Orbital angular momentum- and frequency-dependent high-capacity encrypted hologram through multi-dimensional multiplexing acoustic metasurface

Domain: physics
Published: 2025-11-27
Editorial outcome: major_revision

Abstract
Sound holography has shown great capability in reconstructing the arbitrary complex sound fields, with potential applications in various scenarios. To overcome the limitations of existing technologies in terms of capacity and security, here we propose and experimentally demonstrate a mechanism of orbital angular momentum- (OAM-) and frequency-dependent high-capacity encrypted hologram through multi-dimensional multiplexing acoustic metasurface. By customizing the acoustic response of the metasurface based on a theoretically-derived transfer function, we bestow the metasurface with mode and spectrum selectivity simultaneously such that multiple predesigned images can be stored in parallel and restored with specific combinations of vortex mode and wavelength, thereby realizing the double-key-secured high-capacity holography. A distinct example of image reconstruction of multiple patterns is experimentally demonstrated, achieving as many independent hologram channels as the working frequency number multiplied by the OAM mode number and high-quality reconstruction of sound field information with strong confidentiality. Our methodology offers an avenue for building ultrahigh-capacity holographic information systems in acoustics, with far-reaching impact on encrypted sound communication and beyond.

Introduction
Acoustic holography, a disruptive technique to record and reconstruct the full-wave information of arbitrary targets precisely, has sparked tremendous interest among scientists and engineers of many disciplines, with great potential in diverse applications spanning from ultrasonic transcranial imaging to particle manipulations1–7. Conventional acoustic hologram techniques usually capitalize on sophisticated phased arrays composed of numerous individually addressed transducers to reshape the spatial sound field as required8,9. In addition to the high cost and complexity of fabrication, the bulky size of transducer units, which is usually as large as the wavelength scale, especially in the ultrasonic range, will unavoidably restrict the space-bandwidth product as well as the hologram resolution. In stark contrast, the recently emerging acoustic metamaterials and metasurfaces with exceptional properties unattainable in nature offer a solution to mold the wavefront into almost arbitrary profile with significantly higher degrees of freedom (DoF) than conventional phased arrays at a fraction of the cost and complexity10–17. As a result, the cutting-edge developments combining acoustic holography with metamaterials/metasurfaces revolutionize the technique of achieving high-quality acoustic holograms and demonstrate various fascinating progress, including the creation of complicated patterns of acoustic vortex knots18 and arbitrarily shaped microbubble cavitation fields19, among others20–23.

Nevertheless, most of the existing acoustic metasurfaces can only produce one fixed holographic pattern at a particular imaging plane, leading to limitations in terms of information capability and flexibility24–27. Along with this, the holographic information is also easy to be reproduced or intercepted just by illuminating the metasurface with simple waves. To improve the sound information capacity, recent efforts have focused on frequency-multiplexing sound holography, for which distinct patterns are encoded onto different driving frequencies28–31. However, information capacity in these approaches is still severely limited by spectrum crosstalk and operation bandwidth. Despite spin and polarization multiplexing mechanisms having been proven to be effective in optical holography32–36, their realization in acoustic systems is impossible due to the lack of physical dimensions of acoustic spin and polarization, resulting in the few channels available for storing acoustic hologram information. On the other hand, orbital angular momentum (OAM) provides a new DoF independent of polarization or spin for multiplexing a set of orthogonal modes on the same frequency channel to achieve high spectrum efficiency, which also plays important roles in rotational Doppler shift, edge image processing, and high-speed optical/acoustic communications37–44. In recent investigations within the realms of optics and microwaves, increasing attention has been directed towards OAM-multiplexing holography that superposes OAM of light/microwave into metasurfaces45–48, yet whose direct translation into acoustics will result in unpractical size due to the long wavelength of sound waves. To date, OAM manipulation has still been a missing piece of the puzzle in multi-dimensional multiplexing sound holography. This challenge may be attributed to the lack of an effective mechanism for simultaneously modulating the multiple dimensions of sound, including both OAM and frequency. As a consequence, realizing high-capacity and high-security sound holograms still remains an elusive and unresolved challenge.

In this article, we propose a multi-dimensional multiplexing acoustic hologram mechanism by exploiting OAM as a new manipulation dimension and marrying it with frequency multiplexing to superpose separate patterns into different OAM/frequency modes, which significantly enlarges the information capacity of the holographic sound field. To this end, we develop a novel dual-multiplexing acoustic metasurface, which is systematically designed, which is capable of restoring the high-capacity multiplexed information of holographic sound fields in subwavelength spatial resolution. Two proof-of-concept examples of reconstructing four Arabic numerals and four alphabet letters are demonstrated both numerically and experimentally, which show good agreement with the design and verify the advantage of our proposed mechanism in terms of high-quality, large capacity, and enhanced security.

Results
The theory of multi-dimensional multiplexing sound holography
Figure 1 schematically illustrates our proposed multi-dimensional multiplexing acoustic hologram mechanism, in which acoustic holography information is stored in a monolayered metasurface and can be reconstructed by illuminating the metasurface with incident beams carrying specific keys. In our mechanism, two multiplexing DoFs, OAM and frequency, are simultaneously introduced to increase the information capacity of holographic patterns. To be specific, on the one hand, coexisting orthogonal OAM modes at the same frequency help improve spectral efficiency and relax the requirement for spectrum width. On the other hand, each OAM mode can be broadened to several parallel ones with separate wavelengths, which eliminates the dependence on high-order OAM modes with strong divergence effects and thereby improves the imaging quality of reconstructed patterns. In addition, these two manipulation DoFs also serve as two physical keys, which means that the information hidden in the hologram metasurface can be unveiled solely by OAM beams with specific topological charges and frequencies, as will be theoretically elucidated later. As a result, our proposed OAM- and frequency-dependent mechanism can effectively extend the multiplexing dimension of acoustic holography, making the high-capacity information effectively stored in the double-encrypted holograms and rendering the information exceedingly difficult to decipher, which is beyond the capabilities of existing methods.

To implement our mechanism, we begin by analyzing how customizing the acoustic response of the metasurface (i.e., amplitude, phase, frequency, and OAM) enables the introduction of OAM and frequency multiplexing dimensions, which will be theoretically derived in what follows. For a specific targeted holographic image, a pre-processing of spatial sampling is performed to ensure the OAM selectivity of our mechanism (Fig. 2a), which is commonly used in the holography technique48,49. If spatial sampling is omitted or the sampling interval is too small, donut-shaped artifacts may overlap and cause constructive interference, which in turn degrades OAM selectivity (see Supplementary Note 1 for more details). Here, we discretize the holographic image into N pixels with a sampling interval of 2\lambda, which can effectively prevent spatial overlap of the output while preserving high information density. Al n and \Phi l n are the amplitude and phase of the n-th pixel (n = 1, 2, 3, :::, N) of the holographic image to be encoded into the l-th OAM channel at frequency f. The metasurface is composed of M units. Then, the relationship between the acoustic response of the metasurface and the targeted holographic image can be established by the following matrix equation:

2
6
6
6
6
4
G 11 G 12 . . . G 1M
G 21 G 22 . . . G 2M
. . . . . . . . . . . .
G N1 G N2 . . . G NM
3
7
7
7
7
5
2
6
6
6
6
6
4
p f 1,l
p f 2,l
. . . 
p f M,l
3
7
7
7
7
7
5
=
2
6
6
6
6
6
4
A l 1 exp(j\Phi l 1)
A l 2 exp(j\Phi l 2)
. . . 
A l N exp(j\Phi l N)
3
7
7
7
7
7
5 (1)

where G nm = exp[j(2\pi/c)r nm]/r nm is the Green’s function of the point source in free space, r nm is the distance between the n-th pixel of the image and the m-th unit of the metasurface, and c is the sound speed in air.

Through matrix inversion of Eq. (1), we can obtain the complex amplitude profile p f,l required for generating the corresponding holographic image (Step 1 in Fig. 2b). By encoding p f,l with an additional helical phase distribution of -l\theta (\theta is the azimuth angle and l is the topological charge of the OAM mode), the hologram metasurface is imprinted with OAM and frequency selectivity (Step 2 in Fig. 2b), which means that only emitting an OAM mode with inverse topological charge (l) and corresponding frequency (f) can decode and restore the information hidden in the hologram and consequently recover the fundamental spatial mode with a relatively stronger intensity distribution in each pixel of the holographic image. Then, the OAM- and frequency-multiplexing acoustic hologram can be realized by superposing multiple OAM- and frequency-selective responses on the metasurface (Step 3 in Fig. 2b). Based on this, the general transfer function of the metasurface can be written as follows

P = \sum f,l p f,l exp(-jl\theta)\delta(f, f 0) (2)

where f is the frequency of the incident wave, and \delta is the Kronecker delta function. In the above, by utilizing the Green’s function of a point source in free space and the matrix inversion method, we directly calculate the phase and amplitude mask of the metasurface, without satisfying the paraxial approximation condition. Therefore, the transfer function between the image and hologram metasurface established in our mechanism remains effective even when the metasurface’s size and imaging distance are relatively small. Thanks to this, our mechanism can operate within a compact physical size and demonstrates a remarkable advantage in terms of improved information density. To verify the above theoretical derivations, here we first showcase a simple example of reconstructing four distinct holographic images (alphabet letters A, B, C, and D) stored in two OAM channels and two frequency channels through numerical calculations. In such a case, the detailed process of deriving the phase and amplitude profile on the multiplexing metasurface is illustrated in Fig. 2a, b. The results in Fig. 2c reveal that the multi-dimensional multiplexing holograms accordingly respond to incident OAM beams with different topological charges and working frequencies, which shows the successful reconstruction of the designated holographic patterns in the image plane in a high-quality and low-crosstalk way. The above results verify the effectiveness of our mechanism in extending the hologram multiplexing dimension and boosting information capacity. Besides, our mechanism offers an advantage over traditional phase-only devices in achieving high-quality hologram reconstruction, as demonstrated by a representative example in Supplementary Note 2.

In our theoretical design, OAM selectivity and imaging encoding are accomplished through multiplying the hologram by an ideal helical phase. This means that retrieving the image’s information requires illuminating the metasurface with a perfect vortex beam. However, the vortex beam experiencing propagation in the real-world system cannot hold a perfect pattern. To inspect its influence, we further analyze the tolerance of our mechanism to phase disturbance typically induced by diffraction, medium inhomogeneities, volume scattering, and beyond, and the results in Supplementary Note 3 demonstrate the high robustness of our mechanism against these factors.

Design of meta-pixels
For the purpose of realizing the OAM- and frequency-dependent acoustic holography, the metasurface needs to be customized according to Eq. (2), which calls for the simultaneous modulation of the phase, amplitude, frequency, and OAM characteristics of sound. This is challenging for the existing metamaterial/metasurface unit cells with few controllable DoFs. Here, we propose the design of an acoustical meta-pixel structure capable of all-dimensional sound wave modulation in air, which serves as the basic building block of our multi-dimensional hologram metasurface. The proposed meta-pixel, whose geometry is shown in Fig. 3a, is judiciously designed to contain multiple subunits and each subunit, as the functional parts, are marked by red, blue, and black outlines, respectively, as shown in Fig. 3b. The length, width, and height of the i-th layer of subunit 1 and subunit 2 are W1 i,2, D1 i,2, and H1 i,2, respectively. Here we showcase a structural design with two subunits that work at dual frequencies of f 1 = 13720 Hz and f 2 = 17150 Hz. In such a design, we introduce controllable impedance mismatch to realize full-range amplitude modulation, which is implemented by adjusting the width D1,2 of the impedance modulation layer. Furthermore, to simultaneously control the transmitted phase within the same spatial dimension, the effective refractive index of the units is precisely tailored by lengthening or shortening the equivalent propagation distance of the zig-zag pipe of the phase modulator (i.e., varying H1,2). In addition, the spectral filters on two sides ensure that acoustic waves at the specific frequency interact exclusively with their corresponding subunit, without penetrating into the other, thereby providing strong frequency selectivity. However, the sound waves propagating within these three parts are mutually coupled, resulting in difficulties in traversing the full parameter space of amplitude-phase at different frequencies in a decoupled way. To address this, we theoretically derive the transfer function of the meta-pixel and, based on it, determine the decoupled points within two-dimensional parameter spaces defined by H1,2 and H1,2, at which the meta-pixel enables decoupled amplitude-phase modulation at each frequency component (see Supplementary Note 4 for detailed theoretical derivations and the procedure for determining the decoupling points). The specific geometrical parameters of the meta-pixel are listed in Supplementary Table 1.

Next, we use numerical simulation to demonstrate the amplitude-phase decoupled modulation functionality as well as the frequency multiplexing performance of our designed meta-pixel. Throughout the paper, the simulations are conducted with a finite element method based on COMSOL MULTIPHYSICS 6.1 software. Figure 3c shows the transmission spectra of subunit 1 and subunit 2 within a frequency range of 12 kHz to 20 kHz. It can be seen that at 13720 Hz, the transmission coefficient of subunit 1 (0.99) is much higher than that of subunit 2 (0.18), and the situation is opposite at 17150 Hz (the transmission coefficient is 0.17 for subunit 1 and 0.99 for subunit 2). For a clearer illustration, we also simulate in Fig. 3d the acoustic pressure distribution through the whole meta-pixel at two operation frequencies f 1 and f 2, respectively. These results suggest that the response of subunit 1 (2) is dominant when the driving frequency is f 1 (f 2), while the contribution from the other is negligible thanks to the effective sound-blocking effect by the corresponding spectral filters, enabling independent modulation on each frequency channel with low crosstalk, as demanded in our mechanism.

The numerically simulated results in Fig. 4a demonstrate the ability of our designed meta-pixel in achieving nearly decoupled control of the phase and amplitude DoF, enabling independent tuning of each across its full range by varying only a single structural parameter. This substantially reduces the complexity of the metasurface design. The slight fluctuations observed in the amplitude and phase distributions may result from weak contributions of parameters other than H1,2 and H1,2, which are not considered in the current study for the sake of simplifying the theoretical calculations and structural design. Then, we select 4 discrete amplitude values and 8 discrete phase values to form 4×8 amplitude and phase combinations, thereby establishing a corresponding structural parameter library, as shown in Fig. 4b. Such an amplitude and phase discretization is precise enough to accurately mold the wavefront and achieve high-quality sound field reconstruction (see Supplementary Note 5 for the comparison of imaging quality with different discretization orders). Notice that the number of frequency channels can be further boosted by increasing the state density of the hybrid meta-pixel (see Supplementary Note 6 for another example of 3 frequency channels).

Experimental realization of the high-capacity encrypted holography
Next, we will experimentally demonstrate the proposed mechanism for high-capacity double-key-secured sound holography. Four digits of Arabic numerals from 1 to 4 are selected as the target and encoded in four channels containing two frequency channels (f 1 = 13720 Hz and f 2 = 17150 Hz) and two OAM channels (topological charges of ±1). Specifically, numeral 1 is encoded on the OAM beam with a topological charge of l = +1 at f 1, denoted as the channel (f 1; l = +1). Similarly, numerals 2, 3, and 4 are encoded on channels (f 1; l = -1), (f 2; l = +1), and (f 2; l = -1), respectively. The four images are distributed across two distinct regions on the same imaging plane, with numerals 1 and 3 in one region and numerals 2 and 4 in the other. According to the method discussed above, we can easily calculate the required transmitted amplitude and phase profiles of the hologram metasurface for reconstructing holographic images at different frequencies, without any sophisticated optimization algorithm, as shown in Fig. 5a. Figure 5b shows the fabricated metasurface sample comprising 40×40 meta-pixels via three-dimensional (3D) printing technology at 0.1 mm precision, whose overall geometric size is 0.4×0.4×0.06 m3. This number of unit cells provides sufficient DoFs to enable a high-quality hologram while simultaneously preserving the system’s compactness.

To exclude the influence of unwanted reflection waves, the experiment is conducted in an anechoic chamber. As shown in Fig. 5c, OAM beams are emitted from a circular speaker array with a radial size of 0.12 m and spread towards the receiving metasurface over a distance of 1.2 m. In experiment, we first measured the reconstructed two-dimensional holographic sound field in a 0.4×0.4 m2 square region 0.15 m behind the metasurface. The one-dimensional acoustic intensity profiles at some specific positions marked by the dashed lines in Fig. 5d were also measured. During the measurement, a 1/4-in. free-field microphone (Brüel & Kjær type-4961) fixed on a 3D stepping motor is used to measure the acoustic intensity distribution point by point.

As presented in Fig. 5d, the experimentally measured two-dimensional acoustic intensity distributions and one-dimensional profiles on the image plane agree well with the simulated results, which both demonstrate the high-quality reconstruction of four Arabic numerals under the specific combination of topological charges and frequencies. This not only shows the ability of the metasurface to effectively modulate the multiple DoFs of sound waves including the OAM and frequency, but also manifests the high-security of our mechanism since holographic pattern information stored in the metasurface can only be released by the incident wave with a specific OAM mode and frequency. In addition, the frequency crosstalk is significantly suppressed, as indicated by the fact that nearly no artifacts can be observed in both simulated and experimental results.

Besides, we also showcase in Supplementary Note 7 a more complex case in which all target images fully overlap on the same imaging plane. We select four alphabet letters as the targeted patterns. The simulated and measured results in Supplementary Fig. 14 still exhibit good quality, validating the potential for further increasing information capacity and the mechanism’s independence from the spatial arrangement of the multiplexed images. Moreover, to demonstrate the potential technical advancements enabled by our mechanism, we present in Supplementary Note 7 a representative example of OAM- and frequency-dependent particle assembly, which manifests the significant advantages of our methodology in flexible and diversified sound force manipulation. Beyond this, our mechanism also holds the promise for boosting advances in diverse applications, such as high-capacity encrypted communication and flexible sound wave 3D printing.

Methods
Numerical simulation
Throughout the paper, the simulations are performed with a finite element method based on COMSOL MULTIPHYSICS 6.1 software. Air is set to be the background medium, for which the mass density and sound speed are 1.21 kg/m3 and 343 m/s, respectively. The material for fabricating the metasurface is UV resin whose mass density and sound speed are 1400 kg/m3 and 2000 m/s, respectively.

Experimental configuration
The experiment is performed in an anechoic room to eliminate any disturbance of reflected sound. The metasurface samples are manufactured via 3D printing technology (0.1 mm in precision). At the emitting end, eight HiVi B1S loudspeakers are arranged axisymmetrically along a 12 cm-radius circle and mounted onto an acrylic glass plate, forming a single-ring circular phased array. A single-chip microcomputer, Arduino Mega 2560, is used to generate the drive signal of each loudspeaker. To emit an acoustic vortex beam with l-th order OAM mode, the phase of the drive signal to each speaker should be \phi = l\theta, where \theta is the azimuthal angle between the speaker and the reference. The metasurface is placed in alignment with the single-ring circular phased array at a relative distance of 1.2 m. At the imaging plane, a microphone (Brüel & Kjær type-4961) is mounted on a 3D stepping motor to measure the pressure field. The scanning step is set to sampling interval d for the two-dimensional sound field measurement and to 1 cm for the one-dimensional acoustic pressure profile measurement, respectively. The signals are recorded with a Brüel & Kjær PULSE 3160-A-042 2-channel analyzer. For two-dimensional spatial scanning, the initial scan position is aligned with the center of the first pixel of the hologram image.

Data availability
Source data are provided with this paper.

Code availability
All technical details for implementing the simulation are provided in the Manuscript and Supplementary Information. Codes are available from the corresponding authors J.L., B.L., or J.C. upon request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript reported the theoretical design and experimental demonstration of a dual-multiplexing acoustic metasurface\nwith OAM and frequency selectivity, realizing high-capacity and high-security hologram in acoustic systems for the first time.\nI think introducing a new coding degree of freedom of OAM for sound holography technique is significant and has great\npotential for many important applications, such as sound communication. Besides, the meta-pixel with customized frequency\nresponses is an ingenious design for implementing their mechanism. Overall, I find this is an interesting and nice work that\nrepresents a good example of how new physics concepts can be harnessed to improve existing technology and therefore\nthink it can benefit from the high visibility of Nature Communications. I recommend the paper for publication if the following\npoints can be addressed properly:\n1: Why must holographic images undergo spatial sampling processing, and what are the criteria for determining the optimal\nsampling intervals? I suggest the authors to give some discussions on these issues.\n2: We know that optical holography generally relies on a paraxial approximation condition. Does the operating mechanism\nof this work also need to satisfy the paraxial approximation condition? If not, what is the difference?\n3: More discussions about the robustness of the proposed hologram mechanism are welcome. For example, will the\ndiffraction effects during the wave propagation impact the hologram results?\n4: the authors are suggested to include some numerical demonstrations to compare their metasurface with conventional\nphase-only devices, which would highlight their performance advantages more visually.", "reviewer_2": "This paper proposes a mechanism based on orbital angular momentum (OAM) and frequency-dependent high-capacity\nencrypted hologram through multi-dimensional multiplexing acoustic metasurface. The general idea is interesting although\nthe novelty is extremely limited which makes a major obstacle for this manuscript to meet the high strands of NC especially\nin terms of proposing new conceptual and technical advances which the reviewer did not find in this manuscript, knowing\nthe large amount of work on the topic in the state of the art. The manuscript is poorly written, very confusing and even\nunclear in many parts. The reviewer has major concerns about this manuscript, listed below, which the authors should\naddress before a consideration in a more specialized journal.\n1. What is the meaning of “TRM” in fig.2, the authors did not give any clear explanation on this.\n2. Figure 2a seems to have some unclear meaning. For readers less familiar with the field, it is difficult to understand the\nrelationship between these jumbled diagrams. The authors have to clarify this aspect.\n3. It seems also that the subscript of H in the caption of Fig. 3b should be superscripted ?\n4. Confusion over the use of parameters in Figure 3b in the main text and in Figure 2 in the supplementary material, where H\nin Figure 3b is L in Figure 2?\n5. With the same cross-sectional area of the tube in Fig. 1b of the supplementary material, why does the superscript of p\nchange?\n6. What is the function of Hamp and Hphase ?\n7. Combining Fig. 4 and Fig. S3, it can be seen that the unit is not decoupled control of amplitude and phase but searches\nfor suitable combinations to achieve amplitude-phase traversability, is this somehow limiting the degrees of freedom of\nholographic imaging?\n8. The cells in Figures S2 and 3b seem to be symmetrical structures, but why is there an extra cavity on the left side in\nFigure S1f?\n9. Careful observation of Fig. S2 reveals that the widths between the La and Lc cavities are different, but the authors have\ndirectly replaced this with a 2D structure in Fig. S1, so if this inconsistency in width has no effect on the model, the authors\nhave to explain this in details.\n10.Why did the authors use discrete pixel points for the hologram? Compared to either existing optical holography or\nacoustic holography, the quality of proposed holograms in the manuscript is poor.", "reviewer_3": "This work presents a novel approach to utilizing orbital angular momentum (OAM) and frequency as two coherent secret\nkeys to achieve high-capacity encrypted sound holography. As a practical implementation, the authors design a multiplexing\nacoustic metasurface composed of meta-pixel structures, enabling simultaneous modulation of multiple sound dimensions.\nDue to the high degree of freedom (DoF) characteristic, the information capacity of the holographic sound field is\nsignificantly enhanced, while confidentiality is ensured by the orthogonality of OAM modes and frequencies.\nThe manuscript is well-structured, with a clear presentation of results spanning theoretical modeling, simulations, and\nexperimental realization. While the demonstrated holographic images of four Arabic numerals are relatively simple, the first\nsuccessful demonstration of a high-capacity hologram leveraging new DoF of OAM in acoustic systems is highly significant.\nThis work has crucial implications for various applications, including modern communication technologies. Therefore, I\nrecommend the publication of this manuscript in Nature Communications.\nTo further improve the manuscript, I have the following suggestions:\nThe transverse dimensions of the metasurface are set to 30 cm. What is the rationale behind this choice, and how does this\nsize influence the system’s performance parameters? A more extensive discussion on this aspect would be beneficial.\nThe theoretical derivation of the meta-pixel transfer function in the Supplementary Materials is correct but overly complex,\nparticularly in Fig. S1. Would it be possible to derive the transfer function directly from the whole structure for clarity?\nThe experimental section lacks sufficient detail. Please expand this section, including a more thorough explanation of how\nthe speaker array generates the incident vortex sound beams with a designated mode order."}, "author_response": "1. Detailed response to the report of the reviewers\nThe report of reviewer 1:\nThis manuscript reported the theoretical design and experimental demonstration of a\ndual-multiplexing acoustic metasurface with OAM and frequency selectivity, realizing\nhigh-capacity and high-security hologram in acoustic systems for the first time. I think\nintroducing a new coding degree of freedom of OAM for sound holography technique\nis significant and has great potential for many important applications, such as sound\ncommunication. Besides, the meta-pixel with customized frequency responses is an\ningenious design for implementing their mechanism. Overall, I find this is an interesting\nand nice work that represents a good example of how new physics concepts can be\nharnessed to improve existing technology and therefore think it can benefit from the\nhigh visibility of Nature Communications. I recommend the paper for publication if the\nfollowing points can be addressed properly:\nResponse: We wish to sincerely thank the reviewer for offering positive remarks on our\nwork and valuable suggestions that have helped us to improve our manuscript. In light\nof the reviewer’s report, we have made every effort to carefully revise the manuscript.\nWe have given the response individually to each comment below.\nComment 1: Why must holographic images undergo spatial sampling processing, and\nwhat are the criteria for determining the optimal sampling intervals? I suggest the\nauthors to give some discussions on these issues.\nResponse: Thank you for this valuable suggestion. The reason we perform spatial\nsampling processing on the holographic image is to ensure the orbital angular\nmomentum (OAM) selectivity in our mechanism. The OAM selectivity means that only\nemitting an OAM beam with a specific topological charge (l) can restore the\ninformation hidden in the l-th order OAM channel. When the mode order of the emitting\nbeam is not equal to l, the holographic pixels appear as concentric diffraction rings\nwhose radii increase with the topological charge, which can be regarded as an artifact\n1\n(see Fig. R1). Despite the presence of such artifacts, their null centers and diffuse energy\ndistributions can maintain a significant contrast between themselves and the target\nholographic image, thereby preserving the selectivity of the OAM mode. However, if\nspatial sampling is not performed, or if the sampling interval is too small, these donut-\nshaped artifacts may overlap with each other, resulting in the constructive interference\nof artifacts as well as the degradation of OAM selectivity, as illustrated in a specific\nexample shown in Fig. R2. Theoretically, a larger sampling interval leads to better\nOAM selectivity. However, increasing the sampling interval also results in a reduction\nof spatial pixel density. To investigate their relationship, here we introduce two\nparameters of information density ε and OAM selectivity coefficient η defined as\nm.n\nε=λ/d\nη = Il=m Il=n,\nm,n\nmax max\nwhere d is the sampling interval, λ is the wavelength, and Il is the maximum intensity\nmax\nof holographic pressure patterns for l-th-order OAM incidence. Considering a specific\nsituation where a particular hologram image of a point array with interval d is encoded\nin the -1st-order OAM channel by multiplying the hologram with a 1st-order helical\nphase mask, the sound fields at the imaging plane z = 16 λ can be calculated for different\ninterval d when illuminating the metasurface with 1st-order and -1st-order OAM beams.\nThe curve depicting the relationship between the information density ε and the OAM\nselectivity coefficient η is shown in Fig. R3, highlighting a remarkable trade-off that\n−1,1\ncan be summarized as the following three points: 1. for small sampling intervals,\ndensely packed pixels with high information capacity cause mode overlap, leading to\ndegradation in OAM selectivity (the left inset); 2. rational sampling intervals can strike\na balance between preserving OAM selectivity and maintaining relatively high\ninformation density (the middle inset); 3. overly sparse sampling enhance OAM\nselectivity but result in the reduction of spatial information density (the right inset).\nTherefore, a sampling interval of 2λ is chosen in our scheme to prevent spatial overlap\nof the output while retaining relatively high information density. In the updated version\nof the manuscript, we have added some discussions on this issue based on the\n2\nreviewer’s suggestion.\nFigure R1 | Holographic images and pixel-wise energy distribution under different\nOAM mode incidence. The holography information is loaded in the -1st-order OAM\nchannel by multiplying the hologram with the 1st-order helical phase mask.\nholography information is encoded in the -1st-order OAM channel by multiplying the\nhologram with a 1st-order helical phase mask.\ncoefficient η . The holography information is loaded in the -1st-order OAM channel\n−1,1\nby multiplying the hologram with the 1st-order helical phase mask. The white line in\nthe insets denotes the wavelength.\n2: We know that optical holography generally relies on a paraxial approximation\ncondition. Does the operating mechanism of this work also need to satisfy the paraxial\napproximation condition? If not, what is the difference?\nResponse: We thank the reviewer for pointing out this important question. Our\nproposed mechanism of OAM- and frequency-dependent high-capacity hologram\noperates with no need to satisfy the paraxial approximation condition. As mentioned by\nthe reviewer, optical holography generally relies on a paraxial approximation condition,\nsuch as the commonly used Fourier-transform-based paradigms. However, when it\ncomes to the free-space acoustic system, the paraxial approximation is extremely\ndifficult to satisfy owing to the inherent long wavelength and strong diffraction\ncharacteristics of acoustic waves. To circumvent this fundamental problem, we utilize\nthe Green’s function of a point source in free space and matrix inversion to directly\ncalculate the phase and amplitude mask of the metasurface, which consequently\nbypasses the need for paraxial constraints. Thanks to this, our mechanism also\ndemonstrates a remarkable advantage over the optical counterpart in terms of\n5\ndownsized systematic size and improved information density. In the updated version of\nthe manuscript, we have added a brief discussion to clarify this issue.\n3: More discussions about the robustness of the proposed hologram mechanism are\nwelcome. For example, will the diffraction effects during the wave propagation impact\nthe hologram results?\nResponse: Thank you for this comment. Following the valuable suggestion of the\nreviewer, we have added more discussions and numerical simulations about the\nrobustness of our proposed mechanism against some typical factors, including the\ndiffraction effects as well as the turbulence and scattering during the wave propagation.\nWe first investigate the robustness of our mechanism against the diffraction effects.\nIn\nour proposed OAM- and frequency-dependent acoustic hologram mechanism, the\nOAM selectivity and imaging encoding are accomplished through multiplying the\nhologram with an ideal helical phase, as demonstrated in Eq. (2) and Fig. 2(b) in the\nmanuscript. This means that retrieving the image’s information requires illuminating\nthe metasurface with a perfect vortex beam. However, the vortex beam experiencing\npropagation in the real-world system cannot hold a perfect pattern, as numerically\nsimulated in Fig. R4(a). To inspect the influence of diffraction-induced non-ideal\nhelical phase, we compare the reconstructed holographic sound field under two cases\nof ideal vortex beam incidence and practical vortex beam incidence generated by a\nphased array, and typical results are shown in Fig. R4(b). It can be seen that the two\nresults exhibit high consistency, verifying the robustness of our mechanism. On the\nother hand, considering the complexity of the background medium, many factors, such\nas turbulence, volume scattering, and flow, would distort the helical wavefronts of\nOAM beams and impair the hologram performance (see Fig. R5(a)). This is equivalent\nto imposing the wavefront arriving at the hologram metasurface with a random phase\nperturbation of κϕ, where ϕ is a randomly distributed phase ranging from 0 to 2π,\nr r\nand κ is the perturbation strength. To quantitatively assess the quality of the\n6\nholographic image, a commonly used parameter of correlation coefficients C describing\nthe similarity between two images is introduced [e.g., Nat. Commun. 9, 1632 (2018)],\n∑∑(A −A)(B −B)\nmn mn\nC = m n\n∑∑(A −A)2∑∑(B −B)2\nmn mn\nm n m n\nwhere A and B are the data matrices of the two images, A and B are the mean values\nof the elements in the matrices A and B, respectively. A higher correlation coefficient\nC between the target image and the holography pattern indicates a better imaging\nquality. By varying the perturbation strength κ, we calculate the variation curve of the\ncorrelation coefficient C and plot the result in Fig. R5(b), from which we can observe\nthat a high imaging fidelity (C > 0.8) persists even when the perturbation strength\nreaches 0.3. This demonstrates the significant tolerance of our mechanism to phase\ndisturbance typically induced by factors such as turbulence, medium inhomogeneities,\nand volume scattering. In the updated version of the manuscript, we have added some\ndiscussions and additional simulation results to demonstrate the high robustness of our\nmechanism.\nFigure R4 | (a) The free-space propagation of 1st-order and -1st-order OAM beams\n7\ngenerated by a loudspeaker array. Cross-sectional acoustic pressure distributions\n(grayscale) demonstrate the evolution of interference patterns from near field to far field.\n(b) Reconstructed holographic sound field under two cases of ideal vortex incidence\nand practical vortex incidence generated by a phased array, respectively.\nFigure R5 | (a) Phase profiles of vortex beam without and with distortion. (b) The\nvariation curve of the correlation coefficient C in response to phase perturbation\nstrength κ. Inset: Reconstructed sound field patterns when κ = 0.3 (red box) and κ = 0.8\n(blue box).\n4: the authors are suggested to include some numerical demonstrations to compare their\nmetasurface with conventional phase-only devices, which would highlight their\nperformance advantages more visually.\nResponse: Following your important suggestion, we have provided additional\nsimulations to visually demonstrate the advantages of our multi-dimensional\nmultiplexing metasurface over traditional phase-only devices in terms of the high-\nquality hologram reconstruction. Here, for simplicity without losing generality, we\n8\nselect two simple patterns shown in Fig. R6(a) as the target hologram images that are\nencoded in the 1st-order and -1st-order OAM beams, respectively. The transverse size\nof the metasurface is set to be 160 λ, the imaging distance is 48 λ, and the sampling\ninterval is set to be 2.5 λ. The comparison of the generated hologram image through our\nmetasurface and phase-only device under different OAM incidence is depicted in Figs.\nR6(b) and (c), showing that the OAM holographic patterns generated by our\nmetasurface with simultaneous amplitude and phase modulation exhibit lower inter-\nmode crosstalk and better reconstruction quality. This is also manifested by the\nremarkable value difference of the image correlation coefficient we theoretically\ncalculate for both cases (i.e., 0.90 and 0.92 for our scheme, 0.71 and 0.75 for the phase-\nonly design). In the updated version of the manuscript, we have added Fig. R6 in the\nsupplementary materials, accompanied by some discussions.\nFigure R6 | (a) Target images: star (left) and heart (right), corresponding to 1st-order\n9\nand -1st-order OAM beams. The generated hologram images through (b) our\nmetasurface and (c) phase-only device under different OAM incidence.\nThis paper proposes a mechanism based on orbital angular momentum (OAM) and\nfrequency-dependent high-capacity encrypted hologram through multi-dimensional\nmultiplexing acoustic metasurface. The general idea is interesting although the novelty\nis extremely limited which makes a major obstacle for this manuscript to meet the high\nstrands of NC especially in terms of proposing new conceptual and technical advances\nwhich the reviewer did not find in this manuscript, knowing the large amount of work\non the topic in the state of the art. The manuscript is poorly written, very confusing and\neven unclear in many parts. The reviewer has major concerns about this manuscript,\nlisted below, which the authors should address before a consideration in a more\nspecialized journal.\nResponse: We sincerely appreciate the reviewer’s insightful feedback and assessment\nthat the general idea of our work is interesting. We also apologize for the lack of clarity\nin our previous manuscript, which may have caused the reviewer’s misunderstanding.\nTo address the reviewer's concerns about our work, we have made substantial revisions,\nmainly including:\n1. We have developed an alternative matrix inversion method to calculate the phase\nand amplitude mask of the metasurface, which enables improving the hologram\nquality significantly for a compact system.\n2. We have redesigned the geometrical structure of the meta-pixel to overcome the\nlimitation in amplitude-phase coupling. And the newly designed meta-pixel is\ncapable of achieving exceptional function of amplitude–phase decoupled\nmodulation at each frequency component.\n3. We have derived the transfer function directly from the whole structure rather than\nfrom each component of meta-pixel subunits, which helps understand the structure-\nto-function relationship more clearly.\n4. We have re-fabricated the metasurface sample and conducted new experiments that\nyield significantly improved image quality. A more complex example that all\nhologram patterns are spatially overlapped is also demonstrated both with\n11\nnumerical simulation and experimental measurement, demonstrating our\nmechanism’s potential for further increasing the information density.\n5. Some textual descriptions, graphical presentations, and character annotations have\nbeen revised for better accuracy and clarity.\nWe also provide below a comprehensive clarification to highlight the significance of\nthe results presented in our study, mainly including the conceptual advances to the\nexisting acoustic hologram mechanisms, the performance improvement compared to\nthe optical counterpart, as well as the possible technical advances in a specific field of\nflexible particle manipulation.\n1. The advances of our mechanism over the existing acoustic hologram methods.\nSince the monolithic acoustic hologram using artificial materials was realized in 2016\n[Nature, 537, 518 (2016)], extensive research has been conducted in this field, which is\nmotivated by its remarkable advantages in shaping sound fields with significantly\nhigher degrees of freedom than conventional phased array transducers at a fraction of\nthe cost and complexity. To date, the technique of sound holography has already\nprovided a new paradigm for research in related areas such as metasurface-enabled\nsound wave three-dimensional (3D) printing [Nat. Commun. 15, 6691 (2024)], acoustic\ntweezers [Sci. Adv. 5, eaav1967 (2019)], and beyond. However, due to the inherent lack\nof physical properties such as polarization and spin, acoustic waves only offer limited\ndegrees of freedom (DoF) for manipulation (such as the commonly-used phase,\namplitude, and frequency), resulting in the few channels available for storing acoustic\nhologram information. This constitutes a fundamental bottleneck in existing acoustic\nhologram technology. Taking the above-mentioned sound wave 3D printing as an\nexample, since the existing metasurface can store the information of only one specific\npattern, different printing task requires attaching holographic metasurfaces with distinct\nphase distributions to the transducer, resulting in limited flexibility and channel capacity.\nIn our study, we introduce a new coding DoF of orbital angular momentum (OAM) and\nrealize a high-capacity and high-security acoustic hologram for the first time, which\nbreaks through the above limitations and is expected to advance the development and\npractical implementation of this technology. Our proposed meta-pixel, as the practical\n12\nimplementation of our mechanism, enables synergistic control of the all-dimensional\ninformation of sound waves, including amplitude, phase, frequency, and OAM, which\nis also beyond attainable with existing mechanisms. In Fig. R1, we list the comparison\nbetween our work and the state-of-the-art acoustic hologram techniques in terms of the\ncontrollable DoFs and multiplexing channel, which visually demonstrates the\nsuperiority of our mechanism over the existing methods.\nFigure R1 | Comparison between our work and the state-of-the-art acoustic hologram\ntechniques in terms of the controllable DoFs and multiplexing channel. Distinct\ncontrollable DoFs are annotated with colored arrows.\n2. The performance improvement compared to the optical counterpart. Despite the\ngrowing research focus on OAM-multiplexing holography within the realms of optics,\nwe need to clarify that our work is by no means a direct and simple translation of the\noptical counterpart. Since the acoustic and optical waves differ in speed by\napproximately five orders of magnitude, the acoustic implementation by analogy with\nthe optical system will be unpractically large-scale. In our work, by utilizing the Green’s\nfunction of a point source in free space and the matrix inversion method, we directly\ncalculate the phase and amplitude mask of the metasurface, without satisfying the\nparaxial approximation condition commonly required for optical methods. Therefore,\nthe transfer function between the image and hologram metasurface established in our\nmechanism remains effective even when the metasurface’s size and imaging distance\nare relatively small. Thanks to this, our mechanism can operate within a compact\n13\nphysical size (e.g., a single pixel occupies a spatial footprint of two wavelengths in our\nwork and spans forty wavelengths for an optical system in Nat. Commun. 10, 2986\n(2019)) and demonstrates a remarkable advantage over the optical counterpart in terms\nof improved information density. To quantitatively evaluate the spatial information\ndensity of multiplexed images, here we introduce a parameter ε defined as :\nε=λN /d\nwhere λ represents the wavelength, N represents the total number of OAM channels,\nand d represents the sampling interval. Fig. R2 shows the calculated parameter ε for\nboth our work and representative optical literatures, which obviously demonstrate the\nperformance superiority of our mechanism.\nFigure R2 | Comparison of spatial information density between our work and some\nrepresentative optical literatures.\n3. Possible technical advances in a specific field of flexible particle manipulation. To\nfurther demonstrate the potential technical advancements enabled by our mechanism,\nwe present here a representative example of OAM- and frequency-dependent flexible\nparticle assembly. The existing acoustic force control methods usually rely on an active\nphased array composed of hundreds or even thousands of individually addressed\ntransducers to generate the designated sound fields, whereas the bulky size and complex\ncontrol systems inevitably result in high cost, low precision, and undesired spatial\naliasing effect, making the device's miniaturization and practicability significantly\n14\nchallenging. Despite the recently emerging acoustic metamaterials with exceptional\nproperties not found in nature offering a more compact and precise way to modulate\nthe sound waves at the subwavelength scale, most acoustic metamaterial-based\nmechanical devices explored so far are designed for manipulating the particle in a fixed\nway, with their functionality unchanged once fabricated. Our mechanism enables\nbreaking through the fundamental limitations of both the traditional phased array\nmethod and emerging metamaterial techniques while maintaining their respective\nadvantages simultaneously. Fig. R3(a) demonstrates a schematic of how our OAM-\nfrequency multiplexed holography enables selective manipulation of particles in a\nsimple yet flexible way. The difference in acoustic pressure distribution on the\nhologram, compared to that in the manuscript (i.e., concentric ring here and focusing\npoint in the manuscript), arises from the employment of ±2nd-order spiral phase here.\nThe incident vortex waves of ±1st-order mode generate holographic images featuring\ndoughnut-shaped pixel configurations, as illustrated in Fig. R3(b). Some spheres with\na density of 30 kg/m3 and a radius of 0.5 mm are initially positioned at the center of\nevery pixel in the imaging plane. Upon excitation by OAM beams of specified\ntopological charge l and frequency, each generated doughnut-shaped pixel produces a\npotential well with the acoustic radiation force being directed toward the center of the\npixel (see Fig. R3(c)), which levitates the corresponding particle at the center.\nConsequently, the particles are arranged to spatially match the pre-designed\nholographic patterns. The simulated particle trajectories in Fig. R4 demonstrate the\neffectiveness of our mechanism in flexible and diversified particle sorting, achieved\nsimply by switching the OAM mode and excited frequency of the incident waves,\nwithout altering the metasurface configuration or relying on any complex control\nsystem. This showcases the significantly improved flexibility and simplicity brought\nby our mechanism. Beyond the particle manipulation demonstrated above, our\nmechanism also holds the promise for boosting advances in diverse applications, such\nas high-capacity encrypted communication and flexible sound wave 3D printing.\nOnce again, we sincerely appreciate the reviewer’s constructive feedback and\ninsightful suggestions. And we hope the above explanations can clarify the novelty and\n15\nsignificance of our work. In light of the reviewer’s report, we have made every effort\nto carefully revise the manuscript. We have given the response individually to each of\nyour remaining comments below.\nFigure R3 | (a) Schematic of the OAM- and frequency-dependent particle sorting.\nIncident vortex waves carrying OAM with topological charges l = ±1 at frequencies f\n1\n(13720 Hz) and f₂ (17150 Hz). g denotes the acceleration of gravity. (b) Simulated\nsound intensity distributions under four different input conditions (f₁, l = ±1; f₂, l = ±1).\n(c) Acoustic radiation force (marked as white arrows) generated by one particular pixel\nof the holographic acoustic field. The direction and size of white arrows denote the\ndirection and relative magnitude of the acoustic radiation force.\nexcitations. Color bar: Instantaneous particle velocities (blue: 0 cm/s → red: 16 cm/s).\nLength and direction of arrows denote the instant magnitude and direction of particle\nvelocities.\nexplanation on this.\nResponse: “TRM” in Fig. 2(a) refers to the time-reversal method, which is expressed\nmathematically as in Eq. (2) in our original manuscript. We apologize for the oversight\nin not defining this abbreviation upon its first occurrence. Now we have adopted an\nalternative matrix inversion method to calculate the phase and amplitude profiles of the\nhologram metasurface, with which the quality of the holographic sound field becomes\nbetter when the transverse dimension of the metasurface is relatively small (see details\nin our response to your Comment 10).\nComment 2: Figure 2a seems to have some unclear meaning. For readers less familiar\nwith the field, it is difficult to understand the relationship between these jumbled\ndiagrams. The authors have to clarify this aspect.\nResponse: Thanks for your important suggestion. We have redrawn Fig. 2(a) and\nprovided more detailed descriptions to demonstrate the relationship between different\nsubgraphs in the updated version of the manuscript, which we hope could help the\nreader better understand the processes of image encoding in our proposed OAM- and\nfrequency-dependent acoustic hologram mechanism, as suggested by the reviewer. The\ncomparison of the original and revised diagrams is shown in Figs. R5 and R6, wherein\nthe changes mainly include the following points:\n1. We have split original Fig. 2(a) into Fig. 2(a) and (b) to more clearly demonstrate\nthe workflow of spatial sampling of the target images as well as the derivation of\nOAM- and frequency-multiplexed acoustic hologram.\n2. We have added some necessary annotations, such as “Step 1”, “Step 2”, and “Step\n3”, to clarify sequential procedures and facilitate comprehension of each processing\nstage.\nBesides, some annotations have also been added in the textual description to\nestablish a direct correspondence between the relevant content and the associated\nfigures.\nmultiplexed hologram.\nFigure R6 | Revised diagrams of (a) spatial sampling of the target images and (b)\nderivation of OAM- and frequency-multiplexed hologram.\nComment 3: It seems also that the subscript of H in the caption of Fig. 3b should be\nsuperscripted\nresulting from our oversight, which has been corrected in the revised manuscript.\nComment 4: Confusion over the use of parameters in Figure 3b in the main text and in\nFigure 2 in the supplementary material, where H in Figure 3b is L in Figure 2?\nResponse: We apologize for any confusion caused. We have refined the parameter\nannotations of the redesigned structure in a more consistent and concise manner.\nComment 5: With the same cross-sectional area of the tube in Fig. 1b of the\nsupplementary material, why does the superscript of p change?\nResponse: The change of the superscript in p (e.g., p(1) and p(2)) is to differentiate the\nacoustic pressure at the inlet and outlet of the waveguide with different phases. When\na time-harmonic sound wave propagates in a waveguide with the same cross-section,\nits amplitude remains unchanged, yet the phase evolves with propagation distance. In\nthe original Fig. S1(b) (i.e., Fig. R7 below), the sound field in the tube is composed of\nthe forward and backward waves,\np= p e−jkx + p ejkx\nf b\nwhere the time-dependent term ejωt is omitted, k is the wave number, and x is the\npropagation distance. At the inlet (x=0 ), p(1) = p and p(1) = p . And at the outlet\nf f b b\n(x = L), p(2) = p e−jkL and p(2) = p ejkL. The complex amplitude P(1) (P(1)) and P(2)\nf f b b f b f\n(P(2) ) differ by a phase factor ejkL (e-jkL ). To avoid misunderstanding, we have re-\nb\nderived the transfer function directly from the whole structure rather than from each\ncomponent of the meta-pixel subunits. The updated theoretical derivations are shown\nin Section 4 in the revised supplementary materials.\nComment 6: What is the function of Hamp and Hphase ?\nResponse: In our previously designed meta-pixel, the parameter H refers to the\nphase\nthickness of the phase modulation component in the subunit and does not influence the\nmeta-pixel’s manipulation capabilities. As for another parameter of H (i.e., the\namp\nthickness of the impedance-mismatching tube), it is designed to be about one-quarter\nof the wavelength to ensure full-range amplitude modulation without affecting the\ntransmission phase, as will be elucidated in the following. Our designed impedance-\nmismatching tube, analogous to an optical aperture diaphragm, is a typical acoustic\ncontraction tube. According to the theory of sound waves propagation in the waveguide,\nwhen the thickness of such a contraction tube is one-quarter of the wavelength, the\ntransmission is solely determined by the ratio of the cross-sectional area of the\ncontraction tube (S ) to that of the main tube (S ), mathematically expressed as\n1 2\nt =4/ ( S /S +S /S )2 , while the phase is locked by resonance. This allows us to tune\n1 2 2 1\nthe amplitude of the transmitted wave within the range of 0-1 without influencing the\nphase. To verify this, we provide in Fig. R8 the comparison of transmission amplitude\nand phase responses versus structure parameters D\namp\nfor three different H\namp\n. The\nresults show that when H deviates from one-quarter of the wavelength (left and right\namp\npanels), the phase uniformity degrades and the amplitude modulation range becomes\nsignificantly limited.\nAs pointed out by the reviewer in Comment 7, our initial design exhibited\n21\nincomplete amplitude-phase decoupling. To address this issue, we have redesigned the\nmeta-pixel and realized amplitude-phase decoupled modulation at each frequency\ncomponent. The parameter H (relabeled as H ) of the new structure is a key\namp 4\nparameter that critically affects the amplitude–phase decoupled capability of the meta-\npixel. It is within the two-dimensional parameter space defined by H and H that we\n4 5\nseek the decoupling point of the meta-pixel (see details in our response to your next\nComment). We have added some discussions in the revised manuscript to explain the\nfunction of H (H ).\namp 4\nFigure R8 | Transmission amplitude (blue curves) and phase (yellow curves) responses\nof the subunit of meta-pixel versus structure parameter D at 17150 Hz when H is\namp amp\n1 mm, 5 mm, and 9 mm, respectively.\nComment 7: Combining Fig. 4 and Fig. S3, it can be seen that the unit is not decoupled\ncontrol of amplitude and phase but searches for suitable combinations to achieve\namplitude-phase traversability, is this somehow limiting the degrees of freedom of\nholographic imaging?\nResponse: Thank you for pointing out this important question. As indicated by the\nreviewer, our previously designed meta-pixel can not achieve decoupled control of\namplitude and phase. Its underlying physical cause can be understood as below: despite\nthe internal Helmholtz resonator (HR) of meta-pixel providing strong frequency\nselectivity, the HR embedded in subunit S (S ) cannot effectively block acoustic waves\n1 2\nof frequency f (f ) from entering the respective regions; instead, they merely interrupt\n2 1\n22\nthe propagation path internally, as evidenced by the amplitude distribution of the\ninternal sound field through the unit cell in Fig. 3(d) in the original manuscript. As a\nresult, inevitable coupling arises between the two subunits at both the input and output\nends, leading to the entanglement of amplitude and phase modulation in the transmitted\nwave. To overcome this challenging issue, we propose a new structural design to give\nrise to the amplitude–phase decoupled modulation at each frequency component, which\nis beyond attainable with the existing mechanisms. We theoretically derive the acoustic\nresponse of the new structure and determine the structural parameters that allow\ndecoupled modulation, whose performances are also verified both numerically and\nexperimentally. In the redesigned meta-pixel, the original embedded HR is removed\nand two frequency-specific resonator modules are attached on either side of the subunits,\nas illustrated in the Figs. R9(a) and (b). The resonant cavities on both sides not only\nprovide strong frequency selectivity (see Fig. R9(c)), but also ensure that acoustic\nwaves at the specific frequency interact exclusively with their corresponding subunit,\nwithout penetrating into the other (see Fig. R9(d)). This design enables complete\ncrosstalk-free operation between subunits and allows independent amplitude and phase\ncontrol for each frequency component of the acoustic wave. We theoretically derive the\ntransfer function of the redesigned subunit (see Sec. 4 in the revised supplementary\nmaterials) and reveal that the parameters H 1,2 and H 1,2 play a pivotal role in\n4 5\ninfluencing the decoupling performance. To identify the optimal parameter\ncombinations that allow decoupled amplitude-phase modulation, we define a parameter\nη to characterize the degree of decoupling:\nη=var[A(H1,2)]\n6\nwhere var(A) refers to the variance of A, and A is the different transmission amplitude\nresponse of the meta-pixel when varying H1,2 (i.e., parameter for phase tuning) with\n6\nother structural parameters remaining constant (see Supplementary Table l for detailed\nparameter settings). The parameter η reflects the interplay between the phase and\namplitude modulation, and a smaller value of η indicates better decoupling capacity.\n5\nnamely ( H1,H1 ) =( 7.2mm,4.1mm ) for subunit 1, and ( H2,H2 ) =( 6.5mm,6.2mm )\n4 5 4 5\nfor subunit 2. To verify the effectiveness of our scheme, we numerically calculate the\namplitude-phase modulation performance of the meta-pixel at the determined\ndecoupling point as well as a reference point. The typical results in Fig. R11 show that\nthe newly proposed structure achieves nearly complete decoupling between amplitude\nand phase control at the decoupling point. This allows us to independently control the\ntwo distinct DoFs (amplitude and phase) by using only a single structural parameter for\neach, which substantially reduces the complexity of the metasurface design and helps\nimprove the robustness of our mechanism against fabrication errors. It is worth noting\nthat the slight fluctuations observed in the phase distribution may result from weak\ncontributions of parameters other than H1,2 and H1,2, which are not considered in the\n4 5\ncurrent study for the sake of simplifying the theoretical calculations and structural\ndesign. In the revised manuscript, we have utilized this redesigned meta-pixel as the\nimplementation of our hologram metasurface and the relevant contents have been\nupdated correspondingly.\nplane (c) Transmission frequency response of redesigned meta-pixel from 12 kHz to 20\nkHz (d) Amplitude distribution of internal sound field through the new unit-cell.\nFigure R10 | The magnitude distribution of η withinH1,2–H1,2parameter space.\n4 5\ncalculated decoupling point as well as (b) a reference point.\nComment 8: The cells in Figures S2 and 3b seem to be symmetrical structures, but why\nis there an extra cavity on the left side in Figure S1f?\nResponse: The left side in Fig. S1f you mentioned is not a real cavity but refers to the\nincident region of the sound wave outside the metamaterial unit. In the updated version\nof the manuscript, to avoid misunderstanding, we have redrawn the 2D cross-sectional\nschematic of the redesigned structure in a more explicit way (see Fig. R12).\n0 and Layer 12 are the incident and output regions of the acoustic wave outside the\nmeta-pixel, respectively. (b) 3D diagram of Layer 0 and Layer 1.\nComment 9: Careful observation of Fig. S2 reveals that the widths between the La and\nLc cavities are different, but the authors have directly replaced this with a 2D structure\nin Fig. S1, so if this inconsistency in width has no effect on the model, the authors\nhave to explain this in details.\nResponse: In our previous theoretical derivations, since the area variations between\ndifferent cavity sections are relatively mild, it is reasonable to consider only the plane\nwave modes. Consequently, the transfer matrix depends solely on the areas of cavities,\nirrespective of their shape. And such a plane wave approximation is valid, as proved by\nthe high consistency between the theoretically calculated and numerically simulated\nresults. To achieve decoupled amplitude-phase modulation, we redesign the structure\nof the meta-pixel as we have demonstrated above. As for this new structure, it has more\nvariable cross-sections with larger cross-sectional area ratios. Thus, the higher-order\nmodes should be considered within some specific regions, for which the inconsistency\nin width would influence the model’s acoustic response. Therefore, we have developed\na new theoretical model based on the practical 3D structure, whose effectiveness is also\n27\nverified by comparing the analytical results with the numerical ones. (See Sec. 4 for\ndetails in the revised supplementary materials)\nComment 10: Why did the authors use discrete pixel points for the hologram?\nCompared to either existing optical holography or acoustic holography, the quality of\nproposed holograms in the manuscript is poor.\nResponse: We thank the reviewer for raising these important questions. The discrete\npixel points we use for the hologram are to ensure the OAM selectivity of our\nmechanism. By redesigning the phase and amplitude profile of the metasurface based\non an alternative matrix inverse method, we further improve the quality of the proposed\nholograms. Some new experiments are also conducted, which yield significantly\nimproved image quality. The detailed responses to your questions are provided below.\nThe reason we use discrete pixel points for the hologram is to ensure the OAM\nselectivity of our mechanism. The OAM selectivity means that only emitting an OAM\nbeam with a specific topological charge (l) can restore the information hidden in the l-\nth order OAM channel. When the mode order of the emitting beam is not equal to l, the\nholographic pixels appear as concentric diffraction rings whose radii increase with the\ntopological charge, which can be regarded as an artifact (see Fig. R13). Despite the\npresence of such artifacts, their null centers and diffuse energy distributions can\nmaintain a significant contrast between themselves and the target holographic image,\nthereby preserving the selectivity of the OAM mode. However, if spatial sampling is\nnot performed, or if the sampling interval is too small, these donut-shaped artifacts may\noverlap with each other, resulting in the constructive interference of artifacts as well as\nthe degradation of OAM selectivity, as illustrated in a specific example shown in Fig.\nR14. Theoretically, a larger sampling interval leads to better OAM selectivity. However,\nincreasing the sampling interval also results in a reduction of spatial pixel density. To\ninvestigate their relationship, here we introduce two parameters of information density\nε and OAM selectivity coefficient η defined as\nm,n\n28\nε=λ/d\nη = Il=m Il=n,\nm,n\nmax max\nwhere d is the sampling interval, λ is the wavelength, and Il is the maximum\nmax\nintensity of holographic pressure patterns for l-th-order OAM incidence. Considering a\nspecific situation where a particular hologram image of a point array with interval d is\nencoded in the -1st-order OAM channel by multiplying the hologram with a 1st-order\nhelical phase mask, the sound field at the imaging plane z = 16 λ can be calculated for\ncases of illuminating the metasurface with 1st-order and -1st-order OAM beams at\ndifferent d. The curve depicting the relationship between the information density ε and\nthe OAM selectivity coefficient η-1,1 is shown in Fig. R15, highlighting a remarkable\ntrade-off that can be summarized as the following three points: 1. for small sampling\nintervals, densely packed pixels with high information capacity tend to cause mode\noverlap, leading to degradation in OAM selectivity (the left inset); 2. rational sampling\nintervals can strike a balance between preserving OAM selectivity and maintaining\nrelatively high information density (the middle inset); 3. overly sparse sampling\nenhance OAM selectivity but result in the reduction of spatial information density (the\nright inset). Therefore, a sampling interval of 2 λ is chosen in our scheme to prevent\nspatial overlap of the output while retaining relatively high information density.\nRegarding your other concern about the seemingly poor imaging quality of the\nproposed holograms, we need to clarify that this is not due to any intrinsic limitation of\nour mechanism, but rather results from the compact physical size of the metasurface\nthat is several orders of magnitude smaller than the optical counterparts. In our original\nmanuscript, we employed a time-reversal approach to retrieve the required phase and\namplitude profiles of the metasurface by treating the target holographic pattern as the\nsource. Theoretically, as governed by the time-reversal symmetry of wave equations,\nperfect reconstruction of a holographic acoustic field necessitates access to\nomnidirectional spatial wavevector components. However, such full-space wavevector\ninformation cannot be captured by a metasurface with limited lateral dimensions,\nthereby leading to the observed degradation in imaging quality. As the metasurface size\n29\nincreases, the holographic reconstruction quality improves accordingly, as\ndemonstrated in Supplementary Fig. 13. To further enhance the imaging quality of our\nmechanism within a compact space, here we capitalize on an alternative matrix inverse\nmethod, which is a computational technique used to reconstruct or design acoustic\nfields by solving a system of linear equations that relate the source distribution to the\ndesired sound field. This method establishes a direct mapping between the source and\nthe holographic pattern, exhibiting significant advantages in terms of flexibility and\nhigh-quality reconstruction. In the revised manuscript, we have conducted new\nexperiments using this matrix inverse method and the redesigned meta-pixel, which\ndemonstrates significantly improved holographic quality in comparison with the\nprevious results. We re-fabricated the metasurface sample via 3D printing (Fig. R16(b)),\nwhose amplitude-phase distribution was calculated based on the Green's function and\nmatrix inversion method (Fig. R16(a)). Acoustic field scanning was performed at the\nimaging plane using a stepper motor-controlled setup (overview in Fig. R16(c)). The\nmeasurements were taken at every 4 cm for two-dimensional plane scanning and at 1\ncm for one-dimensional profile scanning. The resulting experimental results (Fig.\nR16(d)) demonstrate significantly improved hologram quality and excellent agreement\nwith the simulated ones, both for two-dimensional sound intensity distribution and one-\ndimensional intensity profiles. The experimental results appear to outperform the\nsimulated ones because, for experimental convenience, we adopted a relatively large\nspatial scanning interval, thereby omitting some artifacts not located at detection points.\nIn our current design, the image pairs (1 and 2; 3 and 4) are spatially separated to\nminimize the impact of holographic artifacts. To demonstrate the possibility of our\nmechanism for further enhancing information capacity, we showcase here a more\ncomplex example in which all target images fully overlap on the same imaging plane.\nWe select four English letters as the targeted patterns. And the simulated and\nexperimental results in Fig. R17 still exhibit good quality, validating the mechanism’s\nindependence from the spatial arrangement of the multiplexed images.\nIn the revised manuscript, we have replaced the original structural designs and\nexperimental results with updated versions. In-depth discussions on the significance of\n30\nspatial sampling are also provided in the Supplementary Materials.\nFigure R13 | Holographic images and pixel-wise energy distribution under different\nOAM mode incidence. The holography information is loaded in the -1st-order OAM\nchannel by multiplying the hologram with the 1st-order helical phase mask.\nThe holography information is encoded in the -1st-order OAM channel by multiplying\nthe hologram with the 1st-order helical phase mask.\ncoefficient η . The holography information is loaded in the -1st-order OAM channel\n−1,1\nby multiplying the hologram with the 1st-order helical phase mask. The white line in\nthe insets denotes the wavelength.\nFigure R16. | (a) Distribution of amplitude and phase of the hologram at 13720 Hz (top)\n33\nand 17150 Hz (bottom). (b) Photograph of the fabricated metasurface sample. The size\nof the sample is 0.4×0.4×0.06 m3. (c) Experimental setup in the anechoic room. (d) Left\npanel: Simulated two-dimensional acoustic intensity distribution under different OAM\nand frequency incidences; Middle panel: measured two-dimensional acoustic intensity\npatterns under different OAM and frequency incidences; Right panel: simulated\n(dashed line) and measured (solid line) one-dimensional acoustic intensity profiles\nalong dashed lines marked in the left panel.\nFigure R17 | Simulated (top) and measured (bottom) holographic intensity patterns of\nletters A, B, C, and D, wherein multiple patterns are spatially overlapped.\nThis work presents a novel approach to utilizing orbital angular momentum (OAM) and\nfrequency as two coherent secret keys to achieve high-capacity encrypted sound\nholography. As a practical implementation, the authors design a multiplexing acoustic\nmetasurface composed of meta-pixel structures, enabling simultaneous modulation of\nmultiple sound dimensions. Due to the high degree of freedom (DoF) characteristic, the\ninformation capacity of the holographic sound field is significantly enhanced, while\nconfidentiality is ensured by the orthogonality of OAM modes and frequencies.\nThe manuscript is well-structured, with a clear presentation of results spanning\ntheoretical modeling, simulations, and experimental realization. While the\ndemonstrated holographic images of four Arabic numerals are relatively simple, the\nfirst successful demonstration of a high-capacity hologram leveraging the new DoF of\nOAM in acoustic systems is highly significant. This work has crucial implications for\nvarious applications, including modern communication technologies. Therefore, I\nrecommend the publication of this manuscript in Nature Communications.\nResponse: We wish to sincerely thank the reviewer for recommending the publication\nof our work in Nature Communications and for offering valuable suggestions that have\nhelped us to improve our manuscript. In light of the reviewer’s report, we have made\nevery effort to carefully revise the manuscript. We have given the response individually\nto each comment below.\nComment 1:\nThe transverse dimensions of the metasurface are set to 30 cm. What is the rationale\nbehind this choice, and how does this size influence the system’s performance\nparameters? A more extensive discussion on this aspect would be beneficial.\nResponse: Thank you for pointing out this important question. The reason we set the\ntransverse dimensions (L) of the metasurface to 30 cm is to provide sufficient degree of\n35\nfreedom for realizing a high-quality hologram, while simultaneously maintaining the\noverall compactness of the system as much as possible. In order to quantitatively\nestimate the influence of transverse dimensions of the metasurface on the system’s\nperformance, here a commonly used parameter of correlation coefficients C describing\nthe similarity between two images is introduced [e.g., Nat. Commun., 9, 1632 (2018)],\n∑∑(A −A)(B −B)\nmn mn\nC = m n\n∑∑(A −A)2∑∑(B −B)2\nmn mn\nm n m n\nwhere A and B are the data matrices of the two images, A and B are the mean values\nof the elements in the matrices A and B, respectively. A higher correlation coefficient C\nbetween the target image and the holography pattern indicates better imaging quality.\nWe numerically calculate the dependence of the correlation coefficient C on the\nparameter L and plot the resulting curve in Fig. R1. From the results, we can observe\nthat the imaging quality increases with L and reaches a plateau around 0.9 at L = 30 cm,\nbeyond which the enhancement becomes marginal. Therefore, we set = 30 cm as the\ntransverse dimension of the metasurface to balance high-fidelity holo𝐿𝐿g raphy with the\ndevice’s simplicity and compactness. In the updated version of the manuscript, we have\nadded some discussions on this issue, following the reviewer’s suggestion.\nFigure R1 | The numerically calculated correlation coefficient as a function of L. Insets:\nthe reconstructed holographic patterns for two specific cases of L=20 cm (left) and\nL=40 cm (right).\nThe theoretical derivation of the meta-pixel transfer function in the Supplementary\nMaterials is correct but overly complex, particularly in Fig. S1. Would it be possible to\nderive the transfer function directly from the whole structure for clarity?\nResponse: Thank you for this important suggestion. To enhance clarity, we have re-\nderived the transfer function directly from the whole structure rather than from each\ncomponent of meta-pixel subunits, as suggested by the reviewer. The updated\nderivation is shown in Section 4 in the revised supplementary materials, which helps\nunderstand the structure-to-function relationship more clearly.\nComment 3\nThe experimental section lacks sufficient detail. Please expand this section, including a\nmore thorough explanation of how the speaker array generates the incident vortex\nsound beams with a designated mode order.\nResponse: We thank the reviewer for offering this suggestion. Our experiment is\nperformed in an anechoic room to eliminate the disturbance of reflected sound. The\nsample is fabricated via 3D printing technology (0.1 mm in precision) using\nphotosensitive resin whose mass density and sound speed are 1400 kg/m3 and 2000 m/s,\nrespectively. At the emitting end, eight HiVi B1S loudspeakers are arranged\naxisymmetrically along a 12 cm-radius circle and mounted onto an acrylic glass plate,\nforming a single-ring circular phased array. A single-chip microcomputer,\nArduino Mega 2560, is used to generate the drive signal of each loudspeaker. To\nemit an acoustic vortex beam with l-th order OAM mode, the phase of the drive\nsignal to each speaker should be φ = lθ, where θ is the azimuthal angle between\nthe speaker and the reference. The metasurface is placed in alignment with the\nsingle-ring circular phased array at a relative distance of 1.2 m. At the imaging plane, a\nmicrophone (Brüel & Kjær type-4961) is mounted on a 3D stepping motor to measure\nthe pressure field. The scanning step is set to sampling interval d for the two-dimensional\nsound field measurement and\nto 1 cm for the one-dimensional acoustic pressure profile measurement, respectively.\nThe signals are recorded with a Brüel&Kjær PULSE3160-A-042 2-channel analyzer.\nIn the updated version of the manuscript, we have expanded the experimental section\nby providing more detailed descriptions.\n37\n2. Detailed changes made to the manuscript are the following:\n(1) Page 2, line 19: We have added a brief statement to highlight that our mechanism\nis independent of the spatial arrangement of multiplexed images: “regardless of\nwhether the patterns are spatially separated or overlapped”.\n(2) Page 3, line 11: We have added some discussions on the advantages of acoustic\nmetamaterials over phased arrays: “In stark contrast, the recently emerging\nacoustic metamaterials and metasurfaces with exceptional properties unattainable\nin nature offer a solution to mold the wavefront into almost arbitrary profile with\nsignificantly higher degrees of freedom (DoF) than conventional phased arrays at\na fraction of the cost and complexity”.\n(3) Page 4, line 3: We have added a brief statement to emphasize the limitation of the\nexisting sound hologram mechanism: “resulting in the few channels available for\nstoring acoustic hologram information”.\n(4) Page 4, line 27: We have added a statement about the amplitude-phase decoupled\nfunction of the designed meta-pixel: “a unit cell of meta-pixels with amplitude-\nphase decoupled modulation function at arbitrary frequency components is\nsystematically designed”.\n(5) Page 5, line 3: We have added the description about the additional hologram\nexample: “Two proof-of-concept examples of reconstructing four Arabic numerals\nand four alphabet letters are demonstrated both numerically and experimentally”.\n(6) Page 6, line 14: We have added a brief explanation about the reason for spatial\nsampling: “For a specific targeted holographic image, a pre-processing of spatial\nsampling is performed to ensure the OAM selectivity of our mechanism (Fig. 2(a)),\nwhich is commonly used in the holography technique48, 49. If spatial sampling is\nomitted or the sampling interval is too small, donut-shaped artifacts may overlap\nand cause constructive interference, which in turn degrades OAM selectivity (see\nSupplementary Note 1 for more details). Here, we discretize the holographic image\n39\ninto N pixels with a sampling interval of 2λ, which can effectively prevent spatial\noverlap of the output while preserving high information density”.\n(7) Page 7, line 8: We have provided more details about the derivation of phase and\namplitude profiles of the metasurface: “The metasurface is composed of M units.\nThen, the relationship between the acoustic response of the metasurface and the\ntargeted holographic image can be established by the following matrix equation:\nG G  G pf, l  Al exp(jΦl) \n11 12 1M 1 1 1\n    \nG G  G pf, l Al exp(jΦl )\n 21 22 2M  2 2  =  2 2  (1)\n         \n    \nG G  G pf, l Al exp(jΦl )\\\nN1 N2 NM M N N\nexp[j(−2πfr c )]\nwhere G = nm 0 is the Green’s function of a point source in free\nnm 4πr\nnm\nspace, r is the distance between the n-th image pixel and the m-th unit of the\nnm\nmetasurface, and c0 is the sound speed in air. Through matrix inversion of Eq. (1),\nwe can obtain the complex amplitude profile pf, l required for generating the\ncorresponding holographic image (Step 1 in Fig. 2(b)). By encoding pf, l with an\nadditional helical phase distribution of lθ (θ is the azimuth angle and l is the\ntopological charge of the OAM mode), the hologram metasurface is imprinted with\nOAM and frequency selectivity (Step 2 in Fig. 2(b)), which means that only emitting\nan OAM mode with inverse topological charge (-l) and corresponding frequency\n(f) can decode and restore the information hidden in the hologram and\nconsequently recover the fundamental spatial mode with a relatively stronger\nintensity distribution in each pixel of the holographic image. Then, the OAM- and\nfrequency-multiplexing acoustic hologram can be realized by superposing multiple\nOAM- and frequency-selective responses on the metasurface (Step 3 in Fig. 2(b)).\nBased on this, the general transfer function of the metasurface can be written as\nfollows\n0\nf, l\nwhere f is the frequency of the incident wave, and δis the Kronecker delta\n0\nfunction”.\n(8) Page 8, line 6: We have added a brief explanation of why our mechanism does not\nrely on paraxial approximation: “In the above, by utilizing the Green’s function of\na point source in free space and the matrix inversion method, we directly calculate\nthe phase and amplitude mask of the metasurface, without satisfying the paraxial\napproximation condition. Therefore, the transfer function between the image and\nhologram metasurface established in our mechanism remains effective even when\nthe metasurface’s size and imaging distance are relatively small. Thanks to this,\nour mechanism can operate within compact physical size and demonstrates a\nremarkable advantage in terms of improved information density”.\n(9) Page 8, line 23: We have added some discussions on the advantage of our\nmechanism over traditional phase-only devices in achieving high-quality hologram\nreconstruction: “Besides, our mechanism offers a remarkable advantage over\ntraditional phase-only devices in achieving high-quality hologram reconstruction,\nas demonstrated by a representative example in Supplementary Note 2. In our\ntheoretical design, OAM selectivity and imaging encoding are accomplished\nthrough multiplying the hologram by an ideal helical phase. This means that\nretrieving the image’s information requires illuminating the metasurface with a\nperfect vortex beam”.\n(10) Page 8, line 25: We have added some discussions about the robustness of our\nmechanism: “In our theoretical design, OAM selectivity and imaging encoding are\naccomplished through multiplying the hologram with an ideal helical phase. This\nmeans that retrieving the image’s information requires illuminating the\nmetasurface with a perfect vortex beam. However, the vortex beam experiencing\n41\npropagation in the real-world system cannot hold a perfect pattern. To inspect its\ninfluence, we further analyze the tolerance of our mechanism to phase disturbance\ntypically induced by diffraction, medium inhomogeneities, volume scattering, and\nbeyond, and the results in Supplementary Note 3 demonstrate the high robustness\nof our mechanism against these factors”.\n(11) Page 9, Figure 2: We have redrawn the schematic diagram in Figs. 2(a) and (b).\n(12) Page 10, Caption of Figures 2(a) and (b): We have updated the caption of Figs.\n2(a) and (b): “(a) Pre-processing of spatial sampling on the target images. (b) The\nschematic diagram of the derivation of phase and amplitude profiles of the\nmetasurface for realizing the OAM- and frequency-multiplexed acoustic hologram.\nThe formula x=G−1y indicates matrix inversion to obtain the amplitude-phase\nprofile of the metasurface, where x and y are the complex amplitude profiles of the\nmetasurface’s acoustic response and target image, and G is Green’s function.”\n(13) Page 10, line 21: We have updated the description about the redesigned meta-\npixel: “The proposed meta-pixel, whose geometry is shown in Fig. 3(a), is\njudiciously designed to contain multiple subunits and each subunit consists of three\nfunctional parts, namely, an amplitude modulator, a phase modulator and two\nspectral filters, whose cross-sections are remarked by red, blue, and black outlines,\nrespectively, as shown in Fig. 3(b). The length, width and height of the i-th layer\nof subunit 1 and subunit 2 are W1,2, D1,2, and H1,2, respectively”.\ni i i\n(14) Page 10, line 28: We have added more discussions about the amplitude-phase\ndecoupling performance of the meta-pixel: “In such a design, we introduce\ncontrollable impedance mismatch to realize full-range amplitude modulation,\nwhich is implemented by adjusting the width D1,2 of the impedance modulation\n4\nlayer. Furthermore, to simultaneously control the transmitted phase within the\nsame spatial dimension, the effective refractive index of the units is precisely\ntailored by lengthening or shortening the equivalent propagation distance of the\nzig-zag pipe of the phase modulator (i.e., varying H1,2). In addition, the spectral\n6\n42\nfilters on two sides ensure that acoustic waves at the specific frequency interact\nexclusively with their corresponding subunit, without penetrating into the other,\nthereby providing strong frequency selectivity. However, the sound waves\npropagating within these three parts are mutually coupled, resulting in difficulties\nin traversing the full parameter space of amplitude-phase at different frequencies\nin a decoupled way. To address this, we theoretically derive the transfer function\nof the meta-pixel and, based on it, determine the decoupled points within two-\ndimensional parameter spaces defined by H1,2 and H1,2, at which the meta-pixel\n4 5\nenables decoupled amplitude-phase modulation at each frequency component (see\nSupplementary Note 4 for detailed theoretical derivations and the procedure for\ndetermining the decoupling points). The specific geometrical parameters of the\nmeta-pixel are listed in Supplementary Table 1”.\n(15) Page 12, Figure 3: We have updated Fig. 3 according to the redesigned meta-pixel.\n(16) Page 12, Caption of Figure 3(b): We have provided clearer annotations for the\ngeometrical parameters of the meta-pixel: “Cross-sectional view of S and S .\n1 2\nW1,2,\ni\nD1,2,\ni\nand H1,2\ni\nare the length, width and height of the i-th layer of S and S ,\n1 2\nrespectively. Specifically, H1,2 is the total length of the zig-zag pipe in Layer 6”.\n6\n(17) Page 13, Line 1: We have added some discussions on the ability of our designed\nmeta-pixel in achieving nearly decoupled control of the phase and amplitude DoF:\n“The numerically simulated results in Fig. 4(a) demonstrate the ability of our\ndesigned meta-pixel in achieving nearly decoupled control of the phase and\namplitude DoF, enabling independent tuning of each across its full range by\nvarying only a single structural parameter. This substantially reduces the\ncomplexity of the metasurface design. The slight fluctuations observed in the\namplitude and phase distributions may result from weak contributions of\nparameters other than H1,2 and H1,2, which are not considered in the current\n4 5\nstudy for the sake of simplifying the theoretical calculations and structural design”.\n(18) Page 13, Figure 4: We have updated Fig. 3 according to the redesigned meta-pixel.\n43\n(19) Page 14, Line 16: We have added a brief description about the spatial arrangement\nof the holographic images: “The four images are distributed across two distinct\nregions on the same imaging plane, with numerals 1 and 3 in one region and\nnumerals 2 and 4 in the other”.\n(20) Page 14, Line 24: We have added a brief explanation about the choice of the\nmetasurface’s transverse size: “This number of unit cells provides sufficient DoFs\nto enable a high-quality hologram while simultaneously preserving the system’s\ncompactness”.\n(21) Page 15, Line 18: We have added some descriptions about the additional\nhologram example: “Besides, we also showcase in Supplementary Note 7 a more\ncomplex case in which all target images fully overlap on the same imaging plane.\nWe select four alphabet letters as the targeted patterns. The simulated and\nmeasured results in Supplementary Fig. 14 still exhibit good quality, validating the\npotential for further increasing information capacity and the mechanism’s\nindependence from the spatial arrangement of the multiplexed images”.\n(22) Page 15, Line 23: We have added some discussions to demonstrate the potential\ntechnical advancements enabled by our mechanism: “Moreover, to demonstrate\nthe potential technical advancements enabled by our mechanism, we present in\nSupplementary Note 8 a representative example of OAM- and frequency-dependent\nparticle assembly, which manifests the significant advantages of our methodology\nin flexible and diversified sound force manipulation. Beyond this, our mechanism\nalso holds the promise for boosting advances in diverse applications, such as high-\ncapacity encrypted communication and flexible sound wave 3D printing”.\n(23) Page 16, Figure 5: We have updated Fig. 5 based on the new experimental results.\n(24) Page 16, Caption of Figure 5: We have updated the caption of Figs. 5(b)-(d).\n(25) Page 17, Line 24: We have provided more details for the experimental section:\n“The metasurface samples are manufactured via 3D printing technology (0.1 mm\nin precision). At the emitting end, eight HiVi B1S loudspeakers are arranged\naxisymmetrically along a 12 cm-radius circle and mounted onto an acrylic glass\n44\nplate, forming a single-ring circular phased array. A single-chip microcomputer,\nArduino Mega 2560, is used to generate the drive signal of each loudspeaker. To\nemit an acoustic vortex beam with l-th order OAM mode, the phase of the drive\nsignal to each speaker should be φ = lθ, where θ is the azimuthal angle between\nthe speaker and the reference. The metasurface is placed in alignment with the\nsingle-ring circular phased array at a relative distance of 1.2 m. At the imaging\nplane, a microphone (Brüel & Kjær type-4961) is mounted on a 3D stepping motor\nto measure the pressure field. The scanning step is set to sampling interval d for\nthe two-dimensional sound field measurement and to 1 cm for the one-dimensional\nacoustic pressure profile measurement, respectively. The signals are recorded with\na Brüel&Kjær PULSE3160-A-042 2-channel analyzer. For two-dimensional\nspatial scanning, the initial scan position is aligned with the center of the first pixel\nof the hologram image”.\n(26) Page 19, References: We have added references 10 and 29. The numbers of the\nrest references have been updated accordingly.\n(27) Supplementary Note 1: We have provided some detailed discussions and\ncalculations to explain the reason we perform spatial sampling processing on the\nholographic image.\n(28) Supplementary Note 2: We have provided a representative example to\ndemonstrate the advantage of our mechanism over traditional phase-only devices.\n(29) Supplementary Note 3: We have added some detailed discussions and\ncalculations to analyze the tolerance of our mechanism to phase disturbance\ninduced by diffraction, medium inhomogeneities, and volume scattering.\n(30) Supplementary Note 4: We have provided detailed theoretical derivations of the\ntransfer function of the meta-pixel as well as the procedure for determining the\ndecoupling points.\n(31) Supplementary Note 7: We have presented a more complex hologram case in\nwhich all target images fully overlap on the same imaging plane.\n45\n(32) Supplementary Note 8: We have demonstrated a representative example of OAM\nand frequency-dependent particle assembly.\n(33) We have improved the English of the manuscript.\n46"}, {"round": 2, "reviewer_comments": {"reviewer_1": "the authors addressed the comments and revise the manuscript well. I agree it for publication.", "reviewer_2": "The authors have done good job to address my previous comments. I think the manuscript could be published in Nature\nCommunications.", "reviewer_3": "The authors have addressed my comments. The article can be accepted."}, "author_response": ""}]

Decision letter