intensity wavenumber spot in 90 deg in the kx-ky domain (Fig. 2g). Additionally, the simulation results agree well with the experimental results (Fig. 2d–f), successfully validating our analytical model, which is for quickly predicting SAWs generated by PIMs with complex-shaped interdigital electrodes and guiding the PIM design and optimization.

By encoding the electrode shape with a phase profile in Supplementary Fig. 3c, which alternates between 0 and π rad with a spatial period of Δd, we developed a PIM to generate pseudo-standing SAWs. Details of the phase profile expression and PIM design can be found in Supplementary Note 2 and Supplementary Fig. 2b. For validation, we manufactured a PIM and performed laser-based SAW field characterization. The measured SAW intensity field (Fig. 2f, right) is similar to the intensity fields of standing SAWs generated by a pair of parallel IDTs, and thus the SAWs generated by our PIM are referred to as ‘pseudo-standing SAWs’ in this study. From the measured intensity field, it is evident that the x-direction spacing between adjacent intensity peaks is Δd/2, which depends on the design of the PIM rather than the generated SAW’s wavelength (λ). Moreover, this spacing is the same as the peak spacing in regular standing SAWs created by the interference of SAWs carrying wavenumbers of ±2π/Δd generated at a frequency f = cSAW(θ)/Δd. To further analyze the ‘pseudo-standing SAWs’, the acquired SAW field is transformed to a wavenumber spectrum in the kx-ky domain. The experimental wavenumber spectrum in Fig. 2i clearly shows two high-intensity spots symmetric to the ky-axis. Our experimental results in the x-y and kx-ky domains prove the effectiveness of generating ‘pseudo-standing SAWs’ using only a PIM to generate traveling SAWs propagating in one direction, rather than the traditional method of using multiple IDTs to generate counter-propagating SAWs.

Generation of long narrow surface acoustic wave (SAW) jets

In addition to generating SAWs propagating in a desired direction and pseudo-standing SAWs, we have developed and validated PIMs (illustrated in Fig. 3a–d) for generating single long SAW jets and twin jets. An acoustic jet is a highly focused acoustic beam with a long beam length in the wave propagation direction while maintaining a subwavelength-scale width68–70, which is defined by the distance between the beam’s full width at half maximum (FWHM) points illustrated in Supplementary Fig. 5. However, few methods can generate one or multiple long SAW jets on piezoelectric chips. Our developed PIMs overcome this limitation, enabling the generation of a long SAW jet and a pair of jets (i.e., twin jets) at high frequencies, such as tens of megahertz. Moreover, key parameters of a SAW jet, such as beam length, orientation, and center position, can be customized by carefully designing the PIM’s electrode configuration.

The key to generating a long SAW jet is to optimize the PIM’s electrode shape so that the PIM can generate a series of focused SAWs whose focal points gradually change from xF(1) to xF(N) as illustrated in Supplementary Fig. 5. Using this approach, a long SAW jet with an energy profile ranging from xF(1) to xF(N) can be generated, as confirmed by the alignment of the generated SAW jet with the focal points shown in Supplementary Fig. 6. Moreover, SAW jets can exhibit different shapes by customizing PIM design parameters (e.g., the number of electrodes N, device span W, focal length dc, and beam center shift Δs), as demonstrated by the simulation results in Supplementary Fig. 7, which also reveal how these parameters affect beam width (Wb). The results show that the beam width gradually increases with increasing focal length and/or decreasing device span. When the number of electrodes is small (e.g., N = 2), this relationship aligns with the prediction Wb = 1.02λdc/W, which is commonly used to estimate the beam widths of focused ultrasound transducers77,78. Second, shifting the jet away from the device center line toward the edge (i.e., increasing Δs) results in a gradual increase in beam width. Third, as the number of electrodes increases, the beam width initially increases and then reaches a plateau. Additionally, by appropriately selecting the design parameters, subwavelength-scale beam widths can be achieved. These relationships observed from simulation results provide valuable guidance for optimizing PIM designs.

For experimental validation, we fabricated three PIMs, whose electrode shapes were designed based on phase profiles in Supplementary Fig. 3d–f, for generating single long SAW jets with different energy distributions (see Fig. 3a–c). When using a PIM (Fig. 3a) designed with focal points close to its electrodes, the measured SAW displacement and energy fields (Fig. 3e, i) clearly show a long SAW jet near the electrodes along the PIM’s centerline. When using a PIM (Fig. 3b) designed with focal points far from the electrodes, a long SAW jet far from the PIM electrodes is generated, as shown by the measured SAW displacement and energy fields (Fig. 3f and j). We also showed the ability to generate an off-center SAW jet (see Fig. 3g and k) by using a PIM whose focal points offset from the PIM centerline. To further analyze the generated SAW jets, the multi-dimensional Fourier transform was applied to the acquired wavefields to reveal their wavenumber components. Compared to the unfocused SAW’s spectrum (Fig. 2g) with only a high-intensity spot, the SAW jets generated along the centerline have wavenumber components widely and symmetrically spread in the ±kx directions (Fig. 3m, n). When a SAW jet is left shifted, its spectrum becomes asymmetric with higher intensity in the region with kx < 0 (Fig. 3o).

In addition to demonstrating single SAW jets, we fabricated a PIM (Fig. 3d) capable of generating a pair of long, narrow jets (referred to as twin jets), which, to the best of our knowledge, has not been achieved before. This PIM minimizes the SAW energy along the centerline through destructive interference, thus allowing for splitting a SAW jet into two lobes to create twin jets. Here, energy minimization is achieved by introducing an additional phase change of π rad to half of the PIM originally designed to generate a single SAW jet. The experimental SAW intensity field in Fig. 3l (right) confirms the generation of a pair of slim SAW jets (i.e., twin jets) with a low energy region confined between them. The wavenumber spectrum in Fig. 3p for twin jets shows two high-intensity regions that are nearly symmetric to the ky-axis, in contrast to the single jet’s wavenumber spectrum in Fig. 3m. Additionally, we found that after splitting a single jet into twin jets, the width of each split jet becomes smaller than that of the original. However, as the device span increases, this effect gradually diminishes, as shown in Supplementary Fig. 7l.

The above analytical and experimental results demonstrate the ability to generate a single long SAW jet and twin jets using PIMs whose electrode shapes are encoded with customized spatial-varying phase profiles. Moreover, by optimizing the interdigital electrode configuration, PIMs can be developed to generate SAW jets with customized beam lengths and positions. Furthermore, compared to a traditional focused IDT with concentric arc-shaped electrodes, a PIM with the same span (W), number of electrodes (N), and center electrode focal length (dc) can generate a significantly longer focused SAW beam, although its width slightly increases, as shown in Supplementary Fig. 8. Additionally, the four SAW fields (Fig. 3i–l, left) simulated using our analytical approach described in Supplementary Note 1 closely resemble their corresponding experimental SAW fields (Fig. 3i–l, right). The beam widths measured from the experimental SAW fields are also close to those obtained from the simulation results (see Supplementary Fig. 9 for beam width comparisons), except for the case in Fig. 3k, which may be attributed to SAW field measurement and device fabrication errors. These results indicate that our analytical model is effective in predicting complex SAW jets generated by PIMs, thereby facilitating the design of PIM devices.

Transformation and ‘diode-like’ transmission of on-chip surface acoustic waves (SAWs)

Acoustic metamaterials are attracting increasing attention for transforming incident acoustic waves by modulating the phase and amplitude profiles of waves transmitted through them, as well as for achieving “diode-like” transmission of acoustic waves, i.e., one-way routing79–82. However, most studies focus on manipulating low-frequency acoustic waves such as bulk waves in air, fluids, and solids. Few studies have realized metamaterials capable of transforming the energy field of SAWs into desired patterns. Moreover, few studies have developed metamaterials that can both transform on-chip SAWs and achieve “diode-like” wave transmission.

To address these limitations, we have developed SAW metamaterial chips (illustrated in Fig. 1f) that can both transform incident SAW fields into desired patterns and achieve ‘diode-like’ routing. Our SAW metamaterial chip is composed of four key modules: (i) a straight electrode IDT, (ii) a lossy material (e.g., high-damping paste or polymer) bonded on the substrate for absorbing SAWs as proven by experimental results in Supplementary Fig. 10, (iii) a one-way gain module (e.g., amplification circuit) for signal amplification and breaking symmetry, as well as (iv) a PIM for emitting transformed SAWs. In the forward direction, an incident SAW is first received by the left-side straight-electrode IDT, generating an electrical signal that is amplified by the one-way gain module. The amplified voltage signal is then sent to the PIM to emit a transformed SAW to the right side of the chip. Through this process, an incident SAW can be transformed into a SAW with the desired wavefront/energy pattern, which depends on the PIM design. In the backward direction, a SAW incoming from the right side of the chip is first received by the PIM, producing an electrical signal. However, this signal cannot be delivered to the left-side straight-electrode IDT, due to the one-way amplification circuit. In addition, the SAW propagating in the −y-direction cannot transmit through the piezoelectric substrate to the left side of the chip because the on-substrate high-damping lossy material absorbs SAWs.

For validation, we fabricated and characterized three SAW transformation chips having different PIM designs (see Fig. 4a, c, and e). When using the design in Fig. 4a, the acquired SAW displacement fields in Fig. 4b and Supplementary Movie. 1 reveal that incident SAWs propagating in the +y-direction can be steered to a direction 10 deg off the +y-axis. When using the chip design in Fig. 4c, the incident straight-wavefront SAWs are transformed into a focused SAW jet, as shown by the acquired displacement fields in Fig. 4d and Supplementary Movie. 2. Moreover, using the chip design in Fig. 4e, the incident SAWs are transformed into twin jets, as shown by the acquired displacement fields in Fig. 4f and Supplementary Movie. 3. In addition, the “diode-like” SAW transmission capability has been validated. The acquired SAW displacement fields in Fig. 4h and Supplementary Movie. 4 show that SAWs generated on the right side of the chip cannot transmit to the left side. The acquired transmission spectra in Supplementary Fig. 11 further confirm the ‘diode-like’ high transmission in the forward direction and nearly zero transmission in the backward direction. By reversing the amplification circuit, backward high transmission can be achieved, while prohibiting forward transmission. Therefore, as validated by our experimental results in Fig. 4 and Supplementary Fig. 11, our developed SAW metamaterial chips enable the transformation of incident SAW fields into desired patterns, as well as the ‘diode-like’ delivery of the information carried by SAWs. Additionally, since the chip uses an IDT positioned to the left of the lossy material for SAW sensing, a small portion of the incident SAW energy (~9.3%, according to experimental results) is reflected. The IDT’s inherent filtering capability allows for frequency-selective SAW transmission. By replacing the sensing IDT with differently shaped sensing electrodes, it may be possible to reduce SAW reflections and enable multi-frequency or broadband SAW transmission—an interesting direction for future investigation.

Phased interdigital metamaterials (PIMs) for acoustofluidics

Acoustofluidics has shown great potential for generating and controlling fluid flows, as well as arranging and transporting micro/nano-objects in fluids. In recent years, these acoustofluidic abilities have been increasingly used for developing lab-on-a-chip devices for biomedical research and applications such as rotating cells, separating bioparticles, concentrating exosomes, as well as transporting and mixing biofluids14–25. Here, we demonstrate that our PIMs can be used for acoustofluidics by enabling acoustic streaming to manipulate fluids and applying forces on micro/nano-particles to transport and arrange them. The acoustofluidic experiments were performed using setups illustrated in Supplementary Fig. 12, with particle distributions and motions recorded by a camera.

When using a pair of PIMs for generating tilted standing SAWs, the acquired microscopic image in Fig. 5b and Supplementary Movie. 5 show that 35 μm SiO2 particles are arranged in a tilted, parallel-line-like distribution, as these microparticles are trapped by the potential valleys of the generated tilted standing SAWs. Interestingly, microparticles can be arranged in a 2D lattice-like pattern (see Fig. 5e and Supplementary Movie. 6), when using a pair of PIMs designed for generating pseudo-standing SAWs. Previous approaches for constructing 2D lattice-like particle patterns typically required orthogonally arranged IDTs that generated SAWs propagating along orthogonal axes; however, our PIM-based approach requires only SAWs propagating along a single axis, which has not been presented before to the best of our knowledge. Since SAWs generated on a piezoelectric wafer are accompanied by dynamic electric fields, we also demonstrated the ability of our PIMs for arranging MWCNTs using the accompanied electric fields. When using tilted standing SAWs, MWCNTs align perpendicular to the node lines, forming multiple parallel groups in the potential valleys of the tilted standing SAWs, as shown by the acquired microscopic image in Fig. 5c. When using pseudo-standing SAWs, MWCNTs align along the SAW propagation direction, forming multiple groups (see Fig. 5f) in SAW potential valleys with a 2D lattice-like distribution. Additionally, the arrangements of microparticles and MWCNTs observed in the above experiments can be predicted using analytically simulated SAW-induced acoustic radiation and dielectrophoretic force fields, respectively (see Supplementary Note 1 and Fig. 13 for the simulation approach and results, respectively).

In addition to arranging the distributions of micro/nano-objects, we demonstrated the ability to transport particles, as well as generate unique fluid streaming fields. First, when using a PIM to generate a long narrow SAW beam, Fig. 5h and Supplementary Movie 7 show that 35 μm SiO2 particles were transported along the SAW beam, primarily by the acoustic radiation force. The acoustic streaming pattern (Fig. 5i), observed through 500nm SiO2 particles influenced mainly by the streaming-induced drag force, revealed a microscale jet flow accompanied by multiple pairs of counter-chirality vortices symmetric to the jet (Supplementary Fig. 14d and Movie. 8). Interestingly, this flow pattern resembles the classical free in-air subsonic jet with an array of vortices83, but here it is observed in a microscale liquid environment on a piezoelectric chip. Second, when using a PIM to generate two long narrow SAW beams, 35 μm particles were transported along the two SAW beams (see Fig. 5k and Supplementary Movie. 9), with some microparticles trapped along the centerline between the two SAW beams due to minimal SAW potential from destructive interference. The streaming pattern (Fig. 5l) revealed by 500nm particles showed that each high-potential SAW beam induced a forward flow with counter-chirality streaming vortices on different sides, and interestingly, the combined effect of inner vortices created a backward flow along the low-potential centerline between the two SAW beams (Supplementary Fig. 14e and Movie. 10). Third, when using a PIM to generate pseudo-standing SAWs with multiple high-intensity beams, 35 μm particles were transported by these high-intensity beams (Fig. 5n and Supplementary Movie. 11). Moreover, each SAW beam induced a forward streaming flow with multiple pairs of counter-chirality streaming vortices on different sides (Supplementary Fig. 14f and Movie. 12). Their combination created an array of periodically arranged streaming vortices (Fig. 5o), having the same chirality along the +y-direction and alternating chirality in the +x-direction.

Methods
Simulation of surface acoustic waves (SAWs) generated by phased interdigital metamaterials (PIMs)

To predict the wavefields of SAWs generated by PIMs with complex-shaped electrodes, an analytical model was established. The details of the analytical model are given in Supplementary Note 1, and a schematic of the analytical model is shown in Supplementary Fig. 1. Since the performance of a SAW device is influenced by both the electrode design and the anisotropic material properties of the piezoelectric substrate, our model considers both the geometries of complex-shaped interdigital electrodes as well as the direction-dependent SAW velocity and electromechanical potential-to-displacement ratio. Based on the analytical model, we can predict the displacement, electric potential, and intensity fields of SAWs generated by PIMs. Using these fields, the SAW-induced acoustic radiation and dielectrophoretic forces acting on small particles can then be analyzed, as detailed in Supplementary Note 1, enabling the prediction of their arrangement within the generated SAW field.

Design and manufacturing of phased interdigital metamaterials (PIMs)

To design a PIM for generating SAWs with desired wavefronts and energy distributions, a subwavelength-resolution spatial-varying phase profile ϕd(x) needs to be determined and then encoded into the electrode configuration. Supplementary Note 2 gives the expressions of the desired phase profiles for generating tilted SAWs, pseudo-standing SAWs, a SAW jet, and twin jets. It also provides the relation to encode the desired phase profile into the interdigital electrode’s profile. Based on the approach in Supplementary Note 2, we designed customized electrode patterns for different PIMs with a working frequency of 18.8 MHz (see Supplementary Fig. 3 for all the phase profiles, electrode configurations, and photos of fabricated devices). To manufacture these PIMs, we deposited a 10 nm layer of chromium (Cr) and then an 80 nm layer of gold (Au) on a 0.5 mm-thick Y128-cut LiNbO3 substrate using procedures including spin coating, lithography, e-beam evaporation, lift-off, etc.

For the transformation and ‘diode-like’ transmission of SAWs, we fabricated devices each integrating a SAW chip and a one-way amplification circuit (see Supplementary Fig. 18 for the schematic and photo of a device). The circuit consists of a BGA2869 amplification chip (NXP Semiconductors), a 220 Ω resistor (R), and four 400 nF capacitors (C1, C2, C3, and C4). Its operational frequency range spans from 0.1 MHz to 2 GHz. To ensure stable amplification without noticeable nonlinear waveform distortion, the input signal at the signal-in port and the direct current (DC) voltage bias at the VDC port were kept below 1 V and 12 V, respectively. The signal-in port is connected to the straight-electrode IDT, allowing the SAW signals acquired by the IDT to be fed into the circuit. The amplified signal at the signal-out port is supplied to the PIM transducer for SAW generation. To generate SAWs with varying displacements, the output voltage at the signal-out port can be adjusted by varying the DC voltage bias applied to the amplification circuit. In this study, a DC voltage bias of 9 V was used to generate SAWs strong enough for reliable measurement using a laser Doppler vibrometer.

Acquisition of surface acoustic wave (SAW) fields

To measure SAW fields generated by a PIM, we used a customized laser Doppler vibrometry setup illustrated in Supplementary Fig. 4. A function generator (DG1022, Rigol) is used to generate a Hanning window modulated multi-cycle sine signal with a center frequency matching the IDT’s resonance frequency (18.8 MHz). This excitation signal is sent to the IDT for SAW generation. A laser Doppler vibrometer (FX-1-130, Polytec Inc.) is used to acquire SAWs propagating on the piezoelectric substrate based on the Doppler effect. The laser beam is focused on the top surface of the piezoelectric wafer, and it is perpendicular to the wafer for measuring the generated SAW’s out-of-plane displacement component. The output voltage signal from the laser Doppler vibrometer is recorded by an oscilloscope (TDS 2014C, Tektronix). By using the laser Doppler vibrometer’s decoder information (10 nm/V), the measured voltages are converted to out-of-plane displacements of SAWs. To acquire a time-space wavefield of generated SAWs, a point-by-point scanning approach is performed, by gradually changing the laser position and acquiring displacement waveforms. At each scanning position, the measurement is repeated eight times to acquire eight waveforms, which are averaged to obtain a denoised waveform. By combining the denoised waveforms acquired at all scanning positions within the desired SAW measurement region, a time-space wavefield u(t, x) containing the information of the SAW propagation can be obtained. By further applying Fourier transform to the time-space wavefield, a frequency-space wavefield U(f, x) is obtained. The SAW intensity field I(f, x) at a selected frequency fi is evaluated using |U(f, x)|2. The cumulative SAW energy field is evaluated with the relation E(x) = ∫ |H(t, x)|2 dt, where H(t, x) is obtained through the Hilbert transform of the wavefield u(t, x). To analyze the wavenumber content of the generated SAWs, the 2D Fourier transform is applied to the frequency-space wavefield U(f, x), yielding the frequency-wavenumber spectrum S(f, k). The spectrum at a selected frequency fi is then extracted and plotted in the kx-ky wavenumber space, as illustrated in Supplementary Fig. 4. From the resulting wavenumber-domain spectrum, the key wavenumber components of the generated SAWs can be analyzed.

Procedures of acoustofluidic experiments

To validate the ability of our PIMs for arranging micro/nano-objects and generating fluid streaming, multiple acoustofluidic experiments were performed. As shown in Supplementary Fig. 12, a continuous sine signal at a frequency matching the PIM’s resonance frequency was generated by an arbitrary function generator (FG3052C, Tektronix), amplified by a power amplifier (A075, E&I), and then sent to the PIM for generating SAWs. A small microfluidic chamber was attached to the LiNbO3 substrate and loaded with deionized water containing micro/nano-objects, such as 35 μm SiO2 microparticles, MWCNTs, and 550 nm SiO2 nanoparticles. The SAW device with the microfluidic chamber is placed on the translation stage of an inverted microscope (TE2000-U, Nikon) to observe the object manipulation process. A camera is used to capture microscopic images and record movies during and after the SAW-based object manipulation. To visualize the fluid streaming, the acquired time-sequential images were stacked together.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This work demonstrates a phased interdigital electrode for controlling surface acoustic waves (SAW). The authors present\nvarious configurations for different SAW fields (tilted, pseudo-standing, long jet, twin jets), diode-like transmission, and\nacoustofluidic applications. Although the underlying physics is well established in other communities, the presented work\nmaximizes the synergy between the well-known concept and the SAW device. This work is interesting, well-written, and has\nthe potential to inspire many innovative applications and new investigations. I recommend this work for publication.\nHowever, I have some comments and suggestions that I hope the authors will address:\n1. The introduction may benefit from further revision to include more relevant work. The authors mention the use of planar\nSAW IDT, but in the SAW community, curved IDTs are commonly used for focusing. Additionally, several papers have\nutilized planar SAW IDTs with supplementary metamaterials/metasurfaces to achieve a similar objective as this study.\nIncluding these references and discussing their relevance would strengthen the introduction.\n2. The authors have conducted impressive experimental work. However, they only present the measured acoustic fields. It\nwould be valuable to include images of the fabricated devices, particularly those with different IDT configurations.\nAdditionally, showing a photo of the diode-like SAW device and its circuit would enhance the clarity of the experimental\nsetup.\n3. In Figure 4, the SAW fields for consecutive time points are plotted using the same data scale. Adjusting the data range\ncould make the transformed SAW field more distinct and visually informative (especially, Fig.4b).\n4. For the acoustofluidic applications presented in the main text, it would be beneficial to compare the focused SAW energy\nbeam with other focused beams from the literature, highlighting differences in focusing ability or potential advantages and\ndisadvantages.\n5. While the demonstration of diode-like SAW behavior is intriguing, its practicality remains limited. Typically, SAW devices\nrequire diode-like characteristics for use in SAW filters. In the presented configuration, incident SAWs can reflect from the\nIDT positioned to the left of the lossy material, resulting in back-propagation towards the source. Additionally, the IDT does\nnot necessarily need to be an IDT; any type of sensor could potentially serve the same purpose. The authors may wish to\ndiscuss these aspects and consider alternative configurations.", "reviewer_2": "The manuscript presents a novel approach to design on-chip phased interdigital metamaterials (PIMs) able to achieve both\nwave transformation and generation of surface acoustic waves (SAWs). Overcoming the limitations of current SAW chips\nthat are only responsible for wave generation, this device encodes subwavelength phase profiles, a characteristic typical of\nmetamaterials.\nThe device introduced combines the functionalities of traditional plane wave transducers and phase-modulating\nmetamaterials on a single module and can thus contribute to both SAW chips and metamaterials.\nThe presented methodology is based on customised interdigital electrodes and has been shown to have applications in\nwave steering, jet generation, fluid and object manipulation, fluid streaming vortices and diode-like wave transmission.\nThe manuscript is generally well-written. Some of the results in the manuscript are demonstrated for the first time, such as\ntwin jets with PIMs and a new setup for diode-like transmission. The findings include simulations based on an analytical\nmodel introduced in this work, and experimental validation of the functionality of manufactured PIMs.\nBelow are some of our major concerns about this work\n• The authors introduce an analytical model for the simulation of SAWs generated by PIMs, which reduces the cost and\nmemory requirements of finite element method (FEM) simulations. Since the analytical method was formulated to substitute\nFEM simulations, a comparison of their differences in assumptions/ accuracy would be helpful.\n• The authors mention that: “When using a PIM (Fig. 3a) designed with focal points close to its electrodes, the measured\nSAW displacement and energy fields (Fig. 3e and 3i) clearly show a long SAW jet near the electrodes.” It would be\nbeneficial to show a comparison between the locations of the intended focal points and the location of the observed jet as\nwell as the location of the electrodes. This will show how close to the electrodes the focal points can be generated and it will\ngive us an idea of the resolution/ accuracy of the reconstructions.\n• In Supplementary Figure 3, the target phase distributions along a line near the electrode are provided. To our\nunderstanding these distributions assume uniform acoustic amplitude along the same line. How is the amplitude of the PIM’s\noutput affected by the electrodes’ complex shapes and what is the effect on the target field reconstruction?\n• Acoustic jets have been defined in page 8 as acoustic beams “maintaining a subwavelength-scale width, which is defined\nby the distance between the beam’s full width at half maximum (FWHM)”. The authors need to comment on the widths of the\nexperimentally acquired jets shown in Fig. 3. It seems that the width of the jets depends on their orientation/position.\nMInor points\n• In page 4, line 9, the unit “rad” is missing after “2π”.\n• For increased clarity, what is being represented in Fig. 1b in red and blue could be specified explicitly. We infer that Fig. 1b\ni depicts the wavefronts of the wave and in Fig. 1b ii-iv a different quantity, which could be intensity, is shown.\n• Can the authors include a photograph of the manufactured PIM?", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts.", "reviewer_4": "In this manuscript, the authors presented on-chip phased interdigital metamaterials (PIMs) and used them for versatile\ntransformation of surface acoustic waves (SAWs), “diode-like” delivery of SAW information, and multi-mode manipulation of\nfluids and micro/nano-objects. Different from traditional SAW devices, they introduced and encoded deep-subwavelength-\nresolution phase profiles into the geometries of interdigital electrodes and combined multiple gain and loss components,\nto achieve multiple unique wave and matter manipulation functions. In addition, they developed a generalized analytical model\nfor quickly simulating SAWs generated by complex-shaped interdigital electrodes. In the reviewer’s opinion, the study in this\nmanuscript is sufficiently innovative and offers a new generalizable scheme and an analytical model to design SAW\ndevices, which can be adopted for generating customized SAW fields, controlling SAW propagation, achieving different\nSAW-based particle manipulation functions, and controlling acoustofluidic streaming fields.\nTo demonstrate the functions of their SAW devices and design approach, the authors performed sufficient amounts of\nexperiments and simulations. First, they designed, simulated, and tested six different PIMs to generate customized SAW\nfields. The SAW fields experimentally acquired by a laser Doppler vibrometer agree with the results of their developed\nmodel. This validates their analytical model and design scheme. Second, they designed and tested four different integrated\nPIMs for transforming incident SAWs into waves with different wavefronts and energy patterns, such as SAWs propagating\nin a specified direction, an SAW jet with energy confined in a wavelength, and twin jets. They also successfully\ndemonstrated “diode-like” SAW transmission, by routing the information carried by SAWs along a forward pathway while\nblocking backward propagation SAWs. Third, they performed serval experiments to demonstrate the manipulation of\nmicroparticles, carbon nanotubes, and fluid streaming. Especially, their fluid streaming fields are quite interesting, as few\nstudies observed on-chip SAW-generated streaming fields with periodically distributed vortices.\nThis manuscript presents multiple ideas (such as phase-encoded SAW devices, SAW transformation and “diode-like”\ntransmission, and analytical models for SAWs generated from complex interdigital electrodes) that are sufficiently innovative\nand the manuscript presents good amounts of experiment and simulation data with good quality. Therefore, I suggest the\npublication of this manuscript after addressing the following comments.\n1. In the Abstract, there is a sentence: “by encoding deep-subwavelength-resolution phase profiles into their interdigital\nelectrodes encoded,”. The authors used “encoding” and “encoded”. Should the “encoded” be deleted?\n2. Figure 2 shows multiple analysis results in the kx-ky wavenumber domain. The authors are suggested to provide a\ndetailed procedure for obtaining those ky-ky characterization results using the generated wave fields in the x-y space.\n3. In this manuscript, most SAW devices operate at a frequency of 18.8 MHz. Can the presented method be used for\ndesigning SAW devices at other frequencies? Are there any limitations in the proposed design method?\n4. Do the substrate properties such as material anisotropy and electromechanical coupling affect the performance of the\ndesigned SAW device with complex electrodes?\n5. In Fig 2 and Fig S3, all the designed SAW devices are facing upward. Is it possible to have more complex designs with\ndevices facing in the x and y directions? This combination may allow for generating more complex SAW fields.\n6. To transform incident waves into complex SAW fields, the authors’ method uses a customized amplification circuit. The\nauthors should provide more details of the circuit design. Are there any frequency and input voltage limitations of the\namplification circuit? A voltage bias is applied to the amplification circuit. What is the allowable voltage range? How to\ndetermine the voltage for achieving the desired SAW transformation ability?\n7. In the particle manipulation section, the authors showed carbon nanotubes (CNTs) can be arranged by their SAW\ndevices. What is the mechanism of arranging CNTs using your designed SAW devices? Is it possible to predict the\nconstructed CNT patterns?"}, "author_response": "Itemized list of response to reviewers’ remarks\n(Black italic: Editor’s remarks; Blue type: Our response; Additions/modifications to the manuscript\nand Supplementary Materials are highlighted in yellow)\nReviewer: #1\nComment 1: This work demonstrates a phased interdigital electrode for controlling surface acoustic\nwaves (SAW). The authors present various configurations for different SAW fields (tilted, pseudo-\nstanding, long jet, twin jets), diode-like transmission, and acoustofluidic applications. Although the\nunderlying physics is well established in other communities, the presented work maximizes the synergy\nbetween the well-known concept and the SAW device. This work is interesting, well-written, and has\nthe potential to inspire many innovative applications and new investigations. I recommend this work\nfor publication. However, I have some comments and suggestions that I hope the authors will address.\n1. The introduction may benefit from further revision to include more relevant work. The authors\nmention the use of planar SAW IDT, but in the SAW community, curved IDTs are commonly used for\nfocusing. Additionally, several papers have utilized planar SAW IDTs with supplementary\nmetamaterials/metasurfaces to achieve a similar objective as this study. Including these references and\ndiscussing their relevance would strengthen the introduction.\nOur response: We thank the reviewer for the positive comments regarding our work. A point-by-point\nresponse to your comments is provided below.\nWe have revised the Introduction section to include more studies related to IDTs with various\ndesigns for different functionalities and metasurfaces/metamaterials for SAW manipulation. Moreover,\nwe discussed their relevance to our study.\nChanges to the manuscript: Below is the revised and newly added content of the Introduction section.\n “Interdigital transducers (IDTs) with parallel straight electrodes and concentric arc-shaped\nelectrodes have long been predominantly used for generating straight-wavefront and focused SAWs\n23-26, 46, 47, respectively. Beyond these traditional designs, IDTs with various unique electrode\ngeometries have been developed to achieve different functionalities — for example, focused IDTs\nwith electrode geometries derived from slowness curves for improved SAW focusing 48-51, spiral-\nshaped IDTs for generating acoustic vortices 19, 42, 52, and angularly slanted IDTs capable of steering\nSAW direction via frequency tuning 53, 54.”\n “Although some studies have explored mechanisms that use components (e.g., thin film transistors,\nmagnetoelastic coupling layers, etc.) placed in the SAW path to modulate transmitted SAWs 55-65,\nPage 1 of 27\nintegrating these components into large arrays for subwavelength-resolution SAW field\ntransformation remains challenging due to the difficulty of scaling down the complex intermediate\nmodulators to subwavelength dimensions and/or fabricating arrays that incorporate many of them.\nRecently, Xu et al. introduced sawtooth-shaped metasurfaces bonded to a substrate for SAW\ndeflection 66, and Wang et al. demonstrated both SAW focusing and deflection by depositing\nmetasurfaces composed of pillar arrays with varied designs onto a substrate 67. However, these\napproaches require complex fabrication processes due to the addition of extra microstructures, and\ntheir functionalities remain limited to SAW deflection and/or focusing.”\n “These PIM offers a unique approach for developing SAW chips that can steer incident SAWs and\ntransform them into waves with diverse wavefronts and energy patterns—without relying on wave\ninteractions with complex-shaped microstructures (such as the micropillar arrays and sawtooth\nmicrostructures used in recently developed metasurfaces 66, 67), which require intricate\nmicrofabrication processes to be integrated onto the substrate. Compared to those metasurfaces\ndesigned for SAW deflection and/or focusing 66, 67, our PIM-based chip can achieve more complex\nSAW field transformations and enable “diode-like” routing of information carried by SAWs.”\nReferences cited\n19. Baudoin M, Gerbedoen JC, Riaud A, Matar OB, Smagin N, Thomas JL. Folding a focalized acoustical\nvortex on a flat holographic transducer: Miniaturized selective acoustical tweezers. Science Advances\n5, eaav1967 (2019).\n23. Friend J, Yeo LY. Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics.\nReviews of Modern Physics 83, 647-704 (2011).\n24. Baudoin M, Thomas J-L. Acoustic tweezers for particle and fluid micromanipulation. Annual Review of\nFluid Mechanics 52, 205-234 (2020).\n25. Meng L, Cai F, Li F, Zhou W, Niu L, Zheng H. Acoustic tweezers. Journal of Physics D: Applied\nPhysics 52, 273001 (2019).\n26. Fu YQ, Luo JK, Nguyen NT, Walton AJ, Flewitt AJ, Zu XT, Li Y, McHale G, Matthews A, Iborra E,\nDu H, Milne WI. Advances in piezoelectric thin films for acoustic biosensors, acoustofluidics and lab-\non-chip applications. Progress in Materials Science 89, 31-91 (2017).\n46. Tan MK, Friend JR, Yeo LY. Interfacial Jetting Phenomena Induced by Focused Surface Vibrations.\nPhysical Review Letters 103, 024501 (2009).\n47. Li P, Ma Z, Zhou Y, Collins DJ, Wang Z, Ai Y. Detachable Acoustophoretic System for Fluorescence-\nActivated Sorting at the Single-Droplet Level. Analytical Chemistry 91, 9970-9977 (2019).\n48. Lanoy M, Pierrat R, Lemoult F, Fink M, Leroy V, Tourin A. Subwavelength focusing in bubbly media\nusing broadband time reversal. Physical Review B 91, 6 (2015).\n49. Li P-Q, Zhou W, Peng B, Zhang C, Zhu X-F, Meng L, Zheng H. Holographic surface-acoustic-wave\ntweezers for functional manipulation of solid or liquid objects. Physical Review Applied 20, 064003\n(2023).\n50. Tian Z, Wang Z, Zhang P, Mai J, Wu Y, Yang S, Gu Y, Bachman H, Liang Y, Yu Z, Huang TJ.\nGenerating multifunctional acoustic tweezers in petri dishes for contactless, precise manipulation of\nbioparticles. Science Advances 6, eabb0494 (2020).\nPage 2 of 27\n51. O'Rorke R, Winkler A, Collins D, Ai Y. Slowness curve surface acoustic wave transducers for optimized\nacoustic streaming. RSC Advances 10, 11582-11589 (2020).\n52. Zhang N, Zuniga-Hertz JP, Zhang EY, Gopesh T, Fannon MJ, Wang J, Wen Y, Patel HH, Friend J.\nMicroliter ultrafast centrifuge platform for size-based particle and cell separation and extraction using\nnovel omnidirectional spiral surface acoustic waves. Lab on a Chip 21, 904-915 (2021).\n53. Gu Y, Chen C, Mao Z, Bachman H, Becker R, Rufo J, Wang Z, Zhang P, Mai J, Yang S, Zhang J, Zhao\nS, Ouyang Y, Wong DTW, Sadovsky Y, Huang TJ. Acoustofluidic centrifuge for nanoparticle\nenrichment and separation. Science Advances 7, eabc0467 (2021).\n54. Kang P, Tian Z, Yang S, Yu W, Zhu H, Bachman H, Zhao S, Zhang P, Wang Z, Zhong R, Huang TJ.\nAcoustic tweezers based on circular, slanted-finger interdigital transducers for dynamic manipulation of\nmicro-objects. Lab on a Chip 20, 987-994 (2020).\n55. Cha J, Kim KW, Daraio C. Experimental realization of on-chip topological nanoelectromechanical\nmetamaterials. Nature 564, 229-233 (2018).\n56. Zhang Z-D, Yu S-Y, Ge H, Wang J-Q, Wang H-F, Liu K-F, Wu T, He C, Lu M-H, Chen Y-F.\nTopological surface acoustic waves. Physical Review Applied 16, 044008 (2021).\n57. Dehghannasiri R, Eftekhar AA, Adibi A. Hypersonic surface phononic bandgap demonstration in a\nCMOS-compatible pillar-based piezoelectric structure on silicon. Physical Review Applied 10, 064019\n(2018).\n58. Ash BJ, Worsfold SR, Vukusic P, Nash GR. A highly attenuating and frequency tailorable annular hole\nphononic crystal for surface acoustic waves. Nature Communications 8, 174 (2017).\n59. Xu M, Yamamoto K, Puebla J, Baumgaertl K, Rana B, Miura K, Takahashi H, Grundler D, Maekawa\nS, Otani Y. Nonreciprocal surface acoustic wave propagation via magneto-rotation coupling. Science\nAdvances 6, eabb1724 (2020).\n60. Shao L, Mao W, Maity S, Sinclair N, Hu Y, Yang L, Lončar M. Non-reciprocal transmission of\nmicrowave acoustic waves in nonlinear parity–time symmetric resonators. Nature Electronics 3, 267-\n272 (2020).\n61. Shen C, Lu S, Tian Z, Yang S, Cardenas JA, Li J, Peng X, Huang TJ, Franklin AD, Cummer SA.\nElectrically tunable surface acoustic wave propagation at MHz frequencies based on carbon nanotube\nthin-film transistors. Advanced Functional Materials 31, 2010744 (2021).\n62. Yan M, Lu JY, Li F, Deng WY, Huang XQ, Ma JH, Liu ZY. On-chip valley topological materials for\nelastic wave manipulation. Nature Materials 17, 993-998 (2018).\n63. Bandhu L, Nash GR. Controlling the properties of surface acoustic waves using graphene. Nano\nResearch 9, 685-691 (2016).\n64. Wang J-Q, Zhang Z-D, Yu S-Y, Ge H, Liu K-F, Wu T, Sun X-C, Liu L, Chen H-Y, He C, Lu M-H,\nChen Y-F. Extended topological valley-locked surface acoustic waves. Nature Communications 13,\n1324 (2022).\n65. Shao L, Zhu D, Colangelo M, Lee D, Sinclair N, Hu Y, Rakich PT, Lai K, Berggren KK, Lončar M.\nElectrical control of surface acoustic waves. Nature Electronics 5, 348-355 (2022).\n66. Xu M, Lee PVS, Collins DJ. Microfluidic acoustic sawtooth metasurfaces for patterning and separation\nusing traveling surface acoustic waves. Lab on a Chip 22, 90-99 (2022).\n67. Wang W, Baranski M, Jin Y, Salut R, Belharet D, Friedt J-M, Pan Y, Xiang Y, Xuan F-z, Khelif A,\nBenchabane S. Experimental Realization of On-Chip Surface Acoustic Wave Metasurfaces at Sub-GHz.\nAdvanced Science 12, 2411825 (2025).\nPage 3 of 27\nComment 2: The authors have conducted impressive experimental work. However, they only present\nthe measured acoustic fields. It would be valuable to include images of the fabricated devices,\nparticularly those with different IDT configurations. Additionally, showing a photo of the diode-like\nSAW device and its circuit would enhance the clarity of the experimental setup.\nOur response: We thank the reviewer for this comment. The fabricated devices and their designs are\nshown in Fig. R1. A photo of our fabricated SAW device with its amplification circuit for “diode-like”\ntransmission is provided in Fig. R2a.\nFig. R1. (Top row) Electrode designs. (Bottom row) Photos of fabricated devices. This figure panel\nincludes the designs of (a) a traditional straight electrode IDT, (b) a PIM for generating SAWs\npropagating in the 80 deg direction, (c) a PIM for generating pseudo-standing SAWs, (d) a PIM for\ngenerating a SAW jet near its electrodes, (e) a PIM for generating a SAW jet far from its electrodes, (f)\na PIM for generating an off-center SAW jet, and (g) a PIM for generating twin jets.\nFig R2. (a) Photo and (b) schematic of a SAW device with its amplification circuit. The amplification\ncircuit includes a 220 Ω resistor (R), four 400 nF capacitors (C , C , C , and C ), and a BGA 2869 chip.\n1 2 3 4\nThe resistor R and capacitors C and C are for providing a stable direct current (DC) voltage bias to\n1 2\nthe BGA 2869 chip from the voltage port V . The capacitors C and C serve as alternating current\nDC 3 4\n(AC) coupling elements, allowing for the transmission of AC signals while blocking DC components.\nChanges to the manuscript: Per the reviewer’s comment, photos of our fabricated PIM chips were\nadded to Supplementary Figure 3. A photo and a schematic of the device for ‘diode-like’ transmission\nof SAWs were added to Supplementary Figure 18.\nPage 4 of 27\nComment 3: In Figure 4, the SAW fields for consecutive time points are plotted using the same data\nscale. Adjusting the data range could make the transformed SAW field more distinct and visually\ninformative (especially, Fig.4b).\nOur response and changes to the manuscript: We thank the reviewer for this comment. We have\nupdated the colormap range of Figure 4b to better see the transformed SAW field.\nComment 4: For the acoustofluidic applications presented in the main text, it would be beneficial to\ncompare the focused SAW energy beam with other focused beams from the literature, highlighting\ndifferences in focusing ability or potential advantages and disadvantages.\nOur response: We thank the reviewer for this comment. We conducted simulations using our\nanalytical model to compare the SAW intensity fields generated by traditional focused IDTs with\nconcentric arc-shaped interdigital electrodes and our PIM device. The devices used for comparison are\nlisted below:\n(a) a traditional focused IDT with the number of electrodes N= 10, arc angle α = 80°, focal length of\nd\nthe central electrode d = 2 mm, and device span 0.9W= 3.15 mm;\nc\n(b) a traditional focused IDT with the number of electrodes N= 10, arc angle α = 90°, focal length of\nd\nthe central electrode d = 2 mm, and device span W= 3.50 mm;\nc\n(b) our proposed PIM with the number of electrodes N=10, focal length of the central electrode d =\nc\n2 mm, and device span W= 3.50 mm.\nFig. R3a-c show the simulated SAW intensity fields of these three cases, respectively. Fig. R3d\ncompares the cross-sectional intensity profiles of the generated SAW beams. Fig. R3e compares the\nintensity profiles along the centerlines of these beams. As shown in Fig. R3, when the traditional\nfocused IDT and our PIM have the same span (W= 3.50 mm), number of electrodes (N= 10), and center\nelectrode focal length (d = 2 mm), our PIM generates a significantly longer focused SAW beam, while\nc\nthe beam width is slightly larger. This difference arises because the interdigital electrodes of our PIM\nare designed with focal points distributed along the y-axis, whereas the electrodes in the traditional\nfocused IDT are designed to focus at the same point.\nChanges to the manuscript: Per the reviewer’s comment, we made the following changes.\n In the Supplementary Materials, we added Supplementary Figure 8: comparison of SAW beams\ngenerated by traditional focused IDTs and our designed PIM.\nPage 5 of 27\n In the Result section of the manuscript, we add the following content: “Furthermore, compared to\na traditional focused IDT with concentric arc-shaped electrodes, a PIM with the same span (W),\nnumber of electrodes (N), and center electrode focal length (d ) can generate a significantly longer\nc\nfocused SAW beam, although its width slightly increases, as shown in Supplementary Figure 8.”\nFig. R3. Comparison of SAW beams generated by traditional focused IDTs and our designed PIM.\n(a) Simulated SAW intensity field for a traditional focused IDT with the number of electrodes N= 10,\narc angle α = 80°, focal length of the central electrode d = 2 mm, and device span 0.9W= 3.15 mm.\nd c\n(b) Simulated SAW intensity field for a traditional focused IDT with the number of electrodes N= 10,\narc angle α = 90°, focal length of the central electrode d = 2 mm, and device span W= 3.50 mm.\nd c\n(c) Simulated SAW intensity field for our PIM with the number of electrodes N=10, focal length of the\ncentral electrode d = 2 mm, and device span W= 3.50 mm. (d) Cross-sectional profiles of the generated\nc\nSAW beams. (e) Intensity profiles along the centerlines of the SAW beams.\nComment 5: While the demonstration of diode-like SAW behavior is intriguing, its practicality remains\nlimited. Typically, SAW devices require diode-like characteristics for use in SAW filters. In the\npresented configuration, incident SAWs can reflect from the IDT positioned to the left of the lossy\nmaterial, resulting in back-propagation towards the source. Additionally, the IDT does not necessarily\nneed to be an IDT; any type of sensor could potentially serve the same purpose. The authors may wish\nto discuss these aspects and consider alternative configurations.\nPage 6 of 27\nOur response: We thank the reviewer for this comment. Although the IDT positioned to the left of the\nlossy material reflects SAWs, we found that the reflected energy is only a small portion (~9.3%) of the\nincident energy, according to the experimental results in Figure 4. We also found that this small amount\nof reflected energy doesn’t substantially affect the transmitted SAW field or the function of our device.\nAdditionally, using an IDT as the sensing unit offers the advantage of frequency selectivity, as the IDT\ninherently functions as a frequency filter. For example, when the incident SAWs contain multiple\nfrequency components, the use of a sensing IDT ensures selective transmission at the desired frequency.\nWe appreciate the reviewer’s suggestion to explore alternative sensor configurations, which could\npotentially reduce SAW reflection. We think this suggestion offers a promising direction for our future\nwork, particularly in designing multi-frequency or broadband devices. We have updated the subsection\n“Transformation and diode-like transmission of on-chip SAWs” to discuss these aspects.\nChanges to the manuscript: Per the reviewer’s comment, we revised the subsection “Transformation\nand diode-like transmission of on-chip SAWs” by adding the following content: “Additionally, since\nthe chip uses an IDT positioned to the left of the lossy material for SAW sensing, a small portion of\nthe incident SAW energy (~9.3%, according to experimental results) is reflected. The IDT’s inherent\nfiltering capability allows for frequency-selective SAW transmission. By replacing the sensing IDT\nwith differently shaped sensing electrodes, it may be possible to reduce SAW reflections and enable\nmulti-frequency or broadband SAW transmission — an interesting direction for future investigation.”\nPage 7 of 27\nReviewer: #2\nComment 1: The manuscript presents a novel approach to design on-chip phased interdigital\nmetamaterials (PIMs) able to achieve both wave transformation and generation of surface acoustic\nwaves (SAWs). Overcoming the limitations of current SAW chips that are only responsible for wave\ngeneration, this device encodes subwavelength phase profiles, a characteristic typical of\nmetamaterials. The device introduced combines the functionalities of traditional plane wave\ntransducers and phase-modulating metamaterials on a single module and can thus contribute to both\nSAW chips and metamaterials.\nThe presented methodology is based on customised interdigital electrodes and has been shown to\nhave applications in wave steering, jet generation, fluid and object manipulation, fluid streaming\nvortices and diode-like wave transmission.\nThe manuscript is generally well-written. Some of the results in the manuscript are demonstrated\nfor the first time, such as twin jets with PIMs and a new setup for diode-like transmission. The findings\ninclude simulations based on an analytical model introduced in this work, and experimental validation\nof the functionality of manufactured PIMs. Below are some of our major concerns about this work\n1. The authors introduce an analytical model for the simulation of SAWs generated by PIMs,\nwhich reduces the cost and memory requirements of finite element method (FEM) simulations. Since\nthe analytical method was formulated to substitute FEM simulations, a comparison of their differences\nin assumptions/ accuracy would be helpful.\nOur response: Our response: We thank the reviewer for the positive comments regarding our work.\nAll the comments have been addressed. A point-by-point response to your comments is given below.\nIn the finite element (FE) method, the governing partial differential equations (PDEs) are\ntransformed into weak formulations (integral representations of the governing PDEs), and the analysis\ndomain is discretized into a collection of finite elements. The FE method then approximates the\nsolutions to the PDEs over these finite elements in the analysis domain. This approach has inherent\nerrors induced by the space discretization of the analysis domain. To achieve acceptable simulation\nresults for wave propagation problems, the element size typically needs to be ≤λ/8, which results in a\nlarge number of elements and requires significant computational resources, such as computation time\nand memory.\nOur analytical simulation approach is based on the assumption that the SAW field generated by a\ntransducer can be approximated by the linear superposition of SAW fields generated by numerous\ninfinitesimal elements that cover the electrode region of the transducer. Although the finite element\nPage 8 of 27\nmethod is a more generalized approach capable of addressing diverse needs and capturing more\ndetailed features (e.g., geometry, material properties, and boundary conditions) of the SAW chip, our\nanalytical approach is more efficient for its intended purpose: predicting SAW fields generated by\nPIMs with complex-shaped electrodes.\nTo compare our analytical approach with the FE method, we conducted simulations for two cases:\n (Case 1) a SAW chip with a traditional straight-electrode interdigital transducer (IDT) that has a\nresonance frequency of 18.8 MHz and N=10 parallel electrodes on a LiNbO substrate (see Fig. R4a);\n3\n (Case 2) a SAW chip with our designed PIM on a LiNbO substrate for generating a focused SAW\n3\nbeam at 18.8 MHz (see Fig. R5a).\nFor each case, FE simulations were performed in COMSOL Multiphysics using models meshed with\nhexahedral elements, which have different in-plane mesh sizes: λ/ , λ/ , and λ/ (see Fig. R4b- 4d for\n2 4 8\nCase 1 and Fig. R5b-5d for Case 2). In the thickness direction, all the models use the same mesh with\na gradient size distribution. In each simulation model, the electrodes are in domain Ω for generating\n2\nreflections. A low-reflection boundary condition is applied to the chip’s bottom surface. All the\nsimulations were run on the same desktop (CPU: Intel Core i7-11700; RAM: 64GB DDR4). As our\ndesktop cannot handle large models, the piezoelectric substrate dimensions were limited to 6×4×0.2mm.\nBy comparing the FE simulation results (Fig. R4e- 4g for Case 1 and Fig. R5e-5g for Case 2),\nwe\nfound that the results become smoother as the mesh size reduces from λ/ to λ/ . However, the simulation\n2 8\nruntime increases from 4 to 39 minutes in Case 1 and from 7 to 101 minutes in Case 2. In contrast, our\nanalytical method requires only 6.4 sec for Case 1 and 7.2 sec for Case 2 to complete simulations with\na spatial resolution of λ/ . Moreover, the analytical SAW fields (Fig. R4i for Case 1 and Fig. R5i for\nFE models with a mesh size of λ/ . Furthermore, we compared FE and analytical results along the y-\n8\naxis (see Fig. R4h for Case 1 and Fig. R5h for Case 2). The comparisons show that the displacement\ndistribution from the analytical method is close to that of the FE model with a mesh size of λ/ .\ncontrast, the results of FE models with coarse meshes fail to capture sinusoidal-like oscillations, resolve\npeak and valley positions, or produce the correct wavelength.\nPage 9 of 27\nFig. R4. FE and analytical simulations of SAWs generated by a traditional straight-electrode IDT.\n(a) Schematic of a SAW chip for simulations. The piezoelectric substrate is divided into domains Ω\n1\nand Ω , where Ω is a perfectly matched layer for reducing end reflections. (b-d) Meshed FE models\n2 2\nwith mesh sizes of λ/2, λ/4, and λ/8 in domain Ω , respectively (top row: entire model; mid row:\n1\nzoomed-in view of the model in the purple box; bottom row: zoomed-in view of the model in the red\nbox). (e-g) Simulated wavefields (out-of-plane displacement u ) when mesh sizes are λ/2, λ/4, and λ/8,\nz\nrespectively (runtimes: 4, 15, and 39 min, respectively). (h) Comparison of displacement u\nz\ndistributions along a green dotted line illustrated in Figure (a). (i) Simulated SAW wavefield of our\nanalytical approach (runtime: 6.4 sec).\nPage 10 of 27\nFig. R5. FE and analytical simulations of SAWs generated by a PIM device. (a) Schematic of a\nSAW chip for simulations. The PIM electrodes are designed with a focal length of d = 2 mm and a\nc\nspan of W= 3.50 mm to generate a SAW jet. (b-d) Meshed FE models with domain-Ω mesh sizes of\n1\nλ/2, λ/4, and λ/8, respectively (top row: complete model; mid row: zoomed-in view of the model in the\npurple box; bottom row: zoomed-in view of the model in the red box). Domain Ω is a perfectly matched\n2\nlayer. (e-g) Simulated wavefields (out-of-plane displacement u ) when mesh sizes are λ/2, λ/4, and λ/8,\nz\nrespectively (runtimes: 7, 47, and 101 min, respectively). (h) Comparison of displacement u\nz\ndistributions along a green dotted line illustrated in Figure (a). (i) Simulated SAW wavefield of our\nanalytical approach (runtime: 7.2 sec).\nPage 11 of 27\nChanges to the manuscript: Per the reviewer’s comment, we made the following changes.\n In the Supplementary Materials, we added Supplementary Figures 16 and 17 to compare FE and\nanalytical simulation methods.\n In Supplementary Note 1, we added the following assumption: “This model is based on the\nassumption that the SAW field generated from a transducer can be approximated as the linear\nsuperposition of SAW fields produced by numerous infinitesimal elements distributed across the\nelectrode region of the transducer.”\n In the Discussion section of the manuscript, we added the following content: “Compared to the\nmultiphysics finite element method, our analytical approach significantly reduces computation\ntime while producing SAW fields that closely match those from finite element models with a fine\nmesh size of λ/8 (see Supplementary Note 3 and Figures 16 and 17). Although the finite element\nmethod is a more generalized approach capable of addressing diverse wave propagation problems\nand capturing more detailed features (e.g., geometry, material properties, and boundary\nconditions) of the SAW chip, our analytical approach is more efficient for its intended purpose:\npredicting SAW fields generated by PIMs with complex-shaped electrodes.”\nComment 2: The authors mention that: “When using a PIM (Fig. 3a) designed with focal points close\nto its electrodes, the measured SAW displacement and energy fields (Fig. 3e and 3i) clearly show a\nlong SAW jet near the electrodes.” It would be beneficial to show a comparison between the locations\nof the intended focal points and the location of the observed jet as well as the location of the electrodes.\nThis will show how close to the electrodes the focal points can be generated and it will give us an idea\nof the resolution/ accuracy of the reconstructions.\nOur response: We thank the reviewer for this comment. In our designed devices (see Fig. R6), the nth\nelectrode is designed to have an intended focal point located at x (n), and the entire device with N\nF\nelectrodes has a series of intended focal points from x (1) to x (N) located along a line. In order to\nF F\nbetter compare the locations of intended focal points and the position of the generated SAW beam,\nwe\nplotted the generated SAW intensity field, the electrode configuration, and the intended focal points in\nthe same coordinate system.\nAs shown in Fig. R6a, the SAW beam generated by a device with N= 10 (see Fig. R6a, bottom)\nspans from the 1st focal point at x (1) to the last focal point x (N). After changing the electrode\nF F\nconfiguration to the design in Fig. R6b (bottom), although the position of the generated SAW beam\nchanges, it still spans along the intended focal points. Fig. R6c and R6d further compare the location\nPage 12 of 27\nof a SAW beam generated by a device with N= 30 (i.e., the SAW beam in the manuscript’s Fig. 3i) to\nthe device’s intended focal point locations, showing good agreement. These comparisons indicate that\nthe location and length of the SAW beam can be predicted using the intended focal point locations.\nFig. R6. Comparison of the designed focal point locations with the location of the generated SAW\nbeam. (a) Simulated SAW intensity field for a device with the number of electrodes N=10, focal length\nd = 3.5 mm, and span W= 3.5 mm. (b) Simulated SAW intensity field for a device with the number of\nc\nelectrodes N=10, focal length d = 2 mm, and span W= 3.5 mm. (c) Simulated SAW intensity field (i.e.,\nc\nresult in Figure 3i of the manuscript) for a device with the number of electrodes N=30, focal length d =\nc\n2 mm, and span W= 3.5 mm. (d) Experimentally acquired SAW intensity field (i.e., result in Figure 3i\nof the manuscript) for a device with the number of electrodes N=30, focal length d = 2 mm, and span\nc\nW= 3.5 mm. The green dots represent the intended focal points of different electrodes.\nChanges to the manuscript: Per the reviewer’s comment, we made the following changes.\n In the Supplementary Materials, we added Supplementary Figure 6: Comparison of the designed\nfocal point locations with the location of the generated SAW beam.\n In the Results section of the manuscript, we added the following content: “Using this approach, a\nlong SAW jet with an energy profile ranging from x (1) to x (N) can be generated, as confirmed\nF F\nby the alignment of the generated SAW jet with the focal points shown in Supplementary\nFigure 6.”\nComment 3: In Supplementary Figure 3, the target phase distributions along a line near the electrode\nare provided. To our understanding these distributions assume uniform acoustic amplitude along the\nPage 13 of 27\nsame line. How is the amplitude of the PIM’s output affected by the electrodes’ complex shapes and\nwhat is the effect on the target field reconstruction?\nOur response: We thank the reviewer for this comment. To investigate the effect of electrode shape\non the generated SAW field, we conducted simulations for the following two cases.\n Case 1: We simulated the SAW field generated by a phased array with two rows of elements\n(indicated by colored dots in Fig. R7a). Each row has 200 elements with the same spacing of 17.5μm\n(~ λ/12 at 18.8 MHz). The spacing between the two rows is half a wavelength (λ/2). The excitation phases for the\narray elements follow a position-dependent phase function ϕ (x), which is determined through the\nd\napproach in Supplementary Note 2 to achieve a focal length d = 2 mm. The excitation voltages of\nc\nthese elements are the same.\n Case 2: We simulated the SAW field generated by our PIM device with complex-shaped electrodes\n(illustrated in Fig. R7c), which are encoded with the position-dependent phase function ϕ (x) by\nd\nintroducing the subwavelength-scale y-position adjustment y(x)c ()/. The phase\nd SAW\nfunction is the same as the one used in Case 1.\nThe SAW intensity fields (Fig. R7b and 7d) of the two cases have similar distributions. For comparison,\nwe plotted their cross-sectional intensity profiles (see Fig. R7e) along multiple lines y = 315 μm\n1\n(~1.5λ), y = 636 mm (~3λ), y = 954 mm (~4.5λ), y = 1272 mm (~6λ), y = 1590 mm (~7.5λ), y = 1908\n2 3 4 5 6\nmm (~9λ), y = 2226 mm (~10.5λ), and y = 2544 mm (~12λ). We also compared the intensity profiles\n7 8\nalong their centerlines (see Fig. R7f). From these comparisons, it can be found that their intensity\ndistributions are close to each other when the distance to the device is greater than three wavelengths\n(3λ). Therefore, although our design uses complex-shaped electrodes, it produces a similar output as a\nphased array with elements along straight lines, especially in the far field.\nChanges to the manuscript: Per the reviewer’s comment, we made the following changes.\n In the Supplementary Materials, we added Supplementary Figure 15: Comparison of SAW fields\ngenerated by our PIM and a phased array.\n In the Discussion section of the manuscript, we added the following content: “Supplementary\nFigure 15 compares the SAW fields generated by our PIM and a phased array composed of\nnumerous elements along straight lines, with excitation phases satisfying the same phase function\nϕ (x). Notably, these SAW fields are nearly identical when the observation distance exceeds three\nd\nwavelengths from the source. This indicates that our PIM, with its complex-shaped electrodes, can\nproduce a SAW field nearly identical to that of the phased array, without requiring the complex\nPage 14 of 27\nmulti-channel excitation system needed to supply different signals to individual elements of the\nphased array.”\nFig. R7. Comparison of SAW fields generated by our PIM and a phased array. (a) Schematic of\nthe simulation setup and (b) simulated SAW intensity field at 18.8 MHz for a phased array with two\nrows of elements (indicated by colored dots). Each row has 200 elements with the same spacing of\n17.5μm (~ λ/12). The spacing between the two rows is half a wavelength (λ/2). The colors of the dots\nindicate different phases applied to individual array elements. (c) Schematic of the simulation setup\nand (d) simulated SAW intensity field at 18.8 MHz for a PIM device with the number of electrodes\nN=2, focal length d = 2 mm, and span W= 3.5 mm. (e, bottom to top) Comparison of intensity\nc\ndistributions along lines y = 315 μm (~1.5λ), y = 636 mm (~3λ), y = 954 mm (~4.5λ), y = 1272 mm\n1 2 3 4\n(~6λ), y = 1590 mm (~7.5λ), y = 1908 mm (~9λ), y = 2226 mm (~10.5λ), and y = 2544 mm (~12λ),\n5 6 7 8\nrespectively. (f) Comparison of intensity distributions along the y-axis.\nPage 15 of 27\nComment 4: Acoustic jets have been defined in page 8 as acoustic beams “maintaining a\nsubwavelength-scale width, which is defined by the distance between the beam’s full width at half\nmaximum (FWHM)”. The authors need to comment on the widths of the experimentally acquired jets\nshown in Fig. 3. It seems that the width of the jets depends on their orientation/position.\nOur response: We thank the reviewer for this comment. We agree with the reviewer that the width of\na jet beam is related to the orientation, position, and other characteristics of the beam. To better\nunderstand how the device design parameters affect the beam widths, we performed simulations using\ndevices with different design parameters (e.g., number of electrodes N, focal length d , device span W,\nc\nand beam center shift Δs). Fig. R8 shows simulated SAW beams and relations of beam widths versus\ndifferent design parameters. From these results, we made the following findings:\n If only increasing the focal length d , the beam width increases (see Fig. R8k). Moreover, the beam\nc\nwidths measured from simulation results agree with the prediction W = 1.02 λ d / W , which is\nb c c\ntypically used for predicting the beam width of a focused ultrasound transducer.\n If only increasing the device span W, the beam width decreases (see Fig. R8l). The beam widths\nmeasured from simulation results again agree with the prediction W = 1.02 λ d / W .\nb c c\n After splitting a single beam into twin beams, the width of each split beam is smaller than that of\nthe original beam (see Fig. R8l). However, with increasing device span, this effect gradually\ndiminishes.\n When shifting a beam from the device centerline to the edge, the beam width gradually increases\n(see Fig. R8m).\n If the number of electrodes increases, the beam width gradually increases at first and then reaches a\nplateau (see Fig. R8n).\n From Fig. R8k to 8n, it can also be seen that by properly choosing the design parameters, the beam\nwidth can be reduced to the subwavelength scale.\nWe believe the above findings from simulation results can be used to better design PIM devices to meet\ndifferent requirements.\nFig. R9 compares the beam widths measured from experimental and simulated SAW fields (i.e.,\nthe SAW fields in Fig. 3 of the manuscript) generated by PIM devices with a large number of electrodes\n(N=30). The beam widths measured from the experimental SAW fields are close to those obtained from\nthe simulation results, except for the case in Fig. 3k, which may be attributed to SAW field\nmeasurement and device fabrication errors. Additionally, if we reduce the number of electrodes down\nfrom N=30, the beam width can be reduced, as shown by the relation in Fig. R8n.\nPage 16 of 27\nFig. R8. Effects of PIM design parameters on SAW beam widths. (a - j) Simulated SAW intensity\nfields for devices with different design parameters (e.g., number of electrodes N, focal length d , device\nc\nspan W, and beam center shift Δs). The values of these parameters are provided at each figure’s top\nleft. The data in (a-f) are normalized using the same scale, whereas a different normalization is applied\nto (g-j). (k) Beam width versus focal length d . (l) Beam width versus span W. (m) Beam width versus\nc\nbeam center shift Δs. (n) Beam width versus number of electrodes N. The predictions in figures (k) and\n(l) are obtained using the relation W = 1.02 λ d / W.\nb c\nPage 17 of 27\nFig. R9. Beam widths measured from experimental and simulated SAW fields in Fig. 3 of the\nmanuscript. The used PIM devices all have a large number of electrodes (N=30). If reducing the number\nof electrodes, the beam width can be reduced, as shown in Fig. R8n.\nChanges to the manuscript: Per the reviewer’s comment, we made the following changes.\n In the Supplementary Materials, we added Supplementary Figure 8: Comparison of SAW beams\ngenerated by traditional focused IDTs and our designed PIM. We also added Supplementary\nFigure 9: Beam widths measured from SAW fields in Figure 3.\n In the Result section of the manuscript, we added the following content. “Moreover, SAW jets can\nexhibit different shapes by customizing PIM design parameters (e.g., the number of electrodes N,\ndevice span W, focal length d , and beam center shift Δs), as demonstrated by the simulation results\nc\nin Supplementary Figure 7, which also reveal how these parameters affect beam width. The results\nshow that the beam width (W ) gradually increases with increasing focal length and/or decreasing\nb\ndevice span. When the number of electrodes is small (e.g., N=2), this relationship aligns with the\nprediction W = 1.02 λ d / W , which is commonly used to estimate the beam width of focused\nb c c\nultrasound transducers77, 78. Second, shifting the jet away from the device centerline toward the\nedge (i.e., increasing Δs) results in a gradual increase in beam width. Third, increasing the number\nof electrodes initially causes the beam width to increase, after which it reaches a plateau.\nAdditionally, by appropriately selecting the design parameters, subwavelength-scale beam widths\ncan be achieved. These relationships observed from simulation results provide valuable guidance\nfor optimizing PIM designs.” “The beam widths measured from the experimental SAW fields are\nalso close to those obtained from the simulation results (see Supplementary Figure 9 for beam\nwidth comparisons), except for the case in Fig. 3k, which may be attributed to SAW field\nmeasurement and device fabrication errors.”\nPage 18 of 27\nComment 5: In page 4, line 9, the unit “rad” is missing after “2π”.\nOur response and changes to the manuscript: Thank you very much for pointing this out. The unit\n“rad” has been added to the manuscript.\nComment 6: For increased clarity, what is being represented in Fig. 1b in red and blue could be\nspecified explicitly. We infer that Fig. 1b i depicts the wavefronts of the wave and in Fig. 1b ii-iv a\ndifferent quantity, which could be intensity, is shown.\nOur response and changes to the manuscript: Thank you very much for pointing this out. We have\nupdated the colors used in Fig. 1 and the figure caption. In the updated figure, the yellow and blue\nparallel lines in (b, c) represent SAW wavefronts, while the SAW intensity patterns in (a, b) are\nillustrated in red.\nComment 7: Can the authors include a photograph of the manufactured PIM?\nOur response and changes to the manuscript: We thank the reviewer for this comment. The\nfabricated devices with their designs are given in Fig. R10. The device photos have been added to\nSupplementary Figure 3.\nFig. R10. (Top row) Electrode designs. (Bottom row) Photos of fabricated devices. This figure panel\nincludes the designs of (a) a traditional straight electrode IDT, (b) a PIM for generating SAWs\npropagating in the 80 deg direction, (c) a PIM for generating pseudo-standing SAWs, (d) a PIM for\ngenerating a SAW jet near its electrodes, (e) a PIM for generating a SAW jet far from its electrodes, (f)\na PIM for generating an off-center SAW jet, and (g) a PIM for generating twin jets.\nPage 19 of 27\nReviewer: #3\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of\nthe Nature Communications initiative to facilitate training in peer review and to provide appropriate\nrecognition for Early Career Researchers who co-review manuscripts.\nOur response: We are grateful for the valuable insights from both the primary reviewer and the Early\nCareer Researcher. We truly thank you for your comments.\nPage 20 of 27\nReviewer: #4\nComment 1: In this manuscript, the authors presented on-chip phased interdigital metamaterials\n(PIMs) and used them for versatile transformation of surface acoustic waves (SAWs), “diode-like”\ndelivery of SAW information, and multi-mode manipulation of fluids and micro/nano-objects.\nDifferent\nfrom traditional SAW devices, they introduced and encoded deep-subwavelength-resolution phase\nprofiles into the geometries of interdigital electrodes and combined multiple gain and loss components,\nto achieve multiple unique wave and matter manipulation functions. In addition, they developed a\ngeneralized analytical model for quickly simulating SAWs generated by complex-shaped interdigital\nelectrodes. In the reviewer’s opinion, the study in this manuscript is sufficiently innovative and offers\na new generalizable scheme and an analytical model to design SAW devices, which can be adopted for\ngenerating customized SAW fields, controlling SAW propagation, achieving different SAW-based\nparticle manipulation functions, and controlling acoustofluidic streaming fields.\nThis manuscript presents multiple ideas (such as phase-encoded SAW devices, SAW\ntransformation and “diode-like” transmission, and analytical models for SAWs generated from\ncomplex interdigital electrodes) that are sufficiently innovative and the manuscript presents good\namounts of experiment and simulation data with good quality. Therefore, I suggest the publication of\nthis manuscript after addressing the following comments.\n1. In the Abstract, there is a sentence: “by encoding deep-subwavelength-resolution phase profiles\ninto their interdigital electrodes encoded,”. The authors used “encoding” and “encoded”. Should the\n“encoded” be deleted?\nOur response and changes to the manuscript: We thank the reviewer for the positive comments\nregarding our work. All the comments have been addressed. A point-by-point response to your\ncomments is given below. Additionally, we revised the Abstract to avoid potential confusion.\nComment 2: Figure 2 shows multiple analysis results in the kx-ky wavenumber domain. The authors\nare suggested to provide a detailed procedure for obtaining those ky-ky characterization results using\nthe generated wave fields in the x-y space.\nOur response: We thank the reviewer for this comment. Fig. R11 illustrates the process to obtain the\nspectrum in the k -k domain using the acquired time-space wavefield u(t, x, y). This wavefield is the\nx y\ncombination of waveforms acquired at a series of laser scanning points. First, the one-dimensional\nFourier transform is applied to all the acquired waveforms, thereby changing the time-space wavefield\nu(t, x, y) to a frequency-space wavefield U(f, x, y). Second, the two-dimensional Fourier transform is\nPage 21 of 27\napplied to the frequency-space wavefield U(f, x, y) to obtain a frequency-wavenumber spectrum\ndenoted as S(f, k , k ).\nx y\nThe above two steps can also be performed in one step through the three-dimensional Fourier\ntransform to directly transform a time-space wavefield u(t, x, y) to a frequency-wavenumber spectrum\nS(f, k , k ). Then, the spectrum in the k -k wavenumber domain at a desired frequency can be selected\nx y x y\nfrom the transformation result S(f, k , k ). For example, the spectrum in Figure 2 of the manuscript was\nx y\nextracted from the transformation result at 18.8 MHz.\nChanges to the manuscript: Per the reviewer’s comment, we added the following content to the\n“Methods” section. “To analyze the wavenumber content of the generated SAWs, the 2D Fourier\ntransform is applied to the frequency-space wavefield U(f, x), yielding the frequency-wavenumber\nspectrum S(f, k). The spectrum at a selected frequency f is then extracted and plotted in the k -k\ni x y\nwavenumber space, as illustrated in Supplementary Figure 4. From the resulting wavenumber-domain\nspectrum, the key wavenumber components of the generated SAWs can be analyzed.”\nFig. R11. Illustration of procedures to obtain a frequency-wavenumber spectrum S(f, k) using the\nacquired time-space wavefield u(t, x).\nComment 3: In this manuscript, most SAW devices operate at a frequency of 18.8 MHz. Can the\npresented method be used for designing SAW devices at other frequencies? Are there any limitations\nin the proposed design method?\nOur response: We thank the reviewer for this comment. Although the devices demonstrated in the\nstudy are designed and fabricated to work at 18.8 MHz, our presented method for designing customized\ndevices with complex-shaped electrodes should be feasible for designing devices working at higher\nfrequencies. Although our current design approach has been demonstrated for generating different\nSAW beams, it still has the following limitations. First, the current designs are for generating SAWs\npropagating in one direction and have not considered the control of SAW propagation in multiple\ndirections. Second, the current designs are limited to generating SAWs at a selected frequency and not\nfor generating SAWs at multiple frequencies or in a wide frequency band. These limitations have been\nPage 22 of 27\nadded to the “Discussion” section. In our future work, we will conduct further investigation to fill these\ngaps.\nChanges to the manuscript: Per the reviewer’s comment, we added the following content to the\n“Discussion” section. “Despite the successful demonstrations described above, the PIMs presented in\nthis study still have several limitations, including the inability to control SAWs propagating in multiple\ndirections and the lack of control over SAWs at multiple frequencies or across a wide frequency band.”\n“Looking ahead, we plan to design PIMs with electrode geometries different from those presented in\nthis paper, aiming to generate other intriguing on-chip SAW patterns, enable wideband operation,\ncontrol SAWs propagating in multiple directions, and extend the operation frequency into the gigahertz\nrange.”\nComment 4: Do the substrate properties such as material anisotropy and electromechanical coupling\naffect the performance of the designed SAW device with complex electrodes?\nOur response: We thank the reviewer for this comment. The performance of the SAW device is\naffected by the substrate’s material anisotropy and electromechanical coupling, which is direction-\ndependent. The reason is that the piezoelectric substrate is an anisotropic material with direction-\ndependent SAW velocity with electromechanical coupling. Because of this dependency, our analytical\nmodel (Supplementary Note 1) considers direction-dependent SAW velocity and electromechanical\npotential-to-displacement ratio for predicting the generated SAW field.\nChanges to the manuscript: Per the reviewer’s comment, we added the following content to the\n“Methods” section. “Since the performance of a SAW device is influenced by both the electrode design\nand the anisotropic material properties of the piezoelectric substrate, our model considers both the\ngeometries of complex-shaped interdigital electrodes as well as the direction-dependent SAW velocity\nand the electromechanical potential-to-displacement ratio.”\nComment 5: In Fig 2 and Fig S3, all the designed SAW devices are facing upward. Is it possible to\nhave more complex designs with devices facing in the x and y directions? This combination may allow\nfor generating more complex SAW fields.\nOur response: We thank the reviewer for this comment. Incorporating SAW devices oriented along\nboth the x and y directions is indeed a promising approach for generating more complex SAW fields\nand controlling SAWs in multiple directions. This study mainly focuses on SAW devices facing in one\nPage 23 of 27\ndirection. Thanks for the suggestion, and in our future work, we will investigate the potential of using\nmultiple devices to control SAWs in different directions and generate more complex SAW patterns.\nChanges to the manuscript: Per the reviewer’s comment, we added the following content to the\n“Discussion” section. “Despite the successful demonstrations described above, the PIMs presented in\nthis study still have several limitations, including the inability to control SAWs propagating in multiple\ndirections and the lack of control over SAWs at multiple frequencies or across a wide frequency band.”\n“Looking ahead, we plan to design PIMs with electrode geometries different from those presented in\nthis paper, aiming to generate other intriguing on-chip SAW patterns, enable wideband operation,\ncontrol SAWs propagating in multiple directions, and extend the operation frequency into the gigahertz\nrange.”\nComment 6: To transform incident waves into complex SAW fields, the authors’ method uses a\ncustomized amplification circuit. The authors should provide more details of the circuit design. Are\nthere any frequency and input voltage limitations of the amplification circuit? A voltage bias is applied\nto the amplification circuit. What is the allowable voltage range? How to determine the voltage for\nachieving the desired SAW transformation ability?\nOur response: We thank the reviewer for this comment. Fig. R12 shows the design of the amplification\ncircuit, which includes a BGA 2869 amplification chip (NXP Semiconductors), a 220 Ω resistor (R),\nand four 400 nF capacitors (C , C , C , and C ). The resistor R and capacitors C and C are for ensuring\n1 2 3 4 1 2\na stable direct current (DC) voltage bias supplied to the BGA 2869 chip from the voltage port V .\nThe\nDC\ncapacitors C and C serve as alternating current (AC) coupling elements for the transmission of AC\n3 4\nsignals while blocking DC components. The circuit’s signal-in port is connected to the straight-\nelectrode IDT (Fig. R12, top left) that is for acquiring SAW signals. The signal-out port is connected\nto the PIM transducer (Fig. R12, top right) for generating SAWs.\nThe operational frequency range of the circuit spans from 0.1 MHz to 2 GHz, which is limited by\nthe BGA2869 chip specification. For stable amplification performance, the AC signal for amplification\nand the DC voltage bias were kept below 1 V and 12 V, respectively. Additionally, even if small\nwaveform distortion occurs at the signal-out port, the IDT can still generate the desired SAWs because\nit also functions as a frequency-selective filter. The amplified signal at the signal-out port is used to\ndrive the PIM transducer (Fig. R12, top right) for SAW generation. To generate SAWs with different\ndisplacements, the signal-out port’s voltage can be adjusted by changing the DC voltage bias applied\nto the circuit. In this study, the DC voltage bias was set to 9 V, so that the generated SAWs were\nsufficiently strong for measurement using the laser Doppler vibrometer.\nPage 24 of 27\nFig R12. Schematic of a SAW device with its amplification circuit. The amplification circuit\nincludes a 220 Ω resistor (R), four 400 nF capacitors (C , C , C , and C ), and a BGA 2869 chip. The\n1 2 3 4\nresistor R and capacitors C and C are for providing a stable direct current (DC) bias to the BGA 2869\n1 2\nchip from the voltage port V . The capacitors C and C serve as alternating current (AC) coupling\nDC 3 4\nelements, allowing for the transmission of AC signals while blocking DC components.\nChanges to the manuscript: Per the reviewer’s comments, we have made the following changes.\n We added Supplementary Figure 18 to show the circuit design.\n In the subsection “Design and manufacturing of PIMs” of the manuscript, we added the\nfollowing details. “For the transformation and ‘diode-like’ transmission of SAWs, we\nfabricated devices each integrating a SAW chip and a one-way amplification circuit (see\nSupplementary Figure 18 for the schematic and photo of a device). The circuit consists of a\nBGA 2869 amplification chip (NXP Semiconductors), a 220 Ω resistor (R), and four 400 nF\ncapacitors (C , C , C , and C ). Its operational frequency range spans from 0.1 MHz to 2 GHz.\n1 2 3 4\nTo ensure stable amplification without noticeable nonlinear waveform distortion, the input\nsignal at the signal-in port and the direct current (DC) voltage bias at the V port were kept\nDC\nbelow 1 V and 12 V, respectively. The signal-in port is connected to the straight-electrode IDT,\nallowing the SAW signals acquired by the IDT to be fed into the circuit. The amplified signal\nat the signal-out port is supplied to the PIM transducer for SAW generation. To generate SAWs\nwith varying displacements, the output voltage at the signal-out port can be adjusted by varying\nthe DC voltage bias applied to the amplification circuit. In this study, a DC voltage bias of 9 V\nwas used to generate SAWs strong enough for reliable measurement using a laser Doppler\nvibrometer.”\nComment 7: In the particle manipulation section, the authors showed carbon nanotubes (CNTs) can\nbe arranged by their SAW devices. What is the mechanism of arranging CNTs using your designed\nSAW devices? Is it possible to predict the constructed CNT patterns?\nPage 25 of 27\nOur response: We thank the reviewer for this comment. SAWs propagating along a piezoelectric\nsubstrate not only generate dynamic mechanical displacements but also electric potentials. Therefore,\nthe multi-walled carbon nanotubes (MWCNTs) in a SAW field are subjected to both the acoustic\nradiation and dielectrophoretic (DEP) forces. According to previous studies (Ma et al., Nanoscale,\n2015, 7, 14047; Zhang et al., Nature Comm., 2021, 12, 3844), the DEP force is much larger than the\nacoustic radiation force. Hence, the MWCNTs should be moved to the region with a stronger electric\nfield due to the positive DEP effect. Moreover, individual MWCNTs are assembled into bundles that\nare oriented along the electric field direction (Hermanson et al., Science, 2001, 294, 1082; Monica et\nal., Nanotech., 2008, 19, 085303). Therefore, when MWCNTs are subjected to standing SAWs, they\nshould form bundles at SAW nodes (as illustrated in Fig. R13a), where the electric field has peaks, and\nthese bundles should orient along the electric field direction that is perpendicular to SAW wavefronts.\nIn order to predict the DEP force field and the MWCNT pattern in a SAW field, we extended our\nanalytical model in Supplementary Note 1. We then used this analytical model to predict the MWCNT\npatterns created by the standing SAW device in Fig 5a (also see Fig R13b for a top view schematic).\nFig. R13c shows the simulated standing SAW intensity field and acoustic radiation force vector field.\nFig. R13d shows the root-mean-squared electric field and DEP force vector field. The zoomed-in view\nof Fig. R13d shows the predicted MWCNT bundle arrangement and orientation, agreeing well with the\nactual MWCNT bundle arrangement and orientation observed from the experimental microscopic\nimage in Fig. R13e. The MWCNT bundles are trapped at SAW potential valleys (i.e., root-mean-\nsquared electric field peaks) and oriented parallel with the electric field direction that is perpendicular\nto the SAW wavefronts.\nChanges to the manuscript: Per the reviewer’ comment, we made the following changes.\n We added Supplementary Figure 13 to show the simulated DEP force vector field and the\npredicted MWCNT pattern.\n We updated the model in Supplementary Note 1 to predict the SAW-induced DEP force fields\nand the resulting MWCNT patterns in SAW fields.\n In subsection “PIMs for acoustofluidics”, we added the following content: “Additionally, the\narrangements of microparticles and MWCNTs observed in the above experiments can be\npredicted using analytically simulated SAW-induced acoustic radiation and dielectrophoretic\nforce fields, respectively (see Supplementary Note 1 and Figure 13 for the simulation approach\nand results, respectively).”\nPage 26 of 27\nFig. R13. Prediction of the arrangement and orientation of MWCNTs in a SAW field.\n(a) Schematic illustrating the arrangement of MWCNTs in a standing SAW field. (b) Schematic\nillustrating the model for simulating standing SAWs generated by a pair of PIMs at 18.8 MHz.\n(c) Simulated standing SAW intensity field. The zoomed-in view shows the simulated acoustic\nradiation force vector field and the predicted distribution of SiO nanoparticles. (d) Simulated root-\n2\nmean-squared electric field and DEP force vector field. The zoomed-in view shows the simulated DEP\nforce vector field and the predicted arrangement and orientation of MWCNT bundles. (e)\nExperimentally acquired microscopic image showing the arrangement and orientation of MWCNT\nbundles.\nPage 27 of 27"}, {"round": 2, "reviewer_comments": {"reviewer_1": "Thank you for addressing all the concerns and comments. I believe the revised manuscript meets criteria of the journal.\nTherefore, I would like to recommend for publication.", "reviewer_2": "The revised manuscript presents a novel approach to designing on-chip phased interdigital metamaterials (PIMs) able to\nachieve both wave transformation and generation of surface acoustic waves (SAWs). Overcoming the limitations of current SAW\nchips that are only responsible for wave generation, this device encodes subwavelength phase profiles, a\ncharacteristic typical of metamaterials. The device introduced combines the functionalities of traditional plane-wave\ntransducers and phase-modulating metamaterials in a single module and can thus contribute to both SAW chips and\nmetamaterials. The presented methodology has applications in wave steering, jet generation, fluid and object manipulation,\nfluid streaming vortices and diode-like wave transmission. The authors have addressed the reviewer comments by adding\nnew investigations and discussions, improving the rigor and clarity of the methodology and results.\nI list a few of the remaining concerns:\n• The authors have responded to the concern regarding the amplitude distribution above the PIMs. Specifically, in Fig. R7\ncomparisons have been added between the intensity along a line above an array of transducers and the proposed PIMs.\nThe y-axis of the plots spans from 0 to 1, showing normalized intensity. However, in almost all plots, the data show values\nmuch less than 1, which makes it hard to observe differences between the intensity as a result of the two methods,\nespecially near the PIMs.\n•In response to Reviewer 4’s question about whether there are limitations prohibiting the scaling of the device to other\nfrequencies, the authors have added to the manuscript their plan to extend the operation frequency into the gigahertz range.\nTheir discussion could have included any fabrication or design-related challenges of scaling the device.\nThe authors have adequately addressed the concern about insufficient comparisons being drawn between FEM simulations\nand their proposed analytical method, which was proposed as a computationally less expensive alternative, and they made\ntheir assumptions more explicit. Comparisons have also been added between the intended positions of the focal points,\nthe\nexperimentally realized beams and the devices, which improved the clarity of the results. Extensive\ninvestigations have been added on the FWHM of jet beams as a function of design parameters and the\nposition/orientation of the beam. The authors have enriched the introduction section with related work on interdigital transducers\nand metasurfaces. They have added photographs of the fabricated devices and of the SAW device, which are instructive of\nthe experimental setup. Comparisons have been added between traditional IDTs and the proposed PIMs, which better\ncapture that advantage of this design in creating longer focused SAW beams.\nOverall, the manuscript has improved significantly after the revision, most our comments have been addressed satisfactorily\nand I believe it is suitable for publication in its current form.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts.", "reviewer_4": "I was pleased to read the revised manuscript. I think the authors have effectively addressed my and other reviewers'\nconcerns. I would recommend accepting this manuscript in its current format."}, "author_response": "Itemized list of response to reviewers’ remarks\n(Black italic: Editor’s remarks; Blue type: Our response; Additions/modifications to the manuscript\nand Supplementary Materials are highlighted in yellow)\nReviewer: #2\nThe revised manuscript presents a novel approach to designing on-chip phased interdigital\nmetamaterials (PIMs) able to achieve both wave transformation and generation of surface acoustic\nwaves (SAWs). Overcoming the limitations of current SAW chips that are only responsible for wave\ngeneration, this device encodes subwavelength phase profiles, a characteristic typical of\nmetamaterials. The device introduced combines the functionalities of traditional plane-wave\ntransducers and phase-modulating metamaterials in a single module and can thus contribute to both\nSAW chips and metamaterials. The presented methodology has applications in wave steering, jet\ngeneration, fluid and object manipulation, fluid streaming vortices and diode-like wave transmission.\nThe authors have addressed the reviewer comments by adding new investigations and discussions,\nimproving the rigor and clarity of the methodology and results.\nI list a few of the remaining concerns:\n The authors have responded to the concern regarding the amplitude distribution above the\nPIMs. Specifically, in Fig. R7 comparisons have been added between the intensity along a line\nabove an array of transducers and the proposed PIMs. The y-axis of the plots spans from 0 to 1,\nshowing normalized intensity. However, in almost all plots, the data show values much less than\n1, which makes it hard to observe differences between the intensity as a result of the two\nmethods, especially near the PIMs.\n In response to Reviewer 4’s question about whether there are limitations prohibiting the scaling\nof the device to other frequencies, the authors have added to the manuscript their plan to extend\nthe operation frequency into the gigahertz range. Their discussion could have included any\nfabrication or design-related challenges of scaling the device.\nThe authors have adequately addressed the concern about insufficient comparisons being drawn\nbetween FEM simulations and their proposed analytical method, which was proposed as a\ncomputationally less expensive alternative, and they made their assumptions more explicit.\nComparisons have also been added between the intended positions of the focal points, the\nexperimentally realized beams and the devices, which improved the clarity of the results. Extensive\ninvestigations have been added on the FWHM of jet beams as a function of design parameters and the\nposition/orientation of the beam. The authors have enriched the introduction section with related work\nPage 1 of 4\non interdigital transducers and metasurfaces. They have added photographs of the fabricated devices\nand of the SAW device, which are instructive of the experimental setup. Comparisons have been added\nbetween traditional IDTs and the proposed PIMs, which better capture that advantage of this design\nin creating longer focused SAW beams.\nOverall, the manuscript has improved significantly after the revision, most our comments have been\naddressed satisfactorily and I believe it is suitable for publication in its current form.\nOur response: We thank the reviewer for the positive comments on our work. A point-by-point\nresponse to your remaining concerns is provided below.\nComment 1: The authors have responded to the concern regarding the amplitude distribution above\nthe PIMs. Specifically, in Fig. R7 comparisons have been added between the intensity along a line\nabove an array of transducers and the proposed PIMs. The y-axis of the plots spans from 0 to 1,\nshowing normalized intensity. However, in almost all plots, the data show values much less than\n1, which makes it hard to observe differences between the intensity as a result of the two\nmethods, especially near the PIMs.\nOur response and changes to the manuscript: Thank you very much for noting this. To better show\nthe differences, we have updated the comparison plots (see Fig R1 below). A smaller y-axis range is\nused when the SAW field intensity is small. We have also updated the y-axis range in Supplementary\nFigure 15.\nPage 2 of 4\nFig. R1. Comparison of SAW fields generated by our PIM and a phased array. (a) Schematic of\nthe simulation setup and (b) simulated SAW intensity field at 18.8 MHz for a phased array with two\nrows of elements (indicated by colored dots). Each row has 200 elements with the same spacing of\n17.5μm (~ λ/12). The spacing between the two rows is half a wavelength (λ/2). The colors of the dots\nindicate different phases applied to individual array elements. (c) Schematic of the simulation setup\nand (d) simulated SAW intensity field at 18.8 MHz for a PIM device with the number of electrodes\nN=2, focal length d = 2 mm, and span W= 3.5 mm. (e, bottom to top) Comparison of intensity\nc\ndistributions along lines y = 315 μm (~1.5λ), y = 636 mm (~3λ), y = 954 mm (~4.5λ), y = 1272 mm\n1 2 3 4\n(~6λ), y = 1590 mm (~7.5λ), y = 1908 mm (~9λ), y = 2226 mm (~10.5λ), and y = 2544 mm (~12λ),\n5 6 7 8\nrespectively. (f) Comparison of intensity distributions along the y-axis.\nPage 3 of 4\nComment 2: In response to Reviewer 4’s question about whether there are limitations prohibiting the\nscaling of the device to other frequencies, the authors have added to the manuscript their plan to extend\nthe operation frequency into the gigahertz range. Their discussion could have included any fabrication\nor design-related challenges of scaling the device.\nOur response: We thank the reviewer for this comment. There are multiple challenges related to the\nfabrication and design of SAW devices, when increasing device frequencies from the megahertz to the\ngigahertz range. The widths and spacings of interdigital electrodes should be reduced to the submicron\nrange, leading to manufacturing challenges. For example, for a SAW device with an operating\nfrequency >1 GHz on a LiNbO wafer, the electrode width and spacing are theoretically <1 μm.\nlithography), which increases process complexity and cost. Additionally, when increasing the design\nfrequency to the gigahertz range, thermal drift (e.g., temperature-induced resonance frequency shift)\nbecomes more pronounced, thereby requiring strategies for operating temperature control. We have\nadded a discussion of these challenges to the Discussion section of the manuscript.\nPage 4 of 4"}, {"round": 3, "reviewer_comments": {"reviewer_2": "I feel the authors have effectively addressed my and other reviewers' concerns. I would recommend accepting this manuscript.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts."}, "author_response": ""}]

Decision letter