Frequency multiplexing with dispersion-engineered elastic metasurfaces

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

Abstract
Distinguishing the frequency components of elastic waves is essential for ensuring the stability of mechanical systems. Herein, a frequency-multiplexed elastic metasurface is introduced, utilizing a dispersion engineering. The proposed dispersion-engineered elastic metasurface multiplexes frequency-dependent phase profiles, enabling frequency-selective wavefront tailoring. The phase profiles of the meta-atoms, derived from Kirchhoff–Love plate theory, are controlled solely by the plate thickness, eliminating the need for a typically required extensive meta-atom library or complex optimization. Integrating the metasurface with a piezoelectric element converts the focused elastic energy into an amplified electrical signal. Therefore, the magnitude of extracted energy alone reveals the underlying spectral information through spatial information. Our finding present the first design methodology for a dispersion-engineered elastic metasurface, overcoming limitations of conventional elastic metasurfaces by encoding multiplex information in a passive framework. Accordingly, the proposed dispersion-engineered elastic metasurface can be incorporated into various plate-based mechanical systems for next-generation technologies.

Introduction
As diffractive elements that enable wave phenomena not readily found in nature through wave-matter interactions, metasurfaces have recently attracted significant attention, in photonics, acoustics, and elastics. As the dimensionally reduced counterpart of metamaterials, metasurfaces offer a promising platform to replace existing system components or imparting additional functionalities to the conventional systems, exhibiting immense potential across diverse domains.

However, conventional metasurfaces have a key limitation: the wave manipulation capabilities and operational frequency range are intrinsically restricted. In particular, owing to the relatively long wavelengths of elastic waves, elastic metasurfaces have traditionally been nearly impossible to adapt in terms of their functionalities or working frequency once fabricated. To overcome this limitation, reconfigurable, tunable and active elastic metasurfaces have been proposed. Nevertheless, these approaches typically involve intricate geometries and require precise control of external stimuli, rendering them impractical for widespread use.

Fundamentally, the challenge of realizing distinct functionalities within a single platform for wavefront modulation across different operating frequencies arises from frequency-dependent variations in wave velocity, i.e., wave dispersion. Consequently, owing to these dispersion effects, diffractive elements designed for a specific frequency inevitably generate undesired wavefields at other frequencies. E.g., diffractive elements used for focusing are affected by chromatic aberration, where the focal length varies with frequency, restricting their functionality to a single or narrow frequency range. Therefore, to broaden the operating frequency range, meta-optics employs strategies that address chromatic aberration by incorporating the wave dispersion of meta-atoms into the design process, i.e., dispersion engineering. However, in elastics, the quadratic dispersion relation causes waves to exhibit a pronounced frequency-dependent dispersion characteristic. Additionally, modal coupling induced by the Poisson effect complicates the design through dispersion engineering nearly infeasible. Furthermore, conventional dispersion engineering requires an extensive meta-atom library, making the process exceedingly resource-intensive and costly. Accordingly, the design of the frequency-multiplexed elastic metasurfaces has typically relied on complex optimization methods. However, such approaches are constrained by intricate algorithms, limiting their applicability to a narrow design flexibility.

Here, for identifying the spectral components of elastic waves, a dispersion-engineered elastic metasurface (DEM) for frequency multiplexing is introduced. The proposed DEM focuses elastic waves at distinct focal positions based on the encoded frequencies, utilizing a frequency-multiplexed elastic metasurface through dispersion engineering. To the best of our knowledge, the present study is the first to demonstrate multi-foci dispersion engineering using a passive elastic metasurface, offering a novel strategy for realizing a frequency-resolved wave focusing platform. At the desired focal position, the confined elastic wave energy corresponding to the target frequency can be converted into electrical energy using a piezoelectric element, enabling spectral component recognition. The proposed DEM is designed by simply controlling the plate thickness. Furthermore, this study represents the first analytical design of an elastic metasurface that encodes diverse information within a passive system, avoiding complex optimization or free-form design algorithms. Moreover, the design methodology of the proposed DEM for frequency-resolved wavefront modulation addresses the limitations of conventional elastic metasurfaces. Accordingly, the methodology is readily transferable to a broad spectrum of plate-based mechanical systems, potentially furnishing a unified platform for structural health monitoring, wireless sensing, and Internet of Things applications.

Results
Phase profile from the Kirchhoff–Love plate theory for off-axis focusing

A DEM for frequency multiplexing is designed to achieve distinct elastic wave functionalities across different frequencies, utilizing a thin plate as the background medium. A compact and lightweight thin plate ensures compatibility with diverse mechanical systems and facilitates elastic wave mode decomposition. Here, we employ the lowest antisymmetric Lamb (flexural) wave mode, the predominant mode propagating in thin plates. Compared to other elastic wave modes, flexural waves exhibit strong dispersion in response to geometric variations, making them an ideal basis for designing metasurfaces capable of expressing a rich and versatile phase shift profile. To investigate the flexural wave dynamics on the thin, homogeneous, and isotropic plate, the biharmonic equation governing the transverse displacement \zeta is expressed, based on Kirchhoff–Love (KL) plate theory, as

D\Delta^2\zeta + \rho t \ddot{\zeta} = 0 (1)

where D = \frac{Et^3}{12(1-\nu^2)}, \Delta, and t represent the flexural rigidity, biharmonic operator, and plate thickness, respectively. The mechanical properties of the material include the elastic modulus E, Poisson ratio \nu, and mass density \rho. In this study, a conventional aluminum (Al-6061) plate is exploited, considering the manufacturability and compatibility, exhibiting mechanical properties of E=70 GPa, \nu=0.33, and \rho=2700 kg m^{-3}. From Eq. (1), the relationship between wavenumber k and working frequency \omega=2\pi f for the given parameters, i.e., the dispersion relation, and operating wavelength \lambda can be derived. Thus, the flexural wave propagating along the plate corresponds to phase shift \phi as

\phi(t, \omega_n) = k(\omega_n)L = \left[ \frac{\rho t}{D} \right]^{1/4} \sqrt{\omega_n} L (2)

where L and n represent the length over which the driving wave propagates along the y-direction and the number of target frequencies to be encoded in the DEM, respectively. Notably, for a fixed L of 0.1 m, the phase shift of the flexural wave varies with \omega_n and t. In other words, Eq. (2) indicates that shaping a desired wavefront at targeted \omega_n can be achieved by simply adjusting t, eliminating the need for designing complex and analytically demanding meta-atoms. Throughout this study, the elastic metasurface is simply designed by defining functional units with varying t as meta-atoms, enabling a concise yet precise representation of the phase shift profile. A background plate with a thickness t of 3 mm is adopted, while meta-atoms with a minimum t of 1 mm are employed to ensure structural stability. With a t interval of 50 \mu m, which ensures the precision achievable with fabrication technology, spectrally varying phase shifts are theoretically derived for a total of 41 meta-atoms. By varying only t, while every other geometric parameter is invariant, impedance discontinuities at meta-atoms are effectively suppressed, ensuring a high transmission ratio average of 0.88 across 0–100 kHz ranges.

For proof-of-concept validation, the spectral components to be regulated by the DEM are selected as 40, 60, and 80 kHz. In this configuration, the proposed meta-atoms achieve a full 2\pi phase shift across all frequencies of interest. Consequently, to represent the interference phenomena of elastic waves accurately, the phase shift profiles obtained from Eq. (2) are wrapped according to the relation \Phi_n(t, \omega_n) = arg(e^{i\phi_n(t, \omega_n)}), where i=\sqrt{-1}. \phi_n(t, \omega_n) becomes increasingly dispersive at higher frequencies, exhibiting a non-linear dependence on \omega_n. Accordingly, even if t remains constant, the meta-atom exhibits different phase shifts depending on \omega_n. Thus, an elastic metasurface designed solely for a single \omega fails to function effectively as a frequency-resolving platform. Meanwhile, to enable the functionality of the frequency-selective wave element, the information of the spectrally distinct wavefront must be encoded onto a single passive metasurface. Thus, a strategy is implemented to focus elastic wave energy at different focal positions depending on the target \omega_n. To achieve this, governed by generalized Snell’s law, the phase shift distribution \psi required for off-axis focusing at each focal position (F_{xn}, F_{yn}) corresponding to the targeted frequencies \omega_n is derived as

\psi_n(x, \omega_n) = k(\omega_n) \left( \sqrt{(x-F_{xn})^2 + F_{yn}^2} - \sqrt{F_{xn}^2 + F_{yn}^2} \right) (3)

According to Eq. (3), off-axis focusing, where the focal position deviates from axis symmetry, enhances the capability of DEM for spatially enriched wave control. In this work, the focal positions corresponding to each frequency are set as (F_{x1}, F_{y1}) = (-0.05, 0.1) m, (F_{x2}, F_{y2}) = (0, 0.15) m, and (F_{x3}, F_{y3}) = (0.05, 0.2) m for 40, 60, and 80 kHz, respectively. The selected configuration highlights the ability of the DEM to address wave dispersion and arbitrarily control focal positions in the vertical and horizontal directions, demonstrating its exceptional versatility. Moreover, separations exceeding one operating wavelength avert crosstalk between the encoded spectral channels. The lateral span of the metasurface is set to x \in (-0.15, 0.15) m, with each meta-atom having a pitch of \Delta x = 2.5 mm, resulting in a total of 121 phase profiles. \Delta x is selected to satisfy the Nyquist sampling criterion and, being markedly sub-wavelength, to enable exquisite wavefront tailoring. For the multiplexing strategy, achieving a rich phase profile within a confined space is crucial. The meta-atom, which is the fundamental unit of the metasurface, is designed by simply controlling t to achieve richer phase profiles, thereby overcoming the challenges of conventional design approaches. Consequently, spectrally and spatially distributed phase shift profiles are wrapped according to the relation \Psi_n(x, \omega_n) = arg(e^{i\psi_n(x, \omega_n)}), enabling the comparison of phase differences. Due to the distinct phase shift profiles demanded by the desired focal position at each target frequency, designing an elastic metasurface exclusively for a single frequency, as in conventional elastic metasurface designs, cannot overcome the inherent wave dispersion. Therefore, to realize a frequency-selective focusing with an elastic metasurface, it is essential to address the wave dispersion caused by variations in the working frequencies.

Frequency multiplexing through dispersion engineering

For DEM to distinguish multiple spectral components, the spatial distribution of meta-atoms composing the passive elastic metasurface must be designed to provide distinct phase shift profiles for each target frequency of interest. Previously, to achieve off-axis focusing at a specific frequency, the phase shift profile was derived from generalized Snell’s law, and suitable meta-atoms were employed to realize the required phase shift profile. However, both phase profiles exhibit frequency-dependent phase dispersion, leading to chromatic aberration in focusing elements and thereby limiting multi-spectral operation. To address the chromatic aberration, the wave dispersion is engineered to simultaneously accommodate the phase shift and phase dispersion profiles. By multiplexing the phase shift profiles associated with the target frequencies, a single representative phase shift profile can then be mapped onto a single metasurface.

As the first step in dispersion engineering, a reference phase shift profile is set at the design frequency \omega_d. By correcting the differences between this baseline and the phase required at other frequencies, multiple data channels can be embedded within a single passive structure. The idea is physically clarified by expanding the analytically derived phase functions \phi_n and \psi_n as Taylor series around \omega=\omega_d as

\phi_n(t, \omega_n) = \phi_n(t, \omega_d) + \left. \frac{\partial \phi_n}{\partial \omega} \right|_{\omega=\omega_d} (\omega_n - \omega_d) + O((\omega_n - \omega_d)^2) (4a)

\psi_n(x, \omega_n) = \psi_n(x, \omega_d) + \left. \frac{\partial \psi_n}{\partial \omega} \right|_{\omega=\omega_d} (\omega_n - \omega_d) + O((\omega_n - \omega_d)^2) (4b)

where O((\omega_n - \omega_d)^2) collects the group delay dispersion and higher-order dispersions. Eq. (4) shows that the phase needed at \omega_n depends on the group delay evaluated at \omega_d; in other words, dispersion engineering amounts to compensating the group velocity differences across the working band. Compensating for the phase difference, it becomes possible to achieve the desired phase shifts for different frequencies within a passive meta-atom. E.g., at f = 40 kHz, \Phi_1 and \Psi_1 serve as the reference phases, with the corresponding phase differences defined as arg(e^{i(\phi_{n=1} - \phi_{n\neq1})}) and arg(e^{i(\psi_{n=1} - \psi_{n\neq1})}), respectively. Consequently, the reference phases and phase differences for \phi and \psi at each target frequency can be distributed along a single dimension, referred to as the phase–phase difference space. The concept of the phase–phase difference space implies that a closer distribution signifies stronger alignment between the implemented and target phase information. Accordingly, the distance d between each point is defined as

d_{p,q} = \sqrt{(\Phi_p - \Psi_p)^2 + (arg(e^{i(\phi_p - \phi_q)}) - arg(e^{i(\psi_p - \psi_q)}))^2} (5)

where p and q represent the index of the reference frequency and the index of the frequency at which phase dispersion occurs, respectively. The objective is to determine t at a specific position x on the metasurface that minimizes d for all selected frequencies. Thus, the frequency-multiplexed plate thickness t_{mux}(x) that minimizes the total d across all cases can be obtained as

t_{mux}(x) = argmin_t \sum_{p=1}^n \sum_{q=1, q\neq p}^n d_{p,q} (6)

To validate the frequency-multiplexing capability of the proposed DEM, focal positions are assigned for three target frequencies. As a result, the distribution of meta-atoms mapped by Eq. (6) exhibits phase shift distributions that closely align with the ideal frequency-multiplexed phase-shift profile, while the corresponding t_{mux}(x) displays a random pattern. Notably, each meta-atom is separated by an interval of 0.5 mm while maintaining its functionality as an independent waveguide. Consequently, the desired frequencies of the incident elastic wave are precisely mapped to distinct focal positions, enabling the effective operation of the frequency-selective wave focusing platform.

Frequency-encoded off-axis focusing

To verify the ability of the proposed DEM to distinguish spectral components accurately, the frequency multiplexing capability, designed via dispersion engineering, is analyzed based on the resulting elastic wave field outcomes. First, full-wave harmonic simulations are conducted to obtain numerically calculated intensity fields |\zeta|^2. An incident plane wave corresponding to the target frequency propagates along the y-direction and passes through the frequency-coded metasurface. The transmitted elastic wave generates the desired focusing field within the background medium. In this study, the intended focusing is considered achieved when the maximum intensity position falls within an error margin smaller than the operating wavelength relative to the target focal position. Specifically, the intensity is maximized at (-0.050, 0.100) m, (0.001, 0.148) m, and (0.048, 0.208) m for 40, 60, and 80 kHz, respectively, demonstrating excellent agreement with the focal positions encoded for the target frequencies. To ensure that the DEM achieves crosstalk-free focusing across adjacent frequency channels, the operating frequencies are deliberately chosen to prevent spectral overlap. Furthermore, the frequency-off-axis focusing capability of the proposed DEM is experimentally measured. To experimentally generate a plane wave, the metasurface is coupled with a parabolic coupler defined by the following relation: L = \sqrt{(x_b + L_b)^2 + y_b^2}. Setting y_n = -0.5 m, an ultrasound transducer is attached at a point located y_m = 0.0391 m from the apex of the parabolic coupler. Consequently, from the experimentally visualized intensity fields, the positions with peak values are observed at (-0.049, 0.103) m, (0.001, 0.149) m, and (0.050, 0.208) m for each frequency of interest, respectively, demonstrating excellent agreement with the numerically expected results. This approach provides experimental evidence that the proposed elastic metasurface, despite being a passive system, effectively distinguishes spectral components. Specifically, examining the point spread function (PSF) passing horizontally and vertically through the desired focal position, it is observed that the focusing fields are formed as encoded in the elastic metasurface. Notably, it is confirmed that for all encoded frequencies, the position of maximum intensity remains within the error margin, indicating successful focusing as intended. Specifically, the full width at half maximum (FWHM) experimentally (numerically) obtained from the PSF at each focal position is determined to be 0.45\lambda (0.49\lambda), 0.55\lambda (0.66\lambda), and 0.64\lambda (0.71\lambda), respectively, confirming that sub-wavelength focusing is consistently achieved across all cases of interest. Moreover, the numerical aperture (NA) serves as a crucial metric for evaluating the specifications of the focusing element and is defined as: NA=sin(tan^{-1}(max(x)/F_n)). For the proposed elastic metasurface-based focusing element, high-NA values of 0.83, 0.71, and 0.60 are achieved at 40, 60, and 80 kHz, respectively. Furthermore, to quantitatively assess the focusing performance from the PSF, the modulation transfer function (MRTF) is derived using the following relation: MTF=|\int(I(x)e^{-i2\pi f_x x}dx)/\int I(x)dx|, where I(x) and f_x represent the intensity distribution and spatial frequency along the x-direction, respectively. In addition, under the configuration devised to validate the frequency-multiplexing capability, the designed DEM retains precise focusing within spectral bandwidth of 12, 11, and 3 kHz centered at the encoded frequencies, whereas frequencies outside these passbands give rise to unintended intensity distributions. Consequently, the proposed elastic metasurface is verified to possess a frequency-coded off-axis focusing capability. Furthermore, by utilizing the proposed DEM, an unprecedented level of design flexibility is achieved, which is unattainable with traditional elastic metasurfaces. Because spatial resolution between encoded focal positions is not a limiting factor, the metasurface can be reconfigured to suit the proper host system. Example, our dispersion-engineering methodology allows for the adjustment of encoded information by varying the number of channels, achieving achromatic focusing at identical focal position for coded frequencies, refraction at the desired angles based on the target frequency, and even the realization of multi-functionality. As a result, dispersion engineering can make the elastic metasurface suitable for a broad range of application scenarios.

Frequency-selective amplified energy conversion

Thus far, a frequency-multiplexed elastic metasurface has been devised through dispersion engineering as a passive system for distinguishing spectral components. Furthermore, we evolve the overall system into a strategy that affords facile discrimination of confined elastic-wave energy according to the encoded frequency. To this end, we incorporate piezoelectric energy conversion technology, able of transducing the strain-energy fields induced by elastic waves in the medium into electrical energy, into the proposed system. A customized disc-type piezoelectric element composed of PZ-PT-PNN is attached to the desired focal positions. The elastic wave energy confined at the focal positions is transformed into electrical energy using a piezoelectric energy converter with diameter d_p of 8 mm and thickness t_p of 2 mm. The d_p is purposefully set far below the shortest encoded wavelength of 19.2 mm at 80 kHz, ensuring that the evaluation reflects only the confined field and is not influenced by spurious amplification or attenuation at antinodes or nodes. The selected t_p maintains impedance compatibility with the background substrate, preserves continuum behavior, and prevents local-strain cancellation, while remaining well within the linear response range for every working wavelength. Moreover, the piezoelectric performance is significantly influenced by the intrinsic resonant frequency of the piezoelectric element. Therefore, the system employs a piezoelectric element with a first resonant frequency of 271.49 kHz to ensure that the operating frequency does not affect the piezoelectric performance within the range of the encoded frequencies of interest. In addition, to facilitate the attachment of the piezoelectric energy converter to the flat surface of the substrate plate, the bottom electrode extends along the lateral sides, allowing the extraction of the generated electrical energy alongside the top electrode. Accordingly, the elastic wave energy is converted into electrical energy with an energy conversion ratio \alpha, and the output voltage V_{out} is extracted under an externally applied load resistance R_{load}. The piezoelectric element attached at the desired focal position exhibits significantly distinguished electrical output compared to the background medium or other focal positions, owing to the confined elastic wave energy. Consequently, by identifying the focal position where the highest electrical energy is extracted, the spectral components of a propagating elastic wave can be inferred.

For spectral identification of propagating waves via an alternative energy channel, the transverse displacement at the desired focal positions corresponding to 40, 60, and 80 kHz, along with the voltage extracted from the piezoelectric energy converter, is analyzed in the time-domain through fast Fourier transform (FFT). In both cases, the FFT intensity peaks precisely at the working frequency of the generated elastic wave, confirming accurate energy conversion at the designated frequencies. Subsequently, the maximum electrical output voltage at the three focal positions corresponding to the three encoded frequencies, resulting in nine cases, is measured while varying R_{load}. The R_{load} is varied from 10^2 to 10^7 \Omega, and it is observed that the encoded focal positions generated a significantly higher electrical voltage compared to the other two focal positions. Under quasi-open circuit conditions of 10^7 \Omega, voltages of 0.57, 0.51, and 0.40 V are extracted at 40, 60, and 80 kHz, respectively. Compared to other cases, these results confirm that the generated electrical energy exhibits superior spatial and spectral selectivity. Furthermore, the output electrical power is calculated using Ohm’s law based on the measured voltage. The maximum output power is found to vary with the working frequency, each having an optimal load resistance, approximately 6 k\Omega at 40 kHz, 4 k\Omega at 60 kHz, and 3 k\Omega at 80 kHz. The key aspect in validating the frequency-selective functionality is assessing the degree of electrical power identification at each focal position. To quantify this, the ratio of the output power at the optimal load resistance is compared across all nine cases. The results reveal notable power ratios of 38.8, 48.7, and 32.1 at the encoded frequencies compared to the minimum output at 60 kHz extracted from F_1, confirming that the proposed DEM successfully functions as a frequency-multiplexing platform.

Methods
Simulation

All numerical simulations are carried out using the finite element method with the commercial software COMSOL Multiphysics 6.2. A unit-cell analysis is performed in the eigenfrequency domain with Floquet periodic boundary conditions, whereas full-wave harmonic simulations are conducted in the frequency domain. The reflected waves at the lateral boundaries are mitigated using perfectly matched layers. Ray tracing analysis is performed in the time-dependent domain using a geometrical acoustic module. The material used in this study consists of conventional aluminum (Al-6061), with a Young’s modulus of 70 GPa, mass density of 2700 kg m^{-3}, and Poisson’s ratio of 0.33. Additional details of the numerical simulations are presented in Supplementary Fig. 22.

Fabrication

A thin aluminum plate measuring 900 \times 500 \times 3 mm^3 is used as the background medium. The metasurface is elaborately fabricated using a computer numerical control milling machine, and the parabolic coupler shape is fabricated by laser cutting the thin plate. An absorbing layer (BLU TACK, BOSTIK) is attached to mitigate the reflected waves at the lateral boundaries. The piezoelectric energy converter is designed such that its resonant frequency does not coincide with the selected frequency (23PbZrO_3–36PbTiO_3–41Pb(Ni_{1/3}Nb_{2/3})O_3, 23PZ–36PT–41PNN). All other components required for experimental validation are purchased from commercial sources.

Measurement

To experimentally measure the behavior of elastic waves propagating along the surface of a thin plate, a continuous sinusoidal wave is generated using a function generator (33500B, KEYSIGHT). To refine the propagating wave and enhance the signal-to-noise ratio, a power amplifier (7224, AETECHRON) amplifies the incident wave. A customized ultrasonic transducer (CERACOMP) made of a single-crystal piezoelectric material is attached to the focal point of a parabolic coupler designed to form a plane wave, providing a point-like source to the substrate plate. The transverse intensity field is scanned using a laser Doppler vibrometer (PSV-400, POLYTEC), and the acquired data are visualized under the control of a data management system (OFV-5000, POLYTEC). Additional details of the experimental setup and plane wave generation within the parabolic coupler are shown in Supplementary Figs. 23 and 24. The resonant frequency of the piezoelectric energy converter is determined by measuring its impedance and phase characteristics (E4990A, KEYSIGHT). The energy conversion capability is evaluated according to the external load resistance, controlled using a resistance substitution box (RS-200, IET LABS). The generated output voltage is then measured using an oscilloscope (HDO4034A, TELEDYNE LECROY).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript introduces a novel concept of vibro-spectrometer to detect frequency components of elastic waves propagating through mechanical systems. It is an elastic metasurface consisting of a thin plate, which multiplexes frequency-dependent phase profiles by suitable selection of the plate thickness. Moreover, piezoelectric energy harvesting technology is exploited to convert focused elastic wave energy into highly amplified electrical energy. As a result, spectral information is inferred from spatial information with energy amplification without the need for postprocessing. Experimental results confirm the numerical predictions.\n\nThe manuscript delivers an interesting and novel contribution to the field, which may have a positive impact in current technology of vibro-spectrometers. Numerical and experimental methods are described at a good level of detail, results are promising.\n\nBefore publication, the reviewer recommends widening the final discussion on the potential of the proposed concept of vibro-spectrometer, with focus on its applicability to plate-based mechanical/structural systems mentioned in the manuscript such as ships and bridges (see authors’ statements at page 4, lines 102-103 and page 13, lines 366-367). Since the frequency range of elastic waves (Hz) in these systems is typically much lower than that addressed in this study (kHz), which should be the actual dimensions/size of the proposed metasurface and would they be compatible with the geometry of the system? More details on these aspects would be beneficial.", "reviewer_2": "This paper describes an approach to design a metasurface capable of frequency-selective plane wave focusing. Essentially, the approach is based on the design of phase shifting atoms (or delay lines) realized by tapering the plate in which the wave-modes are propagating.\n\nThe paper can be considered an extension of what was presented in EML: https://doi.org/10.1016/j.eml.2024.102253. The novelty of this paper is in the different tapering solutions adopted and in the derivation of a design procedure capable of accommodating different focal spots associated with different frequencies. It would be important to clarify if the tapering adopted in the current manuscript is superior in terms of transmission ratio with respect to the one used in the EML paper.\n\nAs a general comment, although the proposed approach seems interesting, the claims concerning the potential of the technology seem exaggerated.\n\nFor example, the fact that the proposed solution can be used as a spectrometer is far from being demonstrated. In the current paper, it is demonstrated that up to 3 waves characterized by different central frequencies can be detected, but the performed characterization is not sufficient to determine what is the spectral resolution (or the frequency selectivity) that one can achieve.\n\nThe following sentence (ll 270-272): \"[...] at frequencies near the target frequency, an unintended intensity field emerges, enabling the frequency multiplexed elastic metasurface to function as a high-resolution vibro-spectrometer\" is quite unclear and does not seem to be justified.\n\nAlso, the sentence (ll 345-346) \"the vibro-spectrometer allows the identification of the frequency components without additional FFT analysis or complex signal postprocessing owing to the significantly amplified electrical energy conversion ratio at the desired focal positions\" would require a deeper analysis and probably should be toned down.\n\nIt would be interesting to analyze the costs and benefits of the complete system including the realization of the metasurface and the piezoelectric sensor front-end circuitry, w.r.t. a conventional solution based on a piezoelectric sensor plus a simple DSP unit.\n\nThere are a lot of practicalities that seem to be completely neglected, for example, the exigency of having a planar wave impinging the metasurface is quite idealistic in typical monitoring setups.\n\nMoreover, the phase shift induced by the metasurface might be changed by temperature and loading conditions variations preventing the proper focusing of the elastic waves.\n\nFinally, the discussion on the effect of the piezotransducer shape and dimensions on the efficiency of the energy harvesting process was deliberately skipped. The authors have used a piezodisk with a diameter significantly smaller than the minimum wavelength, but this is probably a suboptimal choice. Exploiting the transducer shape wavelength tuning effect may improve the system behavior in terms of frequency selectivity and energy harvesting.", "reviewer_3": "In this manuscript, the authors outlined a frequency multiplexed elastic metasurface design than can be leveraged as vibration spectrometer. This reviewer has the following observations:\n\n1. The key idea of the research is to design a metasurface that can effectively map the frequency spectral information to spatial deformation. While this is a direct application of wave focusing based on the generalized Snell’s law, it is indeed an interesting twist.\n\n2.The mathematical derivation of dispersion and wave focusing appear to be rigorous. One question does arise, as the authors employed Taylor series expansion around omega_d. What’s the limitation of this design?\n\n3.What’s not discussed is the frequency sensitivity of this design, and its relation to unit-cell dimensions and numbers. This is a major drawback.\n\n4.The practical significance of the work was not reported correctly and accurately. Extracting frequency spectral information can be done by a variety of methods. The criticisms of current practice in Introduction are far-fetched. The method is only application to very simple plate structure, not those commonly seen in ‘spacecrafts, rail vehicles, and vessels’.\n\n5.There are a number of inaccurate narratives in the manuscript. The authors simply adopted a piezoelectric disc transducer which they claimed as ‘energy harvesting technology allowing wave energy to be converted into highly amplified electrical energy’. There is no energy harvesting, nor is there any amplification.", "reviewer_4": "Review of \"Vibro-spectrometer with frequency multiplexed elastic metasurfaces through dispersion engineering\"\n\nThis manuscript presents a novel and elegant methodology for designing frequency-multiplexed elastic metasurfaces for the purpose of creating a \"vibro-spectrometer.\" The authors report an analytical design framework based on dispersion engineering that enables a passive metasurface to focus different elastic wave frequencies to distinct, pre-programmed spatial locations. The work is well-executed, with strong agreement between theoretical design, numerical simulations, and experimental validation.\n\nThe main contribution lies in the design strategy itself, which cleverly avoids the two prevailing paradigms in the field: computationally demanding inverse design methods (such as topology optimization) and dependence on extensive libraries of intricate, resonant meta-atoms. By adjusting the thickness of a monolithic plate, the authors introduce a much more accessible and intuitive approach to advanced multi-frequency wave control. Furthermore, incorporating piezoelectric energy harvesters for direct, post-processing-free spectral detection provides a compelling and practical demonstration of the metasurface's effectiveness.\n\nThe work is of high quality and represents a significant advance in the field of elastic metamaterials. The conclusions are well-supported by the experiment. The manuscript is well-written, and the figures are nice. I recommend publishing after the authors address a few minor points to strengthen the paper further.\n\nThe manuscript has several noteworthy results: (1) the development of a \"dispersion engineering\" methodology that multiplexes multiple desired phase profiles onto a single, passive elastic metasurface. This method is based on classical plate theory and an intuitive \"phase-phase difference space\" concept, avoiding complex algorithms, (2) The authors successfully designed and fabricated a metasurface that focuses elastic waves at three different frequencies (40, 60, and 80 kHz) to three distinct off-axis locations. The experimental results show good agreement, validating the design approach, and the metasurface achieves high-resolution, diffraction-limited focusing, as quantified by the PSF, a high NA, and Strehl ratios consistently above 0.8. And finally, (3) the authors demonstrate that the integration with piezoelectric energy harvesters provides a clear, practical application. The demonstration that the spectral components of an incident wave can be identified by simply measuring which harvester produces the highest voltage is a powerful result, eliminating the need for complex post-processing.\n\nThe work is significant and will be of interest to researchers in metamaterials, phononics, structural health monitoring, and mechanical engineering. Its primary significance lies in the novelty of its design methodology. Compared to the established literature (e.g., Rong et al., Adv. Funct. Mater. (2020)) that utilizes topology optimization to design frequency-coded multifunctional elastic metasurfaces. This inverse design approach is powerful but computationally expensive and often results in complex, free-form geometries that can be difficult to fabricate and analyze intuitively. The current work by Lee et al. provides a good contrast by achieving a similar outcome with an analytical approach that is computationally simple and uses a simpler geometry (varying thickness). Another relevant example (Zhang et al., Phys. Rev. Applied (2020)) of a frequency-selected acoustic metasurfaces using coding meta-atoms based on dual-layer Helmholtz resonators. That approach required the careful design of a discrete library of resonant unit cells to achieve the desired phase responses (\"0\" and \"1\" codes) at specific frequencies. Lee et al.'s work avoids the need for such a library of complex resonators, instead deriving the required continuous phase from a single, non-resonant meta-atom defined only by its thickness. So, while the concept of frequency-multiplexed wave control is not entirely new (as evidenced by Rong et al. and Zhang et al. and many other studies), the method proposed by Lee et al. is a clear and original contribution. It overcomes key limitations of existing methods, i.e. their computational complexity and the need for complex resonant structures. It offers a more elegant, efficient, and physically intuitive path to designing such devices. The work on the vibro-spectrometer concept with integrated energy harvesting also pushes the field closer to real-world applications.\n\nA revision could clarify or expand upon a few minor points to strengthen the manuscript further. These would improve the paper's completeness.\n\nComments\n\nThe paper demonstrates successful operation at three distinct, well-separated frequencies. A key characteristic of any spectrometer is its resolution. The authors show that at non-encoded frequencies, the focusing is compromised (Supp. Fig. 7), which is a good start. A more quantitative discussion on spectral resolution would be beneficial. How close can two frequencies be before their focal spots are no longer distinguishable? A brief theoretical discussion on how this resolution relates to the metasurface parameters would add significant value.\n\nThe design is based entirely on the KL plate theory, which is an approximation for thin plates. The authors should briefly state the assumptions of this theory (e.g., ignoring shear deformation, etc.) and discuss the range of validity in the context of their design. This provides important context for readers seeking to apply this method to other systems.\n\nThe authors note that the harvester's resonance is far from the operating frequencies. However, the harvester's physical mass and stiffness could locally perturb the elastic field at the focal point. While this is likely a small effect, including a sentence that acknowledges this and justifies why it can be considered negligible would enhance the rigor of the analysis."}, "author_response": "Response to Reviewer #1’s comments\nGeneral Comment: The manuscript introduces a novel concept of vibro-spectrometer to detect\nfrequency components of elastic waves propagating through mechanical systems. It is an elastic\nmetasurface consisting of a thin plate, which multiplexes frequency-dependent phase profiles\nby suitable selection of the plate thickness. Moreover, piezoelectric energy harvesting\ntechnology is exploited to convert focused elastic wave energy into highly amplified electrical\nenergy. As a result, spectral information is inferred from spatial information with energy\namplification without the need for postprocessing. Experimental results confirm the numerical\npredictions. The manuscript delivers an interesting and novel contribution to the field, which\nmay have a positive impact in current technology of vibro-spectrometers. Numerical and\nexperimental methods are described at a good level of detail, results are promising.\nResponse: We extend our heartfelt gratitude to the reviewer for dedicating the reviewer’s\nvaluable time to providing constructive comments, which have greatly enhanced the quality of\nour manuscript and deepened our understanding of study findings. We have carefully\nconsidered and actively incorporated the reviewer’s comments, addressing each point raised\nand resolving the concerns expressed. We have provided a point-by-point response to each\ncomment and have reflected these changes in the revised manuscript. We would like to express\nour gratitude to the reviewer for once again acknowledging the novelty and originality of our\nmanuscript and actively recommending it for publication in Nature Communications.\nComment 1.1: Before publication, the reviewer recommends widening the final discussion on\nthe potential of the proposed concept of vibro-spectrometer, with focus on its applicability to\nplate-based mechanical/structural systems mentioned in the manuscript such as ships and\nbridges (see authors’ statements at page 4, lines 102-103 and page 13, lines 366-367). Since\nthe frequency range of elastic waves (Hz) in these systems is typically much lower than that\naddressed in this study (kHz), which should be the actual dimensions/size of the proposed\nmetasurface and would they be compatible with the geometry of the system? More details on\nthese aspects would be beneficial.\nResponse 1.1: We sincerely thank the Reviewer for the insightful recommendation to broaden\nour discussion on the practical applicability of the proposed vibro-spectrometer to large-scale\nplate-based mechanical structures such as ships and bridges. Below we clarify how the\nmetasurface can be scaled to the much lower elastic-wave frequencies (tens–hundreds of hertz)\ntypical of those systems and how the underlying plate theory remains valid.\nThe vibro-spectrometer is rigorously designed within the linear Kirchhoff–Love (KL) plate\nframework, in which the phase shift imparted by a meta-atom of thickness 𝑡 at angular\nfrequency 𝜔 is:\n1 1\n𝜑 (𝑡,𝜔 ) = 𝑘(𝜔 )𝐿 = ( 𝜌𝑡 )4𝜔2𝐿. (R1.1.1)\nn n n 𝐷 n\nBecause 𝜑 ∝ 𝜔1⁄2, lowering the target frequency band by a factor of 100 (from 40 kHz to\nn\nphase coverage, the propagation length 𝐿 must be increased ten-fold. For our proposed design\n(𝐿 = 0.1 m), this yields 𝐿 = 1 m—well within the characteristic panel dimensions of ship\nhulls and bridge decks. Alternatively, the same phase coverage can be retained at constant 𝐿\nby increasing 𝑡. In our laboratory prototype (𝑡 = 3 mm), scaling 𝐿 back to 0.1 m while\noperating at 400 Hz requires 𝑡 = 23 cm, which falls inside the typical thickness range of\nstiffened steel plates used in maritime and civil structures. Hence the metasurface geometry\ncan be tailored to low-frequency operation by simply adjustment of 𝐿 and/or 𝑡 without\nviolating practical design limits. On the other hand, at the other extreme, when the thickness-\nto-wavelength ratio grows and shear deformation together with rotation inertia become non-\nnegligible, the KL assumptions lose accuracy; in this regime we plane to apply Reissner–\nMindlin (RM) plate theory, which preserves our dispersion engineering strategy while offering\nbetter quantitative fidelity. We have incorporated these quantitative scaling considerations,\nclarified the transition from KL to RM formulations, and replaced overly speculative\nstatements with precise descriptions in the revised Discussion section, thereby providing\npotential readers with an objective assessment of the vibro-spectrometer’s feasibility for real-\nworld plate-based systems and substantially enhancing the manuscript’s clarity and\ncompleteness.\n(Revised manuscript, page 2)\n“Accordingly, the proposed vibro-spectrometer can be incorporated into various plate-based\nmechanical systems for next-generation technologies.”\n(Revised manuscript, page 3)\n“Meanwhile, elastic waves propagating in large-scale mechanical structures can provide\ncritical information on structural integrity. Accordingly, developing compact and passive\nalternatives for extracting their frequency-dependent behavior remains an important\nengineering challenge. Although existing methods provide accurate measurements, their\nlimitations in portability, cost, and energy consumption motivate exploration of\ncomplementary strategies that can be integrated into simple mechanical platforms.”\n(Revised manuscript, page 4)\n“Accordingly, the methodology is readily transferable to a broad spectrum of plate-based\nmechanical systems, potentially furnishing a unified platform for structural health monitoring,\nwireless sensing, and Internet of Things applications.”\n(Revised manuscript, page 13)\n“Interestingly, the analytical framework derived from the KL plate theory exhibits a linear\ndependence on 𝜔1⁄2; therefore, simply scaling the geometric parameters enables effortless\nretuning to alternative frequency ranges. In higher-frequency regimes, where the assumptions\nof KL theory are no longer valid, the methodology remains applicable by replacing that model\nwith the Reissner–Mindlin plate theory, which accounts for shear deformation and rotational\ninertia31,32.”\n[31] Reissner, E., The effect of transverse shear deformation on the bending of elastic plates,\nJ. Appl. Mech. Trans. ASME 12, A69-A77 (1945).\n[32] Mindlin, R. D., Influence of rotatory inertia and shear on flexural motions of isotropic\nelastic plates, J. Appl. Mech. Trans. ASME 18, 31-38 (1951).\n(Revised manuscript, page 13-14)\n“Furthermore, the proposed vibro-spectrometer can be integrated into or replace various plate-\nbased mechanical systems, offering new functionalities for next-generation applications.”\nResponse to Reviewer #2’s comments\nGeneral Comment: This paper describes an approach to design a metasurface capable of\nfrequency-selective plane wave focusing. Essentially, the approach is based on the design of\nphase shifting atoms (or delay lines) realized by tapering the plate in which the wave-modes\nare propagating. The paper can be considered an extension of what was presented in EML:\nhttps://doi.org/10.1016/j.eml.2024.102253. The novelty of this paper is in the different tapering\nsolutions adopted and in the derivation of a design procedure capable of accommodating\ndifferent focal spots associated with different frequencies. It would be important to clarify if\nthe tapering adopted in the current manuscript is superior in terms of transmission ratio with\nrespect to the one used in the EML paper.\nResponse: We sincerely thank the reviewer for investing their valuable time in providing\nconstructive feedback, which has significantly improved the quality of our manuscript and\ndeepened our understanding of the study’s findings. We have carefully considered every point\nraised, provided a detailed, point-by-point response, and incorporated the corresponding\nrevisions throughout the manuscript and supplementary information. We are grateful that the\nreviewer continues to recognize the novelty and originality of our work and potentially supports\nits publication in Nature Communications.\nAs the reviewer noted, our proposed vibro-spectrometer adopts a wavefront-shaping strategy\nthat slightly resembles the meta-atom design reported in the mentioned EML paper: in both\nstudies, phase modulation is achieved solely by tailoring the plate thickness. This approach\nminimizes the impedance mismatching that plagues more intricate meta-atom geometries,\nthereby enabling a high transmission ratio. When revisiting the transmission ratio data in\nresponse to the reviewer’s comment, we discovered an error in the values originally reported\nin the manuscript and supplementary information. We have now recalculated the transmission\nratio with great care over plate thickness of 1−3 mm and operating frequency of 0−100\nkHz for our meta-atom (Fig. R2.0.1).\nFigure R2.0.1. Transmission characteristics of the meta-atom. a Transmission spectrum of\nthe meta-atom as a function of plate thickness and operating frequency. b Transmission of the\nmeta-atom at the frequencies encoded in the vibro-spectrometer as a function of plate thickness.\nThe revised analysis confirms an average transmission of 0.88 across the full frequency range,\nand consistent values of 0.89, 0.89, and 0.87 at the encoded frequencies of 40, 60, and\n80 kHz, respectively. Notably, the background plate thickness of 3 mm yields perfect\ntransmission of 1.00 because it eliminates impedance mismatching. Our meta-atom also\noffers practical advantages over the EML design. In the EML study, the plate thickness varies\ngradually along the propagation direction, inevitably introducing loss, and fabrication requires\ntwo-sided structuring that is difficult to implement at scale, hence the reliance on 3D printing.\nBy contrast, our meta-atom can be fabricated from conventional materials with a single-side\nmilling process, making it both simple and industry-friendly. These merits render our meta-\natom an exceptionally powerful yet elegant building block for frequency-multiplexed elastic\nmetasurfaces that demand broad phase-shift profiles while maintaining high transmission over\na wide bandwidth. Finally, we appreciate the reviewer’s positive assessment of the core novelty\nof our study, namely, the dispersion-engineering framework that precisely manipulates\nfrequency-dependent wave-dispersion characteristics to focus elastic waves at distinct focal\npositions. The reviewer’s insightful comments enable us to identify and correct errors, thereby\nenhancing the manuscript’s completeness and impact. We have revised the text accordingly\nand have added additional citations including the EML paper, to better contextualize and\nunderscore the superiority of the proposed meta-atom.\n(Revised manuscript, page 5)\n“To investigate the flexural wave dynamics on the thin, homogeneous, and isotropic plate, the\nbiharmonic equation governing the transverse displacement 𝜁 is expressed, based on\nKirchhoff–Love (KL) plate theory, as [25-27]:”\n[25] Love, A. E. H. XVI. The small free vibrations and deformation of a thin elastic shell.\nPhilos. Trans. R. Soc. Lond. (A.) 179, 491-546 (1888).\n[26] Shen, Y. et al. Broadband elastic energy harvesting based on achromatic meta-grating.\nExtreme Mech. Lett. 72, 102253 (2024).\n[27] Lee, G. et al. Synthesized dispersion-engineered elastic metasurfaces for achromatic\nfocusing and harvesting across the audible to ultrasound ranges. Proc. Natl. Acad. Sci. USA\n122, e2425407122 (2025).\n(Revised manuscript, page 6)\n“By varying only 𝑡 , while every other geometric parameter is invariant, impedance\ndiscontinuities at meta-atoms are effectively suppressed, ensuring a high transmission ratio\naverage of 0.88 across 0−100 kHz ranges (Supplementary Fig. 2).”\n(Revised supplementary information, page 5)\nSupplementary Figure 2. Transmission characteristics of the meta-atom. a Transmission\nspectrum of the meta-atom as a function of plate thickness and operating frequency. b\nTransmission of the meta-atom at the frequencies encoded in the vibro-spectrometer as a\nfunction of plate thickness.\n“Across the ultrabroad 0−100 kHz frequency range, the mean transmission remains over\n0.88. Notably, when the meta-atom thickness equals that of the background plate of 3 mm,\nimpedance mismatch vanishes and the transmission converges to 1.00 irrespective of\nfrequency (Supplementary Fig. 2b). Specifically, at the encoded frequencies of 40, 60, and\n80 kHz, average transmission values of 0.89, 0.89, and 0.87, respectively, are obtained,\nunequivocally confirming the suitability of the design for high-efficiency wavefront-shaping\napplications.”\nComment 2.1: As a general comment, although the proposed approach seems interesting, the\nclaims concerning the potential of the technology seem exaggerated. For example, the fact that\nthe proposed solution can be used as a spectrometer is far from being demonstrated. In the\ncurrent paper, it is demonstrated that up to 3 waves characterized by different central frequencies can be detected, but the performed characterization is not sufficient to determine what is the spectral resolution (or the frequency selectivity) that one can achieve.\nResponse 2.1: We sincerely appreciate the reviewer’s critical comment regarding the necessity\nof including quantitative evaluations as objective indicators for assessing the performance of\nthe proposed vibro-spectrometer. We fully agree with the reviewer’s concern that without such\nevaluations, the potential of the technology may appear exaggerated. In response, we have\ncarefully revised the manuscript to highlight the proposed vibro-spectrometer’s unique\ncapability for spectral component differentiation and its potential for practical applications. To\nclarify, the proposed system operates based on a mechanism where spectral information is\ninferred from spatial information by employing a frequency-multiplexed elastic metasurface.\nTo rigorously evaluate the performance of the proposed system, we identified and analyzed the\nfollowing four key metrics: 1) Spatial resolution – How tightly the system can spatially confine\nthe wave energy during focusing. 2) Spectral resolution – How well the system can distinguish\nbetween adjacent frequency components. 3) Spectral bandwidth range – The extent of spectral\nbandwidth over which the vibro-spectrometer can effectively encode and recognize frequency\ninformation. 4) Geometric parameters of the meta-atom. We have now included a systematic\nanalysis of these four evaluation metrics in the revised manuscript and supplementary\ninformation to provide a quantitative and objective assessment of the vibro-spectrometer’s\nperformance.\nFigure R2.1.1. Spatial resolution analysis of multiplexed metasurface. a Calculated\nintensity fields of the frequency-multiplexed elastic metasurface for two encoded frequencies\nof 40 kHz and 80 kHz, while varying the spatial gap between the encoded focal positions.\nError distance between the generated and target focal positions for b 40 kHz and c 80 kHz\nas a function of spatial deviation. The dotted line indicates the corresponding working\nwavelength. d Mean phase error between the ideal and mapped phase profile as a function of\nspatial deviation, averaged over the entire metasurface area. Red dot indicates the average of\nthe mean error between the encoded frequencies.\n1) Spatial resolution: The proposed vibro-spectrometer estimates the frequency of propagating\nwaves based on the degree of wave energy confinement at the encoded spatial locations. To\nquantitatively assess this capability, we define the focusing criterion as follows: focusing is\nconsidered successful when the location of the maximum intensity in the field generated by the\ntransmitted wave from the multiplexed elastic metasurface lies within a deviation smaller than\nthe working wavelength from the encoded focal position. For this reason, we selected a\nmaximum working wavelength of 40 kHz, corresponding to a focal spot radius of 27.1 mm,\nto ensure that the encoded focal regions do not spatially overlap. This choice also highlights\nthe system’s capability to freely encode focal positions in both 𝑥- and 𝑦-directions, which is\ncrucial because if multiple focal points were encoded within the same region, the system would\nno longer be able to distinguish the corresponding spectral information. Despite this constraint,\nwe demonstrated that the proposed dispersion-engineered frequency-multiplexed elastic\nmetasurface can generate the intended wave field for the encoded frequencies, regardless of\nwhether the spatial gaps between encoded focal points are small, large, or oven overlapping\n(Fig. R2.1.1a). This includes configurations such as achromatic focusing, where multiple\nfrequencies are focused at the same position. To further verify the spatial resolution, we\nsystematically varied the gap between focal positions in the 𝑥-direction for two input\nfrequencies and assessed whether the system could still maintain spectral distinction. We\nconfirmed that the vibro-spectrometer maintains its functionality from the case of co-located\nfocal positions to focal positions located near the metasurfaces’ edge. Specifically, we\nmeasured the spatial deviation between the actual maximum intensity position in the generated\nfield and the target encoded focal position (Fig. R2.1.1b and Fig. R2.1.1c). For all input\nfrequencies, the spatial errors were within one working wavelength, validating the excellent\nspatial resolution of the proposed vibro-spectrometer. Additionally, we quantitatively\nevaluated how the dispersion-engineered phase profile deviates from the ideal phase profile as\na function of spatial deviation. The average phase error over the entire metasurface are was\nfound to be 0.27 rad, corresponding to only 8.7% of the maximum phase error of 𝜋 (Fig.\nR2.1.1d). This result confirms that the dispersion-engineered phase mapping provides highly\nrobust spatial information coding, even in the presence of spatial deviations.\n(Revised manuscript, page 6)\n“Moreover, separations exceeding one operating wavelength avert crosstalk between the\nencoded spectral channels.”\n(Revised manuscript, page 10)\n“Because spatial resolution between encoded focal positions is not a limiting factor, the\nmetasurface can be reconfigured to suit the proper host system (Supplementary Fig. 10).”\n(Revised supplementary information, page 16-17)\nSupplementary Figure 10. Spatial resolution analysis of multiplexed metasurface. a\nCalculated intensity fields of the frequency-multiplexed elastic metasurface for two encoded\nfrequencies of 40 kHz and 80 kHz, while varying the spatial gap between the encoded focal\npositions. Error distance between the generated and target focal positions for b 40 kHz and c\n80 kHz as a function of spatial deviation. The dotted line indicates the corresponding working\nwavelength. d Mean phase error between the ideal and mapped phase profile as a function of\nspatial deviation, averaged over the entire metasurface area. Red dot indicates the average of\nthe mean error between the encoded frequencies.\n“We quantitatively evaluated the spatial resolution of the proposed dispersion-engineered\nfrequency-multiplexed elastic metasurface. The vibro-spectrometer estimates the frequency of\nincident elastic waves by identifying the spatial positions where the wave energy is confined.\nTo rigorously define this capability, we adopted the following focusing criterion: the system is\nconsidered to achieve successful focusing when the spatial location of the maximum intensity\nin the generated field lies within one working wavelength from the target encoded focal\nposition. For this analysis, we selected the 40 kHz frequency as the reference for defining the\nworking wavelength, corresponding to a focal spot radius of 27.1 mm. This choice ensures\nthat the encoded focal regions in the proof-of-concept design remain spatially distinguishable\nand do not overlap. It also demonstrates the system’s capacity to encode focal positions freely\nin both the 𝑥- and 𝑦-directions. Notably, if multiple focal points were simultaneously encoded\nwithin the same region, the system would lose the ability to separate their respective spectral\ninformation.\nDespite this spatial constraint, the proposed metasurface successfully generated the intended\nspatial mapping across various configurations, including cases where the encoded focal points\nare co-located, closely spaced, or widely separated (Supplementary Fig. 10a). This includes\nachromatic focusing, where multiple frequencies are directed to the same focal point. To further\nverify the system's spatial resolution, we systematically varied the gap between the focal\npositions in the 𝑥-direction for two distinct input frequencies of 40 kHz and 80 kHz. We\nthen assessed the error distance between the generated maximum intensity position in the\nobtained field and the target encoded focal position. The results show that the spatial error\nremains within one working wavelength for all evaluated cases, confirming the system’s robust\nspectral separation performance regardless of focal point spacing (Supplementary Fig. 10b and\nSupplementary Fig. 10c). Additionally, we evaluated the phase fidelity of the dispersion-\nengineered metasurface by calculating the mean phase error between the ideal and mapped\nphase profiles across different spatial deviations. The average phase error was found to be 0.27\nrad, corresponding to only 8.7% of the maximum error bound of 𝜋 (Supplementary Fig. 10d).\nThese results confirm that the dispersion-engineered phase mapping maintains high robustness\nand accuracy, ensuring reliable spatial encoding.”\nFigure R2.1.2. Spectral resolution analysis of multiplexed metasurface. a Calculated\nintensity fields of the frequency-multiplexed elastic metasurface for two encoded frequencies\nof 40 kHz and (40+∆𝑓) kHz, where ∆𝑓 was varied from 1 to 40 kHz. Spatial error\ndistance between the generated and target focal positions for the b reference frequency and c\nmultiplexed frequency as a function of ∆𝑓. The dotted line indicates the corresponding\nworking wavelength. d Mean phase error between the ideal and mapped phase profile as a\nfunction of spectral deviation, averaged over the entire metasurface area. Red dot indicates the\naverage of the mean error between the encoded frequencies.\n2) Spectral resolution: One of the most critical performance indicators for the proposed system\nas a spectrometer is its spectral resolution, since the ability to distinguish closely spaced\nfrequency components directly determines the system’s spectroscopic performance. To\nquantitatively evaluate this metric, we systematically varied the spectral deviation between\nencoded frequencies and analyzed the resulting system behavior. For the dispersion\nengineering, we selected the 40 kHz spectral component as the reference phase profile. We\nthen incrementally increased the frequency for the compensated spectral information to\nexamine whether the system could still implement frequency-separated focusing as intended\n(Fig. R2.1.2a). We encoded the two frequencies to focus at distinct spatial points: (𝐹 ,𝐹 ) =\n𝑥1 𝑦1\n(−0.05,0.15) m for the reference frequency of 40 kHz, and (𝐹 ,𝐹 ) = (0.05,0.15) m\n𝑥2 𝑦2\nfor the multiplexed frequency. Our calculations showed that at the reference frequency,\nperfect focusing at the intended focal position was achieved regardless of spectral deviations,\nconfirming the correctness of the reference phase profile. However, when the spectral deviation\nwas set to 1 kHz, negligible dispersion occurred, resulting in an intensity field nearly identical\nto that of the reference frequency. As the spectral deviation increased to 10 kHz, the phase\ninformation of the two frequencies began to overlap, leading to a distorted and unintended\nwave field. As a deviation of 15 kHz, the intensity field corresponding to 𝐹 began to\n2\nemerge, but with chromatic aberration affecting the original focal position of 𝐹 . Nevertheless,\n1\nwhen the spectral deviation reached 16 kHz or more, the system achieved complete spectral\nseparation and stable multiplexed focusing in all cases. We quantitatively compared the\nlocation of the intensity maxima in the generated field with the desired focal positions for each\nencoded frequency. The results demonstrated that for the reference frequency, the focusing\nerror was significantly smaller than the working wavelength, confirming highly accurate\nfocusing performance (Fig. R2.1.2b). In contrast, for the multiplexed frequency, when the\nspectral deviation was below 15 kHz, the maximum intensity occurred at unintended\nlocations, resulting in unstable wave fields. However, starting from 16 kHz deviation, the\nsystem maintained spatial focusing errors smaller than the operating wavelength, indicating\nrobust and stable spectroscopic performance (Fig. R2.1.2c). Furthermore, we calculated the\naverage phase error over the entire metasurface area as a function of spectral deviation. As 15\nkHz spectral difference, the mean phase error was 0.40 rad, and beyond this point, the error\nconsistently decreased (Fig. R2.1.2d). As a result, the 20 kHz spectral deviation selected for\nthe proof-of-concept demonstration successfully confirms that the proposed vibro-\nspectrometer can robustly separate and detect the encoded frequencies.\n(Revised manuscript, page 9)\n“Under this configuration, the vibro-spectrometer sustains crosstalk-free focusing for\nfrequency separations down to approximately 16 kHz, confirming that the chosen operating\nfrequencies satisfy the spectral resolution requirement and endow the device with obvious\nfrequency-discrimination capability (Supplementary Fig. 6).”\n(Revised manuscript, page 13)\n“Further optimization can enhance the flexibility and spectral resolution of the proposed\ntheoretical approach, enabling a robust metasurface design extensive meta-atom libraries.”\n(Revised supplementary information, page 14-15)\nSupplementary Figure 9. Spectral resolution analysis of multiplexed metasurface. a\nCalculated intensity fields of the frequency-multiplexed elastic metasurface for two encoded\nfrequencies of 40 kHz and (40+∆𝑓) kHz, where ∆𝑓 was varied from 1 to 40 kHz.\nSpatial error distance between the generated and target focal positions for the b reference\nfrequency and c multiplexed frequency as a function of ∆𝑓. The dotted line indicates the\ncorresponding working wavelength. d Mean phase error between the ideal and mapped phase\nprofile as a function of spectral deviation, averaged over the entire metasurface area. Red dot\nindicates the average of the mean error between the encoded frequencies.\n“One of the most critical performance parameters of the proposed frequency-multiplexed\nvibro-spectrometer is its spectral resolution, as the ability to distinguish closely spaced\nfrequencies directly determines its spectroscopic functionality. To quantitatively evaluate this\naspect, we systematically varied the spectral deviation ∆𝑓 between two encoded frequencies\nand analyzed the system’s behavior in terms of focusing performance and phase error. For\ndispersion engineering, the phase profile was designed based on the reference frequency of 40\nkHz. The additional spectral component was then incrementally increased to examine whether\nthe metasurface could maintain frequency-resolved focusing for various ∆𝑓 values\n(Supplementary Fig. 9a). The two target focal positions were set as follows: (𝐹 ,𝐹 ) =\n𝑥1 𝑦1\n(−0.05,0.15) m for the reference 40 kHz component, and (𝐹 ,𝐹 ) = (0.05,0.15) m for\n𝑥2 𝑦2\nthe multiplexed component at (40+∆𝑓) kHz. At the reference frequency of 40 kHz, the\nsystem consistently produced accurate focusing at the intended focal position, regardless of\n∆𝑓, confirming the validity of the designed reference phase profile. In contrast, when ∆𝑓 = 1\nkHz, the dispersion between the two frequencies was negligible, resulting in an overlapping\nintensity distribution nearly identical to that of the 40 kHz component. As ∆𝑓 increased to\n10 kHz, spectral interference became apparent, and the intended dual-frequency focusing\nbehavior started to distort. At ∆𝑓 = 15 kHz, the intensity pattern corresponding to 𝐹 began\n2\nto emerge; however, chromatic aberration simultaneously affected the original focal position\nat 𝐹 . Once ∆𝑓 exceeded 16 kHz, the system achieved clear spectral separation and stable\n1\nmultiplexed focusing without significant cross-talk between the focal positions. We further\nquantified the spatial accuracy by comparing the generated positions of maximum intensity\nwith the target focal positions for each encoded frequency. For the 40 kHz reference\ncomponent, the spatial error remained significantly smaller than the working wavelength,\nconfirming excellent focusing accuracy (Supplementary Fig. 9b). For the multiplexed\nfrequency, stable and accurate focusing wave not achieved when ∆𝑓 was below 15 kHz due\nto insufficient dispersion-induced phase difference. However, beyond ∆𝑓 = 16 kHz, the\nspatial error dropped below the wavelength criterion, indicating robust spectral distinction\n(Supplementary Fig. 9c). Additionally, the average phase error over the entire metasurface was\ncalculated as a function of ∆𝑓. At ∆𝑓 = 15 kHz, the mean phase error reached 0.40 rad, but\nbeyond this threshold, the error consistently decreased with increasing ∆𝑓 (Supplementary\nFig. 9d). These findings validate that a minimum ∆𝑓 of 16 kHz is required for reliable\nspectral separation using the current dispersion design. For the proof-of-concept realization,\nwe selected a ∆𝑓 of 20 kHz, which successfully demonstrated the vibro-spectrometer’s\nrobust multi-frequency detection capability.”\nFigure R2.1.3. Spectral bandwidth characteristics of the multiplexed vibro-spectrometer.\nError distance between the generated focal position and the target encoded position as a\nfunction of frequency for a 40 kHz, b 60 kHz, and c 80 kHz encoded components. The\ndashed vertical lines indicate the encoded frequency, while the horizontal dashed lines\nrepresent the working wavelength criterion.\n3) Spectral bandwidth: The ability of the designed vibro-spectrometer to maintain a certain\nspectral bandwidth around each encoded frequency is also a critical metric for evaluating its\nperformance as a spectrometer. To verify this, we investigated how the location of the\nmaximum intensity changes when input frequencies are varied near each encoded frequency in\nthe proof-of-concept multiplexed metasurface design (Fig. R2.1.3). Specifically, we performed\na frequency sweep from 30 to 90 kHz with 1 kHz intervals, and analyzed the resulting\nintensity field distributions. For each encoded frequency, we measured the spatial deviation\nbetween the position of the maximum intensity and the intended focal position. Our results\nconfirmed that the system maintains the desired focusing capability within the following\nspectral bandwidths: For the 40 kHz encoded frequency, the system maintains functional over\na 12 kHz bandwidth (32-44 kHz), for the 60 kHz encoded frequency, the bandwidth is 11\nkHz (57-68 kHz), and for the 80 kHz encoded frequency, the bandwidth is 3 kHz (77-80\nkHz). The narrowing of the bandwidth at higher frequencies is attributed to the reduction in\nwavelength, which leads to stricter spatial focusing criteria as frequency increases. These\nfindings demonstrate that the proposed multiplexed metasurface-based spectrometer is capable\nof not only detecting discrete, encoded frequency components but also exhibits moderate\ndetection bandwidths around each frequency. This confirms the system’s robustness and\npractical applicability for elastic wave spectral analysis.\n(Revised manuscript, page 10)\n“In addition, the engineered metasurface retains focusing across spectral bandwidths of 12,\n11, and 3 kHz centered on the encoded frequencies, thereby maintaining its functionality as\na spectrometer, whereas frequencies lying beyond these bands generate unintended intensity\npatterns (Supplementary Fig. 9).”\n(Revised supplementary information, page 10-11)\nSupplementary Figure 6. Spectral bandwidth characteristics of the multiplexed vibro-\nspectrometer. a Calculated intensity fields at non-encoded frequencies near the encoded\nfrequencies. Error distance between the generated focal position and the target encoded\nposition as a function of frequency for b 40 kHz, c 60 kHz, and d 80 kHz encoded\ncomponents. The dashed vertical lines indicate the encoded frequency, while the horizontal\ndashed lines represent the working wavelength criterion.\n“Specifically, to quantitatively assess this spectral bandwidth, we conduct a systematic\nfrequency sweep from 30 to 90 kHz in 1 kHz increments and analyze how the focal\nintensity position varied relative to the target encoded positions of the multiplexed metasurface\ndesign (Supplementary Fig. 6b-d). For each encoded frequency, we quantified the spatial\ndeviation between the generated maximum intensity location and the intended focal position.\nThe vibro-spectrometer is considered functional within a bandwidth where this deviation\nremained below one working wavelength. The analysis reveals the following bandwidth: For\nthe 40, 60, and 80 kHz encoded frequency, functional focusing is sustained over 32-44,\n57-68, and 77-80 kHz (12, 11, and 3 kHz bandwidth). The observed reduction in\nbandwidth at higher frequencies is attributed to the shorter wavelengths, which impose stricter\nspatial focusing criteria as the frequency increases. These results demonstrate that the proposed\nmetasurface spectrometer not only enables discrete frequency detection but also provides\nmoderate operational bandwidths around each encoded frequency, validating its flexible and\npractical suitability for elastic wave spectral analysis.”\nFigure R2.1.4. Geometric parameter dependence and phase control of the thickness-\ncontrolled meta-atom. a Focal intensity distribution of three encoded frequencies with the\nmeta-atom pitch ∆𝑥 is varied from 1 to 25 mm. b Phase shift as a function of propagation\nlength for each encoded frequency. c Mean phase error of the multiplexed phase profiles versus\npropagation length. Red line indicates the average error for the encoded frequencies.\n4) Geometric parameter of the meta-atom: Furthermore, the geometric parameters of the meta-\natoms constituting the metasurface are critical design factors that directly affect wavefront\nshaping, and thus functionality of the proposed system as a spectrometer. Our proposed meta-\natom design, based on Kirchhoff–Love plate theory, achieves phase retardation solely by\ncontrolling the plate thickness, and involves two primary geometric parameters. The first\nparameter is the meta-atom width (∆𝑥). Since the metasurface operates as a periodically\narranged spatial sampler, the sampling of the phase profile must satisfy the Nyquist sampling\ncriterion to prevent the generation of undesired diffraction orders and avoid aliasing in the\nsampled wavefront, expressed as:\n𝜆\n∆𝑥 < . (R2.1.1)\nFor the proof-of-concept design, the wavelengths corresponding to 40, 60, and 80 kHz are\n27.1, 22.1, and 19.2 mm, respectively. Considering the encoded focal positions, the\ncorresponding numerical aperture (NA) are 0.83, 0.71, and 0.60, leading to Nyquist\nconditions requiring ∆𝑥 to be smaller than 16.3, 13.3, and 11.5 mm, respectively. We\nverified the system’s focusing ability by varying ∆𝑥 (Fig. R2.1.4a). Our results confirm that\nfor all encoded frequencies, the system maintains accurate focal point separation when ∆𝑥 is\nset to 1 to 10 mm, all of which satisfy the Nyquist criterion. However, when ∆𝑥 was\nincreased to 15 mm, we observed the formation of undesired intensity fields at 80 kHz,\nwhere the sampling condition is severely violated. Moreover, at ∆𝑥 = 25 mm, the system\ncompletely lost the focusing ability across all encoded frequencies. These findings underscore\nthe necessity of choosing a sufficiently small lattice constant relative to the operating\nwavelength and NA for precise dispersion-engineered wavefront shaping. Conventional elastic\nmetasurfaces have struggled to meets this requirement due to the complexity of finely\ndesigning or fabricating meta-atoms that satisfy the Nyquist condition. In contrast, our\nproposed plate-thickness-controlled meta-atom offers a simple yet powerful solution that can\novercome these limitations. Additionally, the meta-atom’s phase shift can be controlled by\nadjusting the propagation length (Fig. R2.1.4b). The phase shift profile shows a linear increase\nwith propagation length, achieving full 2𝜋 phase coverage at propagation lengths of 27.1,\n22.1, and 19.2 mm for 40, 60, and 80 kHz, respectively. This demonstrates that our\nthickness-controlled meta-atom design is compact and effective tool for metasurface\nimplementations operating at the wavelength scale. However, for dispersion engineering,\nachieving full phase coverage alone is insufficient. It is also necessary to compensate for the\ndispersion in the multiplexed phase profiles. To this end, we varied the propagation length and\ncalculated the mean phase error across the entire metasurface for each frequency (Fig. R2.1.4c).\nOur results show that the mean error gradually decreases with increasing propagation length,\nreaching quasi-saturation at around 0.09 m. Based on this analysis, we selected a propagation\nlength of 0.1 m, which yields a low phase-mapping error while ensuring robust focusing\ncapability across all multiplexed frequencies.\nIn summary, we have conducted a systematic and quantitative analysis to evaluate the\nfrequency detection capabilities of the proposed vibro-spectrometer. We have incorporated\nthese additional evaluations into the revised manuscript and supplementary information.\nFurthermore, to address the reviewer’s concern regarding potentially overstated claims in the\noriginal manuscript, we have carefully revised the descriptions related to the vibro-\nspectrometer’s performance. Based on the newly added objective evaluation metrics, we have\nmoderated the previous expressions to provide a more balanced and accurate presentation of\nthe system’s capability. We have made every effort to ensure that potential readers are provided\nwith precise and reliable information.\n(Revised manuscript, page 7)\n“∆𝑥 is selected to satisfy the Nyquist sampling criterion and, being markedly sub-wavelength,\nto enable exquisite wavefront tailoring (Supplementary Fig. 4).”\n(Revised manuscript, page 12)\n“The frequency-dependent focal positioning capability of the analytically designed frequency\nmultiplexed elastic metasurface is validated numerically and experimentally, demonstrating\nclearly distinctive focusing across all encoded frequencies.”\n(Revised supplementary information, page 7-8)\nSupplementary Figure 4. Geometric parameter dependence and phase control of the\nthickness-controlled meta-atom. a Focal intensity distribution of three encoded frequencies\nwith the meta-atom pitch ∆𝑥 is varied from 1 to 25 mm. b Phase shift as a function of\npropagation length for each encoded frequency. c Mean phase error of the multiplexed phase\nprofiles versus propagation length. Red line indicates the average error for the encoded\nfrequencies.\n“One of the key performance indicators of a vibro-spectrometer is its ability to distinguish non-\nencoded frequencies when frequencies other than the encoded ones are generated. Therefore,\nfull wave harmonic simulations are conducted to calculate the intensity fields at non-encoded\nfrequencies near the encoded frequencies (Supplementary Fig. 4a). The results reveal that\nunexpected intensity fields emerge due to the elastic metasurface failing to satisfy the\nprescribed phase profiles for transmitted flexural waves. In general, elastic metasurfaces\ndesigned for a single frequency exhibit chromatic aberration near the target frequency,\nmaintaining a similar trend6. In contrast, a frequency multiplexed elastic metasurface does not\nretain its wavefront characteristics at adjacent frequencies, making it highly suitable for vibro-\nspectrometer applications. Specifically, to quantitatively assess this spectral bandwidth, we\nconduct a systematic frequency sweep from 30 to 90 kHz in 1 kHz increments and analyze\nhow the focal intensity position varied relative to the target encoded positions of the\nmultiplexed metasurface design (Supplementary Fig. 4b-d). For each encoded frequency, we\nquantified the spatial deviation between the generated maximum intensity location and the\nintended focal position. The vibro-spectrometer is considered functional within a bandwidth\nwhere this deviation remained below one working wavelength. The analysis reveals the\nfollowing bandwidth: For the 40, 60, and 80 kHz encoded frequency, functional focusing\nis sustained over 32-44, 57-68, and 77-80 kHz (12, 11, and 3 kHz bandwidth). The\nobserved reduction in bandwidth at higher frequencies is attributed to the shorter wavelengths,\nwhich impose stricter spatial focusing criteria as the frequency increases. These results\ndemonstrate that the proposed metasurface spectrometer not only enables discrete frequency\ndetection but also provides moderate operational bandwidths around each encoded frequency,\nvalidating its flexible and practical suitability for elastic wave spectral analysis.”\nComment 2.2: The following sentence (ll 270-272): \"[...] at frequencies near the target\nfrequency, an unintended intensity field emerges, enabling the frequency multiplexed elastic\nmetasurface to function as a high-resolution vibro-spectrometer\" is quite unclear and does not\nseem to be justified.\nResponse 2.2: We fully agree with the reviewer’s constructive comment that the statements in\nthe original manuscript were quite unclear and insufficiently defined, and we sincerely\nappreciate the reviewer’s careful attention to these details. Regarding Comment 2.2, we would\nlike to note that our response is provided in conjunction with the analysis presented in Figure\n2.1.3 of Response 2.1 (revised supplementary information, Supplementary Fig. 6).\nIn the revised manuscript, we have corrected the previously unclear statements about the\nperformance of the vibro-spectrometer by introducing quantitative analyses. Specifically, one\nof the original bases for our claim of “high-resolution” was the full width at half maximum\n(FWHM) of the wave field formed by the metasurface. From the focusing fields generated at\nthe encoded frequencies, we obtained FWHM values of 0.49𝜆 , 0.66𝜆 , and 0.71𝜆\nnumerically, and 0.45𝜆, 0.55𝜆, and 0.64𝜆 experimentally. These results demonstrate sub-\nwavelength focusing ability across all encoded frequencies, which was the reason we initially\ndescribed the system as high-resolution. However, we acknowledge the reviewer’s point that\nthis characterization reflects spatial resolution, not spectral resolution, which is the more\nappropriate metric for evaluating the spectrometer’s performance. This distinction could\npotentially confuse readers. Therefore, we have revised the statements to describe the vibro-\nspectrometer’s performance more accurately and quantitatively, ensuring clarity and avoiding\nmisinterpretation. We greatly appreciate the reviewer’s insightful feedback, which has allowed\nus to refine the manuscript and enhance its overall quality through the removal of imprecise\nclaims and the addition of quantitative evaluations.\n(Revised manuscript, page 10)\n“Specifically, the full width at half maximum (FWHM) experimentally (numerically) obtained\nfrom the PSF at each focal position is determined to be 0.45𝜆 (0.49𝜆), 0.55𝜆 (0.66𝜆), and\n0.64𝜆 (0.71𝜆), respectively, confirming that sub-wavelength focusing is consistently achieved\nacross all cases of interest.”\nComment 2.3: Also, the sentence (ll 345-346) \"the vibro-spectrometer allows the\nidentification of the frequency components without additional FFT analysis or complex signal\npostprocessing owing to the significantly amplified electrical energy conversion ratio at the\ndesired focal positions\" would require a deeper analysis and probably should be toned down.\nResponse 2.3: We sincerely thank the reviewer for providing such a thoughtful and\nconstructive comment, which has helped us improve the overall quality and clarity of the\nmanuscript. We fully agree with the reviewer’s observation that the original statement could\nbe perceived as somewhat exaggerated. As the reviewer correctly pointed out, certain\nexpression in our previous manuscript might have unintentionally appeared to undermine\nresearch efforts in other related fields. We acknowledge that those studies indeed have their\nown distinct merits, and likewise, the proposed vibro-spectrometer system has its unique\nadvantages. To address this concern and ensure a more objective and balanced presentation,\nwe have revised the relevant statements in the manuscript to clearly distinguish these aspects\nand convey the information with improved clarity.\n(Revised manuscript, page 2)\n“Therefore, the magnitude of extracted energy alone reveals the underlying spectral\ninformation through spatial information.”\n(Revised manuscript, page 3)\n“Existing instruments, however, are expensive, bulky, and contact-based sensors are\nadditionally susceptible to signal distortion arising from their direct mechanical coupling to the\nstructure.”\n(Revised manuscript, page 12)\n“The vibro-spectrometer introduced here, however, is purposely configured to identify the\noperating frequency solely by comparing the amplitudes of the converted electrical signals.”\n(Revised manuscript, page 13)\n“Furthermore, by integrating a piezoelectric energy conversion technology, the individual\nfrequency components can be discerned exclusively from the magnitude of the extracted\nelectrical output, since each encoded focal position generates a markedly distinct voltage\nsignature.”\nComment 2.4: It would be interesting to analyze the costs and benefits of the complete system\nincluding the realization of the metasurface and the piezoelectric sensor front-end circuitry,\nwith respect to a conventional solution based on a piezoelectric sensor plus a simple DSP unit.\nResponse 2.4: We are grateful to the reviewer for the insightful comment that prompted us to\nhighlight the strengths of the proposed vibro-spectrometer while transparently disclosing its\nshortcomings, thus enabling potential readers to obtain a more balanced view.\nAs recommended, we have now provided a detailed comparison between our metasurface-\nbased system and the conventional digital signal processor (DSP) approach. In the traditional\nscheme, a piezoelectric sensor is paired with a DSP unit that performs time- or frequency-\ndomain analysis on the sampled waveform. This pathway demands high-resolution, real-time\nanalog-to-digital conversion, a continuous power supply, and software or firmware support for\ncomputation and data storage. Although such systems offer considerable flexibility in\ninformation processing, they become increasingly complex when continuous frequency\nmonitoring is required in resource-constrained environments.\nBy contrast, the proposed metasurface platform uses a dispersive phase profile engraved into\nthe plate to realize fully passively spatial frequency demultiplexing. Each spectral component\nis focused to a unique spatial spot, where a dedicated piezoelectric sensor directly measures\nthe energy without any additional spectral computation. The sensors are connected only to\nsimple rectification and energy conversion circuits, eliminating the need for micro-controllers\nor DSPs; consequently, the system can operate intermittently or in a self-powered mode with\nnegligible power demand.\nWhile the metasurface incurs an upfront fabrication cost, this cost is amortized in large-scale\nproduction. Moreover, in multi-frequency or multi-mode applications that require additional\nsensors or broader sampling bandwidth, the advantages of our approach become even more\npronounced.\nIn summary, the metasurface-based strategy trades electronic complexity for structural\ncomplexity, offering a scalable, low-power alternative for distributed frequency sensing in\nembedded or autonomous systems with stringent computational and power budgets. We have\nincorporated this balanced assessment—covering both strengths and limitations—into the\nrevised Discussion section to help readers evaluate the proposed vibro-spectrometer\nobjectively.\n(Revised manuscript, page 13)\n“Although the vibro-spectrometer incurs an initial expense to fabricate the metasurface, this\ninvestment consolidates the power supplies and signal-processing electronics demanded by\nconventional digital processors into the passive structure itself, endowing the device with self-\npowered spectroscopic functionality.”\nComment 2.5: There are a lot of practicalities that seem to be completely neglected, for\nexample, the exigency of having a planar wave impinging the metasurface is quite idealistic in\ntypical monitoring setups. Moreover, the phase shift induced by the metasurface might be\nchanged by temperature and loading conditions variations preventing the proper focusing of\nthe elastic waves.\nResponse 2.5: We appreciate the constructive comment provided by the reviewer, which has\nhelped us strengthen the practicality of our study. We fully concur that our original manuscript\ndid not explicitly evaluate how non-ideal conditions might influence the performance of the\nproposed vibro-spectrometer. In response, we have systematically analyzed environmental\nvariations that are likely to occur in real-world monitoring scenarios.\nFigure R2.5.1 Evaluation of the elastic metasurface focusing performance under varying\nangles of incidence. a Calculated intensity fields for three encoded frequencies of 40, 60,\nand 80 kHz under various angle of incidence (AOI). As the AOI deviates from normal\nincidence, the focal spot gradually shifts due to angle-dependent dispersion. b 𝑥-position and\nc 𝑦-position of the focal spot as a function of AOI for each encoded frequency, showing\nmonotonic trends and increasing deviation with AOI. The shaded region denotes the tolerance\nrange. d Maximum intensity values of the focused field across AOIs, normalized to the value\nat normal incidence.\nFirst, as suggested by the reviewer, we evaluated the focusing performance of the elastic\nmetasurface under variations in the angle of incidence (AOI), 𝜃, for each encoded frequency\n(Fig. R2.5.1a). As the AOI changes, the phase shift profile encoded in the passive elastic\nmetasurface for a planar incident wave experiences angular-dependent dispersion, which\nintroduces deviations from the desired intensity field. Specifically, we analyzed the spatial\nposition of the maximum intensity in the resulting intensity field (Fig. R2.5.1b and Fig.\nR2.5.1c). Overall, we observed that the focusing trends remained consistent across various\nAOIs compared to the baseline case of 𝜃 = 0°. However, in the case of the 80 kHz working\nfrequency, the focal point began to shift noticeably from the designed location when the AOI\nexceeded approximately 6°. Nevertheless, for all encoded frequencies, the metasurface\nmaintained a practical field-of-sensing range within ±10°. Notably, at 40 kHz, the position\nof the maximum intensity remained within a single wavelength of the target location for AOIs\nup to ~20°, indicating robust focusing ability against angular variations. Furthermore, as\nshown in Fig. R2.5.1d, the maximum intensity values remained comparable to those at normal\nincidence, suggesting minimal degradation in focusing performance. Although designing a\nmetasurface that compensates for AOI-dependent dispersion is beyond the current scope, we\nnote that such compensation can be theoretically achieved by employing the following phase\nshift profile as:\n𝜓 (𝑥,𝜔 ,𝜃 ) = −𝑘(𝜔 )(√(𝑥 −𝐹 )2 +𝐹2 −√𝐹2 +𝐹2 +𝑥sin𝜃 ). (R2.5.1)\nn n n n xn yn xn yn n\nEq. R2.5.1 implies that additional momentum along the 𝑥-direction must be considered\ndepending on the AOI. Consequently, the required phase shift profile varies with AOI, and\ndeviation from the encoded angle leads to undesired field formation due to angular-dependent\ndispersion. While extending the AOI tolerance further would be an interesting direction for\nfuture work, such design considerations fall outside the scope of the present study. Nonetheless,\nEq. R2.5.1 provides a pathway to implement AOI-multiplexed wave control when required.\nThese results demonstrate the high degree of freedom inherent to our elastic metasurface-based\nmultiplexing strategy in controlling diverse dispersive characteristics. More importantly, they\nprovide strong evidence that the proposed vibro-spectrometer is not overly sensitive to AOI\nfluctuations, affirming its feasibility for practical implementation. We have included the above\nevaluations and discussions regarding AOI robustness in the revised supplementary\ninformation to enhance the completeness of the manuscript.\n(Revised supplementary information, page 28-29)\nSupplementary Note 19. Evaluation of angular robustness of metasurface\nSupplementary Figure 19. Evaluation of the elastic metasurface focusing performance\nunder varying angles of incidence. a Calculated intensity fields for three encoded frequencies\nof 40, 60, and 80 kHz under various angle of incidence (AOI). As the AOI deviates from\nnormal incidence, the focal spot gradually shifts due to angle-dependent dispersion. b 𝑥-\nposition and c 𝑦-position of the focal spot as a function of AOI for each encoded frequency,\nshowing monotonic trends and increasing deviation with AOI. The shaded region denotes the\ntolerance range. d Maximum intensity values of the focused field across AOIs, normalized to\nthe value at normal incidence.\n“To assess the angular robustness of the elastic metasurface, we characterized the focusing\nperformance under varying angles of incidence (AOI), for all three encoded frequencies: 40,\n60, and 80 kHz (Supplementary Fig. 19a). Due to the passive and spatially encoded nature of\nthe metasurface, deviations in AOI introduce AOI-dependent dispersion, as the phase\ncompensation originally designed for normal incidence becomes mismatched under oblique\nincidence. This mismatch results in a shifted or distorted intensity field, particularly evident in\nthe lateral displacement of the focal spot. We extracted the spatial coordinates corresponding\nto the maximum intensity in each obtained intensity field to quantify this effect (Supplementary\nFig. 19b and Supplementary Fig. 19c). Across the broad AOI range considered (AOI = ±10°),\nthe metasurface maintains a consistent focusing trend, with gradual displacement of the focal\nspot observed. This behavior is particularly stable at lower frequencies; at 40 kHz, the lateral\ndeviation of the focal point remains within one wavelength from the encoded position even for\nAOIs approaching ±10°, demonstrating exceptional resilience to AOI variation. At higher\nfrequencies; at 80 kHz, the focal position begins to deviate more noticeably for AOIs beyond\napproximately 6°, indicating increased sensitivity due to shorter wavelength and greater\naccumulated phase mismatch. Nonetheless, even in this case, the metasurface maintains\nfunctional focusing within a practical range of 10°, which defines a useful field-of-sensing\nwindow for the proposed vibro-spectrometry application. To further evaluate performance\ndegradation, we analyzed the peak intensity as a function of AOI (Supplementary Fig. 19d).\nThe maximum intensity remains relatively constant across all AOIs and frequencies, with only\nminor fluctuations, suggesting that the focusing efficiency is largely preserved despite the\nangular dispersion. This implies that while the focal position may shift under oblique incidence,\nthe integrity of the focal energy remains robust. These results highlight the versatility of the\nproposed metasurface multiplexing scheme and confirm the feasibility of the vibro-\nspectrometer for practical scenarios where AOI alignment may not be perfectly maintained.\nAlthough the current design does not explicitly account for AOI-dependent phase\ncompensation, we note that such correction can be theoretically incorporated by modifying the\nphase shift profile to:\n𝜓 (𝑥,𝜔 ,𝜃 ) = −𝑘(𝜔 )(√(𝑥 −𝐹 )2 +𝐹2 −√𝐹2 +𝐹2 +𝑥sin𝜃 ). (S9)\nn n n n xn yn xn yn n\nEq. S9 introduces an additional linear phase term, 𝑥sin𝜃 , to compensate for the extra\nn\nmomentum component along the 𝑥-direction introduced by the oblique wavefront. This\nimplies that the optimal phase profile must be dynamically adapted for each AOI to suppress\nangular dispersion and maintain perfect focusing; such an approach would enable the\nimplementation of AOI-multiplexed or angular-adaptive wave control using elastic\nmetasurfaces.”\nFigure R2.5.2 Evaluation of the elastic metasurface focusing performance under various\nworking temperature. Temperature dependence of the conventional aluminum’s a Young’s\nmodulus, b mass density, and c Poisson’s ratio. d Calculated intensity distributions at different\ntemperatures and driving frequencies. e 𝑥-position and f 𝑦-position of the maximum intensity\nas a function of temperature. g Maximum field intensity at the encoded focal position across\ndifferent temperatures and frequencies.\nSecond, we investigated the influence of temperature on the focusing performance of the\nproposed vibro-spectrometer. As the elastic properties of the medium vary with temperature,\nchanges in operating temperature inevitably impact wave propagation characteristics (Fig.\nR2.5.2a, Fig. R2.5.2b, and Fig. R2.5.2c). Specifically, with increasing temperature, the\nYoung’s modulus and mass density generally decrease, while the Poisson’s ratio tends to\nincrease. As a result, the behavior of elastic waves, which is governed by these three\nmechanical properties, is also affected by temperature. In our original manuscript, both the\nnumerical calculations and experimental validations were conducted under ambient conditions\n(300K). To thoroughly assess temperature dependence, we performed full-wave harmonic\nsimulations across a wide temperature range, from cryogenic (50K) to high-temperature (550K)\nconditions (Fig. R2.5.2d). Remarkably, even under such extreme environments, the simulated\nintensity fields show that focusing still occurs at the encoded focal positions. More specifically,\nthe 𝑥-position of the maximum intensity remained nearly unchanged across the entire\ntemperature range, demonstrating strong robustness of lateral focusing (Fig. R2.5.2e). In\ncontrast, some deviations in the 𝑦-position were observed, particularly beyond ~400K, likely\ndue to changes in the elastic wave dispersion relation induced by elevated temperatures (Fig.\nR2.5.2f). Nevertheless, the shift in focal position remained relatively small at lower\ntemperatures and within acceptable bounds for most operating conditions. Furthermore, we\ncompared the maximum intensity values of the focused field at different temperatures and\nfound that they remained comparable to those at room temperature, even under extreme thermal\nconditions (Fig. R2.5.2g). This indicates that the vibro-spectrometer maintains stable\nperformance in terms of focusing efficiency despite significant changes in the elastic medium’s\nmechanical properties. These findings suggest that the elastic metasurface can retain optimal\nfocusing performance if it is designed with consideration of the specific temperature\nenvironment in which the vibro-spectrometer is deployed. We have included this analysis in\nthe revised supplementary information to demonstrate that the proposed system maintains\nrobust functionality even under extreme temperature conditions.\n(Revised supplementary information, page 30-31)\nSupplementary Note 20. Evaluation of temperature robustness of the metasurface\nSupplementary Figure 20. Evaluation of the elastic metasurface focusing performance\nunder various working temperature. Temperature dependence of the conventional\naluminum’s a Young’s modulus, b mass density, and c Poisson’s ratio. d Calculated intensity\ndistributions at different temperatures and driving frequencies. e 𝑥-position and f 𝑦-position\nof the maximum intensity as a function of temperature. g Maximum field intensity at the\nencoded focal position across different temperatures and frequencies.\n“To evaluate the thermal robustness of the proposed elastic metasurface-based vibro-\nspectrometer, we systematically analyzed the focusing behavior across a broad temperature\nrange, from cryogenic (50 K) to elevated (550 K) conditions. Since elastic wave propagation\nis inherently dependent on the medium’s mechanical properties, temperature-induced changes\nin these parameters directly influence the focusing performance. The temperature dependence\nof three key mechanical properties8—Young’s modulus, mass density, and Poisson’s ratio—\nare presented in Supplementary Fig. 20a–c. As temperature increases, both Young’s modulus\nand mass density exhibit a decreasing trend, while the Poisson’s ratio slightly increases. These\nvariations alter the elastic wave dispersion relation, which can affect focal positions and\nintensities. To investigate these effects in detail, we performed full-wave harmonic simulations\nat three encoded frequencies over the full temperature range. The resulting intensity fields\n(Supplementary Fig. 20d) reveal that the metasurface maintains clear focusing behavior at the\nencoded focal positions throughout the broad temperature span. The 𝑥-position of the focal\npoint, corresponding to the encoded spatial location, remains nearly invariant with temperature\n(Supplementary Fig. 20e), indicating excellent robustness of lateral focusing. On the other hand,\nmoderate deviations in the 𝑦 -direction focal position are observed, especially above\napproximately 400 K (Supplementary Fig. 20f). These deviations are likely attributable to\ntemperature-induced changes in the elastic wave velocity and resulting modifications in phase\nmatching conditions. Nonetheless, the variations remain small and within tolerable limits for\npractical implementation. Furthermore, the intensity of the focused field at each frequency\nremains relatively stable over the temperature range (Supplementary Fig. 20g), demonstrating\nthat the focusing efficiency is largely preserved, even under substantial shifts in the working\ntemperature. These findings confirm that the proposed vibro-spectrometer design maintains\nrobust focusing behavior and signal intensity over a wide range of operating temperatures. This\nthermal insensitivity supports the system’s applicability in various practical environments,\nincluding those with fluctuating or extreme thermal conditions.”\n[8] Muraca, R. F. et al. Materials data handbook: Aluminum alloy 6061. NASA-CR-123772\n(1972).\nFigure R2.5.3 Effect of material damping on elastic wave focusing. a Calculated field\nintensity distribution for the vibro-spectrometer under increasing isotropic structural loss factor\n𝜂 from 0 to 0.01, evaluated at three operating frequencies. b 𝑥-positions and c 𝑦-positions\nof the maximum intensity as a function of 𝜂. d Maximum field intensity at the encoded focal\nposition across different 𝜂 and frequencies.\nLastly, we evaluated the focusing ability of the elastic metasurface by considering the\npropagation loss of elastic waves caused by material damping, which is inevitable in practical\nsystems. To assess how material damping affects the frequency-dependent focusing capability\nof the vibro-spectrometer, we conducted full-wave harmonic simulations by varying the\nisotropic structural loss factor, 𝜂, from 0 to 0.01 (Fig. R2.5.3a). For this analysis, we used\nthe following governing equation implemented via the finite element method to calculate the\nwave manipulation behavior of the elastic metasurface as:\n∇∙[(1+𝑖𝜂)ℂ:𝜖(𝒖)]+𝜌𝜔2𝒖 = 𝟎. (R2.5.2)\nHere, ℂ, 𝒖, and 𝜖(𝒖) = 1 (∇𝒖+(∇𝒖)𝑇) represent the linear elastic stiffness tensor, the\n2\ncomplex displacement vector, and the strain tensor, respectively. The calculated results\nrevealed that, although the overall field intensity decreases as damping increases, the phase\nshift induced by the metasurface remains well-preserved within the linear regime. Since this\nphase-based wavefront modulation is inherently governed by the system’s linear response, the\nfocal position remains virtually unaffected by changes in 𝜂. In other words, while damping\nleads to energy loss and reduced amplitude, it does not influence the metasurface’s phase\nmodulation mechanism—hence, the spatial focusing position is maintained. As a result, even\nwhen damping is considered, the elastic metasurface retains its frequency-distinguishable\nfocusing ability, confirming that the proposed vibro-spectrometer is capable of functioning\nreliably in real-world, lossy environments.\nIn summary, we have demonstrated that the frequency-multiplexed elastic metasurface-based\nvibro-spectrometer maintains its intended functionality robustly under variations in angle of\nincidence, working temperature, and structural damping. We have included these additional\ninvestigations and findings in the revised manuscript and supplementary information, thereby\nreinforcing the practical applicability of our approach.\n(Revised supplementary information, page 32-33)\nSupplementary Note 21. Effect of material damping on focusing performance\nSupplementary Figure 21. Effect of material damping on elastic wave focusing. a\nCalculated field intensity distribution for the vibro-spectrometer under increasing isotropic\nstructural loss factor 𝜂 from 0 to 0.01, evaluated at three operating frequencies. b 𝑥-\npositions and c 𝑦-positions of the maximum intensity as a function of 𝜂. d Maximum field\nintensity at the encoded focal position across different 𝜂 and frequencies.\n“To assess the practical viability of the proposed elastic metasurface-based vibro-spectrometer,\nwe evaluated its performance under the influence of material damping, which causes inevitable\npropagation loss in real-world elastic systems. This loss arises from internal friction and\nstructural dissipation, often characterized by a frequency-dependent isotropic structural loss\nfactor 𝜂. To investigate how damping affects the frequency-resolved focusing capability, we\nperformed full-wave harmonic simulations by systematically varying 𝜂 from 0 to 0.01, a\nrealistic range for many engineering materials (Supplementary Fig. 21a). These calculations\nwere conducted using the finite element method with the following governing wave equation\nthat incorporates complex modulus damping as:\n∇∙[(1+𝑖𝜂)ℂ ∶ 𝜖(𝒖)]+𝜌𝜔2𝒖 = 𝟎. (S10)\nHere, ℂ is the linear elastic stiffness tensor, 𝒖 is the complex displacement field, and\n𝜖(𝒖) = 1 (∇𝒖+∇𝒖𝑇) denotes the strain tensor. The calculated results indicate that increasing\n2\ndamping leads to a monotonic decrease in the absolute field intensity, as expected due to\ndissipative energy loss. However, critically, the metasurface-induced phase modulation\nremains intact across all damping conditions considered. Since the focusing behavior of the\nvibro-spectrometer is governed by spatial phase encoding rather than amplitude, the focal spot\nremains at the intended location regardless of damping (Supplementary Fig. 21b and\nSupplementary Fig. 21c). The maximum intensity for all encoded frequencies remains nearly\nunchanged as 𝜂 increases, demonstrating the robustness of spatial focusing (Supplementary\nFig. 21d). This confirms that although damping reduces the overall signal strength, it does not\ndegrade the ability of designed metasurface to distinguish frequencies via spatial encoding.\nTherefore, the proposed vibro-spectrometer maintains its functional integrity and frequency\nresolution in the presence of practical levels of structural loss, ensuring reliable performance\nin lossy or imperfect material environments.”\n(Revised manuscript, page 14)\n“Operating entirely within the linear regime, the device shapes wavefronts with flexible fidelity\nand thus remains resilient to oblique incidence (Supplementary Fig. 19). The intrinsic\nmechanical robustness of the elastic substrate further ensures immunity to temperature\nvariations (Supplementary Fig. 20) and material losses (Supplementary Fig. 21), which\nadvantages rarely attainable in another wave physics.”\nComment 2.6: Finally, the discussion on the effect of the piezotransducer shape and\ndimensions on the efficiency of the energy harvesting process was deliberately skipped. The\nauthors have used a piezodisk with a diameter significantly smaller than the minimum\nwavelength, but this is probably a suboptimal choice. Exploiting the transducer shape\nwavelength tuning effect may improve the system behavior in terms of frequency selectivity\nand energy harvesting.\nResponse 2.6: We would like to express our sincere gratitude to the reviewer for providing\ncritical comments that helped us significantly improve the quality of our manuscript. We\nacknowledge that our previous version did not sufficiently address the reviewer’s concerns\nregarding the influence of the shape and dimensions of the piezoelectric element on the\nefficiency of the energy harvesting process. Importantly, we would like to emphasize that our\ndesign choices were not arbitrary nor deliberately skipped, but rather the result of a rigorous\nand rational pre-analysis process, as detailed below.\nTypically, disc-type piezoelectric elements are characterized by two geometric parameters:\ndiameter and thickness. Both parameters undoubtedly influence piezoelectric performance and\ncan exhibit complex dependencies when coupled with variables such as the working\nwavelength and properties of the attached medium. Crucially, in order for the proposed vibro-\nspectrometer to function reliably, the piezoelectric performance must be evaluated in a manner\nthat isolates the influence of wave focusing, free from the confounding effects of other\nconditions.\nFigure R2.6.1 Open-circuit piezoelectric voltage responses as a function of geometric\nparameters. a Output voltage as a function of sensor diameter at fixed thickness of 2 mm for\nexcitation frequencies of 40, 60, and 80 kHz. Voltage peaks appear near the excitation\nwavelengths, confirming resonance-induced amplification. A dotted line indicates selected\ndiameter of piezoelectric element. b Output voltage as a function of sensor thickness at fixed\ndiameter of 8 mm, showing significant voltage suppression when sensor thickness approaches\nthat of the host plate of 3 mm, due to impedance matching and strain energy dissipation. A\ndotted line indicates selected thickness of piezoelectric element.\nTo quantitatively compare piezoelectric performance based on diameter, it is essential to ensure\na linear relationship between the output voltage of the piezoelectric element and the intensity\nof the generated elastic wave. However, when the diameter becomes comparable to the\nwavelength of the elastic wave, resonant or interference effects may induce non-linear\namplification or attenuation. Generally, when the sensor size is much smaller than the\nwavelength, it behaves as a point detector and the output is almost linearly proportional to the\nincident wave amplitude. Conversely, if the sensor is larger than the wavelength, phase\nvariations across the sensor surface lead to partial signal cancellation (aperture effect), which\nreduces the output. As such, if the diameter is sufficiently smaller than the wavelength, the\noutput increases linearly with input amplitude, with minimal waveform distortion. On the other\nhand, when the diameter approaches or exceeds the wavelength, interference between\nwavefronts across the sensor reduces or even cancels the output signal. Therefore, larger\nsensors result in decreased and distorted outputs, complicating direct comparisons of\npiezoelectric responses to identical inputs. Additionally, the output voltage in a piezoelectric\nsensor-plate system can be affected by mechanical resonance, aperture effects, and diffraction.\nResonance occurs when the excitation frequency matches the sensor’s natural frequencies (e.g.,\nradial or thickness modes), leading to sharp output increases. For instance, the sensor may act\nas a resonator and exhibit amplified output even at low excitation levels, particularly when its\ndiameter approximates the operating wavelength—resulting in maximum energy transfer for\nspecific Lamb wave modes. To ensure that the sensor behaves like a point detector and thus\nmaintains linearity between input and output, the piezoelectric element must be smaller than\nthe wavelength, and the design should avoid resonance. Based on these considerations, our\nselection of an 8 mm diameter piezoelectric element is highly rational, given that it is\nsignificantly smaller than the encoded wavelengths of 27.1, 22.1, and 19.2 mm. To\nsubstantiate this, we performed an open-circuit analysis where we varied the element diameter\nwhile fixing the thickness at 2 mm (Fig. R2.6.1a). We observed that the output voltage peaked\nat diameters of 24, 20, and 18 mm for 40, 60, and 80 kHz, respectively—closely\nmatching the respective operating wavelengths. Furthermore, in the sub-wavelength regime,\noutput voltage increased linearly with diameter across all frequencies. Interestingly, we also\nobserved secondary peaks at 54 , 43 , and 35 mm—roughly twice the operating\nwavelengths—indicating harmonic phenomena. This strongly supports the conclusion that\npiezoelectric performance is intrinsically tied to the operating wavelength.\nThe thickness of the piezoelectric element is another important factor influencing performance.\nIn general, increasing thickness improves piezoelectric performance. However, in the proposed\nvibro-spectrometer—where the piezoelectric element is attached to a thin plate—the\nperformance is predominantly governed by the plate's thickness. While no phase reversal\noccurs in strain distribution through the thickness, a mismatch in mechanical impedance can\ninfluence wave transmission efficiency. When the sensor and host plate have identical\nthickness, their similar flexural rigidity and mass distribution result in near impedance\nmatching. In this case, bending waves transmit smoothly across the piezoelectric patch with\nminimal reflection, resulting in reduced local strain and hence lower voltage output. In essence,\nthe sensor behaves like a continuous medium with the host plate, failing to accumulate\nsufficient strain energy. Conversely, if the sensor’s thickness deviates from that of the host\nplate (either thicker or thinner), wave scattering (reflection and transmission) becomes\nsignificant. For instance, a thicker patch increases local stiffness and mass, leading to stronger\nreflection and complex strain profiles. This can induce local strain peaks and enhance output\nvoltage. Therefore, to enable quantitative comparisons of elastic waves with varying intensities,\nthe sensor thickness must lie within the linear regime of piezoelectric performance. We verified\nthis through simulations under open-circuit conditions while varying the thickness and fixing\nthe diameter of 8 mm (Fig. R2.6.1b). We observed a significant drop in output voltage when\nthe sensor’s thickness approached that of the background plate of 3 mm, due to impedance\nmatching and reduced internal strain energy. Notably, the minimum output thickness shifted\nwith the operating wavelength, reinforcing the influence of wavelength on sensor behavior.\nConsequently, we selected a 2 mm thickness that ensures sufficient linearity in piezoelectric\nresponse while avoiding impedance matching with the host plate. This confirms that our chosen\nthickness enables quantitative evaluation without compromising the vibro-spectrometer's\nintended function.\nIn conclusion, we systematically determined the piezoelectric element’s geometric parameters\nto minimize their influence and isolate the elastic wave’s effects, as the reviewer rightly\nemphasized. We have now included this rationale in the revised manuscript and supplementary\ninformation, which not only enhances the clarity of our design choices but also improves the\noverall completeness of our study for potential readers.\n(Revised manuscript, page 11)\n“The elastic wave energy confined at the focal positions is transformed into electrical energy\nusing a piezoelectric energy converter with diameter 𝑑 of 8 mm and thickness 𝑡 of 2\np p\nmm (Supplementary Fig. 15). The 𝑑 is purposefully set far below the shortest encoded\np\nwavelength of 19.2 mm at 40 kHz, ensuring that the evaluation reflects only the confined\nfield and is not influenced by spurious amplification or attenuation at antinodes or nodes. The\nselected 𝑡 maintains impedance compatibility with the background substrate, preserves\np\ncontinuum behavior, and prevents local-strain cancellation, while remaining well within the\nlinear response range for every working wavelength.”\n(Revised supplementary information, page 22-24)\nSupplementary Note 15. Geometric optimization of piezoelectric element\nSupplementary Figure 15. Open-circuit piezoelectric voltage responses as a function of\ngeometric parameters. a Output voltage as a function of sensor diameter at fixed thickness of\n𝟐 mm for excitation frequencies of 𝟒𝟎, 𝟔𝟎, and 𝟖𝟎 kHz. Voltage peaks appear near the\nexcitation wavelengths, confirming resonance-induced amplification. A dotted line indicates\nselected diameter of piezoelectric element. b Output voltage as a function of sensor thickness\nat fixed diameter of 𝟖 mm, showing significant voltage suppression when sensor thickness\napproaches that of the host plate of 𝟑 mm, due to impedance matching and strain energy\ndissipation. A dotted line indicates selected thickness of piezoelectric element.\n“The performance of disc-type piezoelectric sensors is strongly governed by two geometric\nparameters: diameter and thickness. While both parameters affect the sensitivity and dynamic\nbehavior of the sensor, their interdependence with the excitation wavelength and coupling\nmedium can introduce complex, nonlinear effects. In the context of the proposed vibro-\nspectrometer, which relies on quantitative comparisons of elastic wave responses, it is essential\nto isolate the influence of wave-induced strain from confounding artifacts such as geometric\nresonances and aperture effects6.\nTo enable reliable wave sensing, the output voltage of the piezoelectric element should ideally\nvary linearly with the incident elastic wave amplitude. This linearity is best achieved when the\npiezoelectric diameter is significantly smaller than the excitation wavelength. In this regime,\nthe sensor approximates a point detector, exhibiting minimal phase variation across its surface\nand thus avoiding destructive interference. In contrast, when the sensor diameter becomes\ncomparable to or larger than the wavelength, interference and phase cancellation across the\nsensing area reduce the output signal—a phenomenon referred to as the aperture effect.\nFurthermore, resonant amplification can occur when the piezoelectric diameter approaches the\noperating wavelength, introducing nonlinear behavior. This is particularly relevant for radial\nor Lamb wave excitation modes, where the sensor acts as a mechanical resonator, exhibiting\nenhanced output due to constructive modal coupling. To quantify these effects, we performed\nopen-circuit simulations in which the thickness was fixed at 2 mm while the diameter was\nvaried across excitation frequencies of 40, 60, and 80 kHz (Supplementary Fig. 15a). The\noutput voltage peaks were observed at diameters of 24, 20, and 18 mm—closely matching\nthe respective wavelengths of 27.1, 22.1, and 19.2 mm. These peaks correspond to resonant\nenhancement conditions. Notably, secondary maxima were also observed at approximately\ndouble the wavelength (54, 43, and 35 mm), indicative of harmonic responses. In the\nsubwavelength regime, the output increased nearly linearly with diameter across all frequencies,\nconfirming that small sensors maintain predictable performance. Based on this analysis, we\nselected a sensor diameter of 8 mm, which lies well within the subwavelength regime for all\noperational frequencies. This ensures minimal aperture and resonant effects, preserving\nlinearity and allowing for consistent comparison of incident wave intensities.\nThe thickness of the piezoelectric patch is another key parameter influencing its performance,\nparticularly in systems where the element is surface-mounted on a host plate. While thicker\nelements generally produce higher voltages due to larger internal strain fields, the interface\ndynamics with the host structure also play a crucial role. When the piezoelectric thickness\nmatches that of the background plate of 3 mm, the impedance mismatch is minimized,\nenabling efficient wave transmission but reducing internal strain accumulation within the\nsensor. This results in diminished voltage output, as the sensor becomes mechanically\nindistinguishable from the plate and fails to localize energy. To examine this effect, we varied\nthe sensor thickness while fixing the diameter at 8 mm, again under open-circuit conditions\n(Supplementary Fig. 15b). The results show a pronounced drop in output voltage when the\nthickness approached that of the host plate. This output minimum shifted depending on the\nexcitation frequency, reflecting the wavelength-dependent nature of wave-structure\ninteractions. When the sensor was either significantly thinner or thicker than the plate, output\nvoltage increased due to enhanced strain localization and scattering at the interface. As a result,\nwe chose a thickness of 2 mm—sufficiently distinct from the plate thickness—to avoid\nimpedance matching while maintaining linear piezoelectric response. This geometry ensures\nadadequate strain localization without introducing modal artifacts, enabling accurate\nquantification of elastic wave energy.\nTogether, these findings validate the chosen dimensions, diameter of 8 mm and thickness of\n2 mm, as optimal for minimizing geometry-induced distortion while ensuring high-fidelity\nelastic wave detection in the vibro-spectrometer system.”\n[6] Lee, T. G. et al. Enhanced energy transfer and conversion for high performance phononic\ncrystal-assisted elastic wave energy harvesting. Nano Energy 78, 105226 (2020).\nResponse to Reviewer #3’s comments\nGeneral Comment: In this manuscript, the authors outlined a frequency multiplexed elastic\nmetasurface design than can be leveraged as vibration spectrometer. This reviewer has the\nfollowing observations.\nResponse: We extend our heartfelt gratitude to the reviewer for dedicating the reviewer’s\nvaluable time to providing constructive comments, which have greatly enhanced the quality of\nour manuscript and deepened our understanding of the study findings. We have carefully\nconsidered and actively incorporated the reviewer’s insightful comments, addressing each\npoint raised and resolving the concerns expressed. We have provided a point-by-point response\nto each comment and have reflected these changes in the revised manuscript. We have made\nefforts to enhance the quality and professionalism of the manuscript based on the comments\nprovided by the reviewer, and through this revision process, we have strived to bring our\nmanuscript to a level worthy of publication in Nature Communications. We would like to\nexpress our gratitude to the reviewer for once again actively acknowledging the novelty and\noriginality of our manuscript.\nComment 3.1: The key idea of the research is to design a metasurface that can effectively map\nthe frequency spectral information to spatial deformation. While this is a direct application of\nwave focusing based on the generalized Snell’s law, it is indeed an interesting twist.\nResponse 3.1: We sincerely thank the reviewer for recognizing the novelty and excellence of\nthe metasurface we designed to realize the proposed vibro-spectrometer. As the reviewer has\nnoted, mapping spectral information to spatial deformation through dispersion engineering\nrepresents the first wave-control strategy of its kind in elastics. Guided by the simplest elastic-\nplate theory, we designed the meta-atoms and applied the generalized Snell’s law to focus\nwaves at the encoded focal positions; by integrating these elements, we introduced a new\nfunctionality. We again appreciate the reviewer’s high appraisal of the originality and\nversatility of our study, and we are confident that the proposed vibro-spectrometer will not only\npush beyond current design-methodology limits in elastic metasurfaces but also pave the way\nfor practical real-world applications.\nComment 3.2: The mathematical derivation of dispersion and wave focusing appear to be\nrigorous. One question does arise, as the authors employed Taylor series expansion around 𝜔 .\n𝑑\nWhat’s the limitation of this design?\nResponse 3.2: We thank the reviewer for the constructive comment that prompted us to clarify\nthe limitations of the mathematical frameworks employed in designing our vibro-spectrometer\nand, in doing so, to present a perspective for future work.\nIn this study, we adopted a dispersion engineering strategy that encodes multiple spectral\nfeatures within a single passive platform. Dispersion engineering in the metasurface field is\ntypically achieved by selecting meta-atoms that compensate the frequency-, polarization-, or\nangle-dependent differences in wave dispersion. Our work is the first to introduce dispersion\nengineering into elastics for a multiplexed metasurface, proposing a design strategy that\ntranscends the limits of conventional approaches on the basis of classical elastic dynamics.\nStarting from Kirchhoff–Love plate theory, we expressed the phase shift profile attainable with\na meta-atom as a function of a single geometrical parameter, allowing a direct link to the\nrequired phase profile derived from the generalized Snell’s law. By expanding both phase shift\nprofiles into Taylor series, we obtained an intuitive physical interpretation: the 0th-order term\nserves as the reference for multiplexing, while the 1st-order term is directly related the\nderivative of the phase shift with respect to frequency; that is, the group delay arising from\ndifferences in group velocity. Consequently, wave dispersion originates from group-velocity\nvariations, and dispersion engineering can be viewed as a strategy to compensate group delay.\nHowever, when the group delay varies steeply, this first-order description is insufficient. In\nsuch cases, higher-order terms must be considered. For example, the 2nd-order term,\n1𝜕2𝜓n (𝑥,𝜔) | (𝜔 −𝜔 )2 and 1𝜕2𝜑n (𝑡,𝜔) | (𝜔 −𝜔 )2, captures the group delay\n2 𝜕𝜔2 𝜔=𝜔 n d 2 𝜕𝜔2 𝜔=𝜔 n d\nd d\ndispersion caused by frequency detuning. Thus, phase differences across frequency can be\nascribed to a combination of group delay, group delay dispersion, and higher-order dispersion,\nand compensating all of these allows the design of a multiplexed elastic metasurface.\nIn the original manuscript we discussed only the 0th- and 1st-order terms; in the revision we\nhave generalized the explanation by incorporating higher-order dispersion and expanded the\nDiscussion section to outline future research directions that exploit this broader framework.\nThese additions, we believe, enhance both the completeness of the manuscript and the\ntransparency of the accompanying design methodology.\n(Revised manuscript, page 7-8)\n“As the first step in dispersion engineering, a reference phase shift profile is set at the design\nfrequency 𝜔 . By correcting the differences between this baseline and the phase required at\nd\nother frequencies, multiple data channels can be embedded within a single passive structure.\nThe idea is physically clarified by expanding the analytically derived phase functions 𝜑\nn\nand 𝜓 as Taylor series around 𝜔 = 𝜔 as:\nn d\n𝜑 (𝑡,𝜔 ) = 𝜑⏟ ( 𝑡, 𝜔 )+ 𝜕𝜑n (𝑡,𝜔) | (𝜔 −𝜔 )+𝑂((𝜔 −𝜔 )2), (4a)\nn n n d n d n d\n⏟ 𝜕 𝜔 𝜔 = 𝜔\nreference d\nphase group delay\n𝜓 (𝑥,𝜔 ) = 𝜓⏟ (𝑥 , 𝜔 )+ 𝜕𝜓n (𝑥,𝜔) | (𝜔 −𝜔 )+𝑂((𝜔 −𝜔 )2), (4b)\nn n n d n d n d\n⏟ 𝜕 𝜔 𝜔 = 𝜔\nreference d\nphase group delay\nwhere 𝑂((𝜔 −𝜔 )2) collects the group delay dispersion and higher-order dispersions. Eq.\nn d\n4 show that the phase needed at 𝜔 depends on the group delay evaluated at 𝜔 ; in other\nn d\nwords, dispersion engineering amounts to compensating the group velocity differences across\nthe working band.”\n(Revised manuscript, page 13)\n“Furthermore, introducing higher-order contributions of the Taylor series expansion, including\nthe group delay dispersion, into the dispersion relation enables a more precise physical analysis\nof phase mapping.”\nComment 3.3: What’s not discussed is the frequency sensitivity of this design, and its relation\nto unit-cell dimensions and numbers. This is a major drawback.\nResponse 3.3: We sincerely appreciate the reviewer’s critical comment regarding the necessity\nof including quantitative evaluations as objective indicators for assessing the performance of\nthe proposed vibro-spectrometer. We fully agree with the reviewer’s comment that including\nthe resolution of the vibro-spectrometer and the influence of the metasurface parameters would\nsignificantly enhance the value of the manuscript. In response, we have carefully revised the\nmanuscript to highlight the proposed vibro-spectrometer’s unique capability for spectral\ncomponent differentiation and its potential for practical applications. To clarify, the proposed\nsystem operates based on a mechanism where spectral information is inferred from spatial\ninformation by employing a frequency-multiplexed elastic metasurface. To rigorously evaluate\nthe performance of the proposed system, we identified and analyzed the following four key\nmetrics: 1) Spatial resolution – How tightly the system can spatially confine the wave energy\nduring focusing. 2) Spectral resolution – How well the system can distinguish between adjacent\nfrequency components. 3) Spectral bandwidth range – The extent of spectral bandwidth over\nwhich the vibro-spectrometer can effectively encode and recognize frequency information. 4)\nGeometric parameters of the meta-atom. We have now included a systematic analysis of these\nfour evaluation metrics in the revised manuscript and supplementary information to provide a\nquantitative and objective assessment of the vibro-spectrometer’s performance.\nThe detailed response to Comment 3.3 is identical to the response provided for Comment 2.1\nfrom Reviewer #2. As the content is rather extensive, we kindly request the reviewer to refer\nto Response 2.1 instead of reiterating the same explanation here. We sincerely appreciate your\nunderstanding.\nComment 3.4: The practical significance of the work was not reported correctly and accurately.\nExtracting frequency spectral information can be done by a variety of methods. The criticisms\nof current practice in Introduction are far-fetched. The method is only application to very\nsimple plate structure, not those commonly seen in ‘spacecrafts, rail vehicles, and vessels’.\nResponse 3.4: We sincerely appreciate the reviewer’s valuable comment regarding the\npractical scope and positioning of our work. We agree that the original manuscript may have\noverstated the generalizability of the proposed method and insufficiently acknowledged the\neffectiveness of existing spectral analysis techniques. In response, we have made the following\nrevisions to the Introduction and Discussion sections to more accurately reflect the context and\ncontribution of our work.\nFirst, we have revised the statements about conventional accelerometer- and LDV-based\nspectral analysis methods to acknowledge their precision and proven reliability, while\nhighlighting that our work is intended as a complementary, lower-cost and passive alternative\nin certain application scenarios. Next, we have adjusted the previous claims regarding real-\nworld deployment scenarios. Rather than suggesting direct application to large-scale or\ncomplex structures (e.g., spacecraft or vessels), we now cautiously describe the method’s\npotential applicability to plate-like subsystems within such systems. Finally, the concluding\nsentence of the original manuscript, which claimed direct utility in ‘spacecrafts, rail vehicles,\nand vessels,’ has been revised to emphasize conceptual contribution and clearly delimit the\nphysical scope of the demonstration. We believe these modifications appropriately address the\nreviewer’s concerns and help improve the clarity, accuracy, and practical framing of the\nmanuscript. We thank the reviewer again for the constructive feedback that helped improve the\nquality and precision of our work.\n(Revised manuscript, page 2)\n“Accordingly, the proposed vibro-spectrometer can be incorporated into various plate-based\nmechanical systems for next-generation technologies.”\n(Revised manuscript, page 3)\n“Meanwhile, elastic waves propagating in large-scale mechanical structures can provide\ncritical information on structural integrity. Accordingly, developing compact and passive\nalternatives for extracting their frequency-dependent behavior remains an important\nengineering challenge. Although existing methods provide accurate measurements, their\nlimitations in portability, cost, and energy consumption motivate exploration of\ncomplementary strategies that can be integrated into simple mechanical platforms.”\n(Revised manuscript, page 4)\n“Accordingly, the methodology is readily transferable to a broad spectrum of plate-based\nmechanical systems, potentially furnishing a unified platform for structural health monitoring,\nwireless sensing, and Internet of Things applications.”\n(Revised manuscript, page 13-14)\n“Furthermore, the proposed vibro-spectrometer can be integrated into or replace various plate-\nbased mechanical systems, offering new functionalities for next-generation applications.”\nComment 3.5: There are a number of inaccurate narratives in the manuscript. The authors\nsimply adopted a piezoelectric disc transducer which they claimed as ‘energy harvesting\ntechnology allowing wave energy to be converted into highly amplified electrical energy’.\nThere is no energy harvesting, nor is there any amplification.\nResponse 3.5: We thank the reviewer for the constructive comments that have helped us\nimprove the clarity of our manuscript. We fully agree that several terms we previously used to\ndescribe the operating principles of the vibro-spectrometer were inappropriate. As the reviewer\npointed out, the elastic waves that are spatially focused to separate spectral information cannot\ntruly be regarding as being “amplified.” To prevent confusion, we have replaced every\noccurrence of “amplification” with wording that conveys the “confined” energy.\nIn addition, we fully concur that the phrase “energy harvesting’ may be misleading because of\nits broad connotations. To describe the operation of the vibro-spectrometer more accurately,\nwe have replaced “energy harvesting” with “energy conversion,” emphasizing that the\npiezoelectric element directly converts elastic energy into an electrical signal without\naccumulation or gain. We greatly appreciate the reviewer’s meticulous feedback, which has\nenabled us to enhance the clarity, accuracy, and overall quality of the manuscript.\n(Revised manuscript, page 13-14)\n“Furthermore, integrating piezoelectric energy conversion technology allows the focused\nelastic wave energy to be converted into a readily distinguishable electrical energy.”\n(Revised manuscript, page 11)\n“Therefore, piezoelectric energy conversion technology, capable of transforming the strain\nenergy field induced by elastic waves in the medium into electrical energy, is incorporated into\nthe proposed system (Fig. 4a).”\n(Revised manuscript, page 11)\n“The elastic wave energy confined at the focal positions is transformed into electrical energy\nusing a piezoelectric energy converter with diameter 𝑑 of 8 mm and thickness 𝑡 of 2\np p\nmm (Supplementary Fig. 15).”\n(Revised manuscript, page 11)\n“In addition, to facilitate the attachment of the piezoelectric energy converter to the flat surface\nof the substrate plate, the bottom electrode extends along the lateral sides, allowing the\nextraction of the generated electrical energy alongside the top electrode.”\n(Revised manuscript, page 11-12)\n“To verify whether spectral components can be distinguished through measured signals, the\ntransverse displacement at the desired focal positions corresponding to 40, 60, and 80 kHz,\nalong with the voltage extracted from the piezoelectric energy converter, is analyzed in the\ntime-domain using FFT (Fig. 4b and Supplementary Fig. 17).”\n(Revised manuscript, page 13)\n“Furthermore, by integrating a piezoelectric energy conversion technology, the individual\nfrequency components can be discerned exclusively from the magnitude of the extracted\nelectrical output, since each encoded focal position generates a markedly distinct voltage\nsignature.”\n(Revised manuscript, page 14)\n“The piezoelectric energy converter is designed such that its resonant frequency does not\ncoincide with the selected frequency (23PbZrO –36PbTiO –41Pb(Ni Nb )O , 23PZ–36PT–\n3 3 1/3 2/3 3\n41PNN).”\n(Revised manuscript, page 15)\n“The resonant frequency of the piezoelectric energy converter is determined by measuring its\nimpedance and phase characteristics (E4990A, KEYSIGHT).”\n(Revised manuscript, page 22)\n“Figure 1. Frequency component detection capability of the vibro-spectrometer through\npiezoelectric energy conversion. a Piezoelectric energy converter with a diameter 𝒅 = 𝟖\n𝐩\nmm and thickness 𝒕 = 𝟐 mm attached to the desired focal positions, along with the electrical\n𝐩\ncircuit model representing the energy conversion process. b Fast Fourier transform results of\nthe time-domain signals of displacement and piezoelectricity at each focal position for the\ntarget frequencies. c Maximum electrical output voltage extracted from piezoelectric energy\nconverter at each focal position (𝑭 , 𝑭 , and 𝑭 at 𝟒𝟎, 𝟒𝟎, and 𝟖𝟎 kHz) as a function of\n𝟏 𝟐 𝟑\nload resistance 𝑹 for the target frequencies (𝒇 , 𝒇 , and 𝒇 at 𝟒𝟎, 𝟔𝟎, and 𝟖𝟎 kHz).\n𝐥𝐨𝐚𝐝 𝟏 𝟐 𝟑\nd Electrical power ratio at the encoded focal positions for the target frequencies.”\n(Revised supplementary information, page 25)\n“Supplementary Note 16. Piezoelectric characteristics of the energy converter”\n(Revised supplementary information, page 25)\n“Supplementary Figure 2. Piezoelectric characteristics of the energy converter. a\nImpedance and b phase spectrum of the customized piezoelectric energy converter as functions\nof frequency.”\n(Revised supplementary information, page 25)\n“In this study, piezoelectric energy conversion technology is introduced to enable vibro-\nspectrometer functionality by transforming confined elastic wave energy at encoded focal\npositions into electrical energy. The piezoelectric energy converter is a 2 mm-thick, 8 mm-\ndiameter disc composed of 23PbZrO –36PbTiO –41Pb(Ni Nb )O (23PZ–36PT–41PNN).”\n3 3 1/3 2/3 3\n(Revised supplementary information, page 25)\n“To identify the resonant frequency of the fabricated piezoelectric energy converter, an\nimpedance analyzer is used to measure the impedance and phase spectrum as functions of\nfrequency (Supplementary Fig. 16). The first effective resonant frequency is found to be\n271.49 kHz, confirming that the encoded frequencies do not interfere with the energy\nconverter’s piezoelectric performance and thereby enabling rigorous signal comparisons\nregardless of the vibro-spectrometer’s operating frequencies.”\n(Revised supplementary information, page 26)\n“Supplementary Figure 17. Time signals of displacement and piezoelectricity. a\nDisplacement-time signal measured using a laser Doppler vibrometer, and b Piezoelectricity-\ntime signal measured using a piezoelectric energy converter at the encoded frequencies.”\n(Revised supplementary information, page 26)\n“Furthermore, to determine if the operating frequency is preserved when the focused elastic\nwave energy is converted by the piezoelectric element, the output voltage from the\npiezoelectric energy converter was also measured over time (Supplementary Fig. 17b).”\n(Revised supplementary information, page 27)\n“The output voltage 𝑉 extracted from the piezoelectric energy converter is converted into\nelectrical power 𝑃 based on Ohm’s law, given by:”\n(Revised supplementary information, page 35)\n“To enable signal detection in the vibro-spectrometer, a customized piezoelectric energy\nconverter is attached at the desired focal positions. The generated electrical energy is measured\nthrough an external load resistance, and the corresponding output voltage is monitored by an\noscilloscope, thereby allowing the power ratio of the electrical signal to be evaluated.”\n(Revised manuscript, page 4)\n“At the desired focal position, the confined elastic wave energy corresponding to the target\nfrequency can be converted into electrical energy using a piezoelectric element, enabling\nspectral component recognition without requiring complex additional analysis.”\n(Revised manuscript, page 11)\n“Vibro-spectrometer using piezoelectric energy conversion”\n(Revised manuscript, page 11)\n“However, to function as a vibro-spectrometer, the overall system requires a method for\ndirectly detecting the confined elastic wave energy.”\n(Revised manuscript, page 11)\n“The piezoelectric element attached at the desired focal position exhibits significantly\ndistinguished electrical output compared to the background medium or other focal positions\nowing to the confined elastic wave energy.”\n(Revised manuscript, page 12)\n“The key aspect in validating the vibro-spectrometer functionality is assessing the degree of\nelectrical power identification at each focal position. To quantify this, the ratio of the output\npower at the optimal load resistance is compared across all nine cases (Fig. 4d). The results\nreveal notable power ratios of 38.8, 48.7, and 32.1 at the encoded frequencies compared to\nthe minimum output at 60 kHz extracted from 𝐹 , confirming that the proposed frequency\n1\nmultiplexed elastic metasurface successfully functions as a vibro-spectrometer.”\nResponse to Reviewer #4’s comments\nGeneral Comment: Review of \"Vibro-spectrometer with frequency multiplexed elastic\nmetasurfaces through dispersion engineering\". This manuscript presents a novel and elegant\nmethodology for designing frequency-multiplexed elastic metasurfaces for the purpose of\ncreating a \"vibro-spectrometer.\" The authors report an analytical design framework based on\ndispersion engineering that enables a passive metasurface to focus different elastic wave\nfrequencies to distinct, pre-programmed spatial locations. The work is well-executed, with\nstrong agreement between theoretical design, numerical simulations, and experimental\nvalidation. The main contribution lies in the design strategy itself, which cleverly avoids the\ntwo prevailing paradigms in the field: computationally demanding inverse design methods\n(such as topology optimization) and dependence on extensive libraries of intricate, resonant\nmeta-atoms. By adjusting the thickness of a monolithic plate, the authors introduce a much\nmore accessible and intuitive approach to advanced multi-frequency wave control.\nFurthermore, incorporating piezoelectric energy harvesters for direct, post-processing-free\nspectral detection provides a compelling and practical demonstration of the metasurface's\neffectiveness. The work is of high quality and represents a significant advance in the field of\nelastic metamaterials. The conclusions are well-supported by the experiment. The manuscript\nis well-written, and the figures are nice. I recommend publishing after the authors address a\nfew minor points to strengthen the paper further. The manuscript has several noteworthy results:\n(1) the development of a \"dispersion engineering\" methodology that multiplexes multiple\ndesired phase profiles onto a single, passive elastic metasurface. This method is based on\nclassical plate theory and an intuitive \"phase-phase difference space\" concept, avoiding\ncomplex algorithms, (2) The authors successfully designed and fabricated a metasurface that\nfocuses elastic waves at three different frequencies (40, 60, and 80 kHz) to three distinct off-\naxis locations. The experimental results show good agreement, validating the design approach,\nand the metasurface achieves high-resolution, diffraction-limited focusing, as quantified by the\nPSF, a high NA, and Strehl ratios consistently above 0.8. And finally, (3) the authors\ndemonstrate that the integration with piezoelectric energy harvesters provides a clear, practical\napplication. The demonstration that the spectral components of an incident wave can be\nidentified by simply measuring which harvester produces the highest voltage is a powerful\nresult, eliminating the need for complex post-processing. The work is significant and will be\nof interest to researchers in metamaterials, phononics, structural health monitoring, and\nmechanical engineering. Its primary significance lies in the novelty of its design methodology.\nCompared to the established literature (e.g., Rong et al., Adv. Funct. Mater. (2020)) that utilizes\ntopology optimization to design frequency-coded multifunctional elastic metasurfaces. This\ninverse design approach is powerful but computationally expensive and often results in\ncomplex, free-form geometries that can be difficult to fabricate and analyze intuitively. The\ncurrent work by Lee et al. provides a good contrast by achieving a similar outcome with an\nanalytical approach that is computationally simple and uses a simpler geometry (varying\nthickness). Another relevant example (Zhang et al., Phys. Rev. Applied (2020)) of a frequency-\nselected acoustic metasurfaces using coding meta-atoms based on dual-layer Helmholtz\nresonators. That approach required the careful design of a discrete library of resonant unit cells\nto achieve the desired phase responses (\"0\" and \"1\" codes) at specific frequencies. Lee et al.'s\nwork avoids the need for such a library of complex resonators, instead deriving the required\ncontinuous phase from a single, non-resonant meta-atom defined only by its thickness. So,\nwhile the concept of frequency-multiplexed wave control is not entirely new (as evidenced by\nRong et al. and Zhang et al. and many other studies), the method proposed by Lee et al. is a\nclear and original contribution. It overcomes key limitations of existing methods, i.e. their\ncomputational complexity and the need for complex resonant structures. It offers a more\nelegant, efficient, and physically intuitive path to designing such devices. The work on the\nvibro-spectrometer concept with integrated energy harvesting also pushes the field closer to\nreal-world applications. A revision could clarify or expand upon a few minor points to\nstrengthen the manuscript further. These would improve the paper's completeness.\nResponse: We extend our heartfelt gratitude to the reviewer for dedicating the reviewer’s\nvaluable time to providing constructive comments, which have greatly enhanced the quality of\nour manuscript and deepened our understanding of study findings. We have carefully\nconsidered and actively incorporated the reviewer’s comments, addressing each point raised\nand resolving the concerns expressed. We have provided a point-by-point response to each\ncomment and have reflected these changes in the revised manuscript. We would like to express\nour gratitude to the reviewer for once again acknowledging the novelty and originality of our\nmanuscript and actively recommending it for publication in Nature Communications.\nComment 4.1: The paper demonstrates successful operation at three distinct, well-separated\nfrequencies. A key characteristic of any spectrometer is its resolution. The authors show that\nat non-encoded frequencies, the focusing is compromised (Supp. Fig. 7), which is a good start.\nA more quantitative discussion on spectral resolution would be beneficial. How close can two\nfrequencies be before their focal spots are no longer distinguishable? A brief theoretical\ndiscussion on how this resolution relates to the metasurface parameters would add significant\nvalue.\nResponse 4.1: We sincerely appreciate the reviewer’s critical comment regarding the necessity\nof including quantitative evaluations as objective indicators for assessing the performance of\nthe proposed vibro-spectrometer. We fully agree with the reviewer’s comment that including\nthe resolution of the vibro-spectrometer and the influence of the metasurface parameters would\nsignificantly enhance the value of the manuscript. In response, we have carefully revised the\nmanuscript to highlight the proposed vibro-spectrometer’s unique capability for spectral\ncomponent differentiation and its potential for practical applications. To clarify, the proposed\nsystem operates based on a mechanism where spectral information is inferred from spatial\ninformation by employing a frequency-multiplexed elastic metasurface. To rigorously evaluate\nthe performance of the proposed system, we identified and analyzed the following four key\nmetrics: 1) Spatial resolution – How tightly the system can spatially confine the wave energy\nduring focusing. 2) Spectral resolution – How well the system can distinguish between adjacent\nfrequency components. 3) Spectral bandwidth range – The extent of spectral bandwidth over\nwhich the vibro-spectrometer can effectively encode and recognize frequency information. 4)\nGeometric parameters of the meta-atom. We have now included a systematic analysis of these\nfour evaluation metrics in the revised manuscript and supplementary information to provide a\nquantitative and objective assessment of the vibro-spectrometer’s performance.\nThe detailed response to Comment 4.1 is identical to the response provided for Comment 2.1\nfrom Reviewer #2. As the content is rather extensive, we kindly request the reviewer to refer\nto Response 2.1 instead of reiterating the same explanation here. We sincerely appreciate your\nunderstanding.\nComment 4.2: The design is based entirely on the KL plate theory, which is an approximation\nfor thin plates. The authors should briefly state the assumptions of this theory (e.g., ignoring\nshear deformation, etc.) and discuss the range of validity in the context of their design. This\nprovides important context for readers seeking to apply this method to other systems.\nResponse 4.2: We want to extend our heartfelt thanks to the reviewer for offering insightful\ncomments that can greatly improve the expertise and quality of our manuscript. As correctly\nnoted, the vibro-spectrometer is designed on the basis of the classical Kirchhoff–Love (KL)\nthin-plate theory, which neglects shear deformation and rotational inertia. This approximation\nis valid when the elastic wavelength propagating in the plate is smaller than the plate thickness;\nin our study, the 3 mm-thick plate used as the background medium yields a cut-off frequency\nof 1.1 MHz, below which the bending-wave dynamics are accurately captured by the KL\nmodel. For higher frequencies—or for plates whose thickness is no longer negligible compared\nwith the wavelength—the KL theory progressively accumulates error, and the Reissner–\nMindlin (RM) theory, which accounts for both shear deformation and rotational inertia, should\nbe employed for precise prediction. Interestingly, the phase shift derived under the RM\nframework can still be tuned by varying only the plate thickness. Nevertheless, the key novelty\nof the proposed elastic-metasurface platform lies in its ability to encode multiple spectral\nsignatures into a passive system through dispersion engineering. Consequently, by selecting\nthe classical elastic theory best suited to the target operating regime, our vibro-spectrometer\ncan be readily implemented while preserving the originality and robustness of the design. As a\nresult, in line with the reviewer’s constructive suggestion, we have added these theoretical\nassumptions and future perspectives to the Discussion section of the revised manuscript.\n(Revised manuscript, page 13)\n“Interestingly, the analytical framework derived from the KL plate theory exhibits a linear\ndependence on 𝜔1⁄2; therefore, simply scaling the geometric parameters enables effortless\nretuning to alternative frequency ranges. In higher-frequency regimes, where the assumptions\nof KL theory are no longer valid, the methodology remains applicable by replacing that model\nwith the Reissner–Mindlin plate theory, which accounts for shear deformation and rotational\ninertia31,32.”\n[31] Reissner, E., The effect of transverse shear deformation on the bending of elastic plates,\nJ. Appl. Mech. Trans. ASME 12, A69-A77 (1945).\n[32] Mindlin, R. D., Influence of rotatory inertia and shear on flexural motions of isotropic\nelastic plates, J. Appl. Mech. Trans. ASME 18, 31-38 (1951).\nComment 4.3: The authors note that the harvester's resonance is far from the operating\nfrequencies. However, the harvester's physical mass and stiffness could locally perturb the\nelastic field at the focal point. While this is likely a small effect, including a sentence that\nacknowledges this and justifies why it can be considered negligible would enhance the rigor of\nthe analysis.\nResponse 4.3: We would like to express our sincere gratitude to the reviewer for providing\ncritical comments that helped us significantly improve the quality of our manuscript. We\nacknowledge that our previous version did not sufficiently address the reviewer’s concerns\nregarding the influence of the geometric configurations of the piezoelectric element on the\nefficiency of the energy harvesting process. Importantly, we would like to emphasize that our\ndesign choices were not arbitrary nor deliberately skipped, but rather the result of a rigorous\nand rational pre-analysis process.\nThe detailed response to Comment 4.3 is identical to the response provided for Comment 2.6\nfrom Reviewer #2. As the content is rather extensive, we kindly request the reviewer to refer\nto Response 2.6 instead of reiterating the same explanation here. We sincerely appreciate your\nunderstanding.\nResponse to Reviewer #1’s comments\nGeneral Comment: Reviewer #1 did not leave comment, and just told us he/she was happy\nwith the revision.\nResponse: We sincerely thank Reviewer #1 for carefully evaluating our revised manuscript\nand for expressing satisfaction with the revisions. We greatly appreciate the reviewer’s\nsupportive assessment and are encouraged by their positive feedback.\nResponse to Reviewer #2’s comments\nGeneral Comment: I have to thank the authors because they have put a lot of effort in\naddressing my comments of the first revision round. However, I didn’t find some of their\nexplanations convincing. (…) In conclusion, I would again recommend to better describe the\nlimitations of the proposed solution, tone down its description as a \"spectrometer\" (in the title\nas well) and focus on the wave control capability.\nResponse: We sincerely thank Reviewer #2 for the thorough re-evaluation of our revised\nmanuscript and for acknowledging our efforts in addressing the earlier comments. We deeply\nappreciate the constructive feedback, which has helped us further improve the clarity and\nbalance of our work. We understand the reviewer’s concern that describing our device as a\n“spectrometer” could be misleading. In response, we have revised the title and the relevant\nparts of the manuscript to avoid overstating its capability. Instead, we now emphasize the\nfrequency-selective wave control and detection functionality of the metasurface, which better\nreflects the essence of our work. Furthermore, we have expanded the discussion of the\nlimitations highlighted by the reviewer, including the requirements on frequency separation,\nincidence angle, and frequency resolution, and we now present these constraints more\nexplicitly. By clearly stating these limitations, we believe the manuscript provides a more\nbalanced description of the proposed approach. We are grateful to the reviewer for these\nvaluable suggestions, and we believe that the revisions have significantly strengthened the\nmanuscript in both accuracy and presentation.\nComment 2.1: My fundamental concern is that defining what is introduced here as a\n\"Spectrometer\" seems to be misleading. A spectrometer is an apparatus used for recording and\nmeasuring spectra, here it was just demonstrated that signals confined in three frequency bins\ncan be detected with these additional limitations: 1) the separation between the frequencies\nmust be wider than 16kHz; 2) the impinging wave must be planar and the incident angle must\nbe almost perfectly perpendicular to the metasurface (since the focal point is changed by the\nangle of arrival, the piezodisk used as detector will miss the focused wave otherwise); 3) we\nhave to tolerate an ambiguity in the frequency detection of 12, 11, and 3 kHz, for the 40, 60\nand 80kHz wave detectors, respectively. This last point seems to be quite annoying, because\nthis implies having the worst frequency resolution at the lowest frequency, which is the\nopposite of what you want in typical signal processing tasks (think about the multiresolution\ntime-frequency decomposition of the wavelet transform which is the opposite: high frequency\nresolution at low frequencies).\nAnother aspect that I don't find convincing is the claimed superiority with respect to a purely\nelectronic signal processing. First, it must be clarified that what the metasurface can do is much\nmore similar to a filter bank with three narrowband filters rather than a DFT. Implementing\nsuch filterbank with passive analog (electrical) filters is simple, low-power and low-cost. It is\nalso claimed that the proposed solution allow \"to identify frequency solely by comparing the\namplitudes of the converted electrical signals\" but it would be nice to specify with greater detail\nwhat would be the signal conditioning circuit that would allow this result. My concern is that\nthe different waves will impinge upon the different piezodisk locations in different time instants\nso, in order to re-align the signals and compare them, it seems to be necessary to digitalize,\nstore and process them. This means multiple ADC converters and dedicated DSP routines.\nResponse 2.1: We sincerely appreciate the reviewer for providing such critical comments that\nhave significantly contributed to improving the objectivity and completeness of our manuscript.\nWe fully agree with the reviewer’s observation that our claims regarding the proposed\nfrequency-selective wave-control device being described as a “spectrometer”, as well as those\nrelated to sensing through confined energy conversion, were somewhat overstated.\nAccordingly, we have withdrawn the claim of “spectrometer” and revised the manuscript\nthroughout to emphasize the wavefront modulation capability of the frequency-multiplexed\nelastic metasurface. In addition, we have made effort to enhance the objectivity of the\nmanuscript by incorporating the limitations and future perspectives of the proposed platform,\nas the reviewer kindly suggested as follows.\n1) The separation between the frequencies must be wider than 16 kHz – As mentioned in\nSupplementary Fig. 6, the minimum spacing of 16 kHz between the encoded frequencies in the\nproposed elastic metasurface applies only to the designed system for the proof-of-concept\ndemonstration. This value naturally varies depending on factors such as the target frequency,\nsystem size, number of channels, and numerical aperture (NA), and therefore does not represent\nthe intrinsic capability of the metasurface. However, as the reviewer rightly pointed out, once\nthe system is claimed as a “spectrometer”, this value becomes highly critical. Therefore, we\nhave withdrawn the claim of “spectrometer” and thoroughly revised the manuscript, including\nthe title, to focus on the frequency-selective wavefront-tailoring capability rather than spectral\nresolution. Furthermore, we have added the relevant limitations and future perspectives\nconcerning spectral resolution in the Discussion section, in order to clarify the boundaries of\nthe proposed system and enable potential readers to evaluate our work more objectively.\n(Revised manuscript, page 1)\n“Title: Frequency multiplexing with dispersion-engineered elastic metasurfaces”\n(Revised manuscript, pages 3-4)\n“Here, for identifying the spectral components of elastic waves, a dispersion-engineered elastic\nmetasurface (DEM) for frequency multiplexing is introduced. The proposed DEM focuses\nelastic waves at distinct focal positions based on the encoded frequencies, utilizing a frequency-\nmultiplexed elastic metasurface through dispersion engineering (Fig. 1a).”\n(Revised manuscript, page 9)\n“To ensure that the DEM achieves crosstalk-free focusing across adjacent frequency channels,\nthe operating frequencies are deliberately chosen to preserve a distinct and robust frequency-\ndiscrimination capability (Supplementary Fig. 6).”\n(Revised manuscript, page 13)\n“Additionally, further optimization can enhance the flexibility and spectral resolution of the\nproposed theoretical approach.”\n2) The impinging wave must be planar and the incident angle must be almost perfectly\nperpendicular to the metasurface – We agree with the reviewer’s insightful comment that the\nproposed elastic metasurface performs encoded focusing only under normal incidence and\ngenerates unintended wavefields under oblique incidence, thereby presenting practical\nlimitations in functioning as a spectrometer. However, we would like to note that maintaining\nwavefront-shaping functionality over wide angular ranges is itself a distinct research topic, and\nin particular, studies addressing wide-angle operation for elastic metasurfaces remain\nextremely limited. Thus, exploring this aspect lies somewhat beyond the central focus of the\npresent work, which emphasizes the frequency-multiplexing capability imparted by dispersion\nengineering. For this reason, we have withdrawn the claim regarding spectrometer functionality,\nclarified the angle-dependent limitations of the demonstrated elastic metasurface, and, to\nfurther enhance the objectivity and completeness of the manuscript, proposed as a future\nperspective the potential extension of metasurface design strategies widely studies in photonics\nand acoustics—such as quadratic metasurfaces—into elastodynamics.\n(Revised manuscript, page 13)\n“The demonstrated DEM is presently encoded only for normal incidence, thereby imposing a\nlimitation on the angle-of-incidence (Supplementary Fig. 19). Nevertheless, it is in principle\nextendable to multiplex wavefronts under oblique incidence, and by adopting strategies such\nas quadratic phase profile, it can be further advanced into a metasurface operable over wide-\nangle conditions26.”\n3) We have to tolerate an ambiguity in the frequency detection of 12, 11, and 3 kHz, for the 40,\n60 and 80kHz wave detectors, respectively – We agree with the reviewer that, from the\nperspective of typical signal processing, the spectral bandwidths of 12, 11, and 3 kHz at the\nencoded frequencies of 40, 60, and 80 kHz, respectively, appear to yield a rather paradoxical\noutcome. However, this uncommon trend arises because the signal processing mechanism\nenabled by the proposed DEM operates on a principle fundamentally different from that of a\nconventional spectrometer. In the original manuscript, we argued that frequency-selective\nwavefront shaping enables the formation of distinct focal spots, thereby mapping spectral\ninformation into spatial information. Under such circumstances, the spectral bandwidth and\nresolution are determined by factors different from those of conventional spectrometers,\nincluding the target frequency, system size, number of channels, and NA. For the proof-of-\nconcept demonstration, the encoded focal positions were designed such that the NA decreases\nas the frequency increases. Since a smaller NA exacerbates chromatic aberration with respect\nto changes in the target frequency, this in turn results in more restricted spectral bandwidth at\nhigher frequencies. Taken together, these coupled effects lead to trends that deviate from those\nof conventional spectrometers, making it inappropriate to quantitatively evaluate the proposed\nDEM using the same performance metrics. Consequently, we fully agree with the reviewer’s\nconcern that claiming spectrometer functionality for the DEM-based signal processing is\noverstated. To address this, we have withdrawn such claims, revised the manuscript to focus\non elastic-wave modulation, and explicitly clarified the limitations concerning spectral\nselectivity, thereby making every effort to resolve the reviewer’s concern.\n(Revised manuscript, page 10)\n“In addition, under the configuration devised to validate the frequency-multiplexing capability,\nthe designed DEM retains precise focusing within spectral bandwidth of 12, 11, and 3 kHz\ncentered at the encoded frequencies, whereas frequencies outside these passbands give rise to\nunintended intensity distributions (Supplementary Fig. 9).”\nFurthermore, we also fully agree with the reviewer’s comment regarding the comparison with\nelectronic signal processing. Our intention was merely to introduce a new mechanism for\nspectral information detection, demonstrating that spectral information can also be inferred\nthrough energy conversion into a different form of energy. Therefore, we have removed the\nprevious claims regarding the superiority of spectrometer-like functionality and revised the\nmanuscript to tone down the discussion, focusing instead on frequency-selective modulation\nbased on wave dynamics.\n(Revised manuscript, page 10)\n“Furthermore, we evolve the overall system into a strategy that affords facile discrimination of\nconfined elastic-wave energy according to the encoded frequency. To this end, we incorporate\npiezoelectric energy conversion technology21,22, capable of transducing the strain-energy fields\ninduced by elastic waves in the medium into electrical energy, into the proposed system (Fig.\n4a).”\nComment 2.2: Finally, the selection of the piezodisk diameter and thickness seems to be\nsuboptimal: by looking at Supplementary Figure 15, it is clear that a larger piezodisk in\ndiameter and/or a thinner one would generate stronger amplitude response, and consequently\nbetter signal to noise ratios. The motivations presented by the authors are not justified by\nexamples, or sufficiently detailed physical analysis, specifically for what concerns the\nlimitations of the \"linear response range\".\nResponse 2.2: We sincerely thank the reviewer for this insightful comment that allows us to\nclarify the rationale behind the chosen geometry of the piezoelectric disk. We fully\nacknowledge, as correctly noted, that increasing the disk diameter or reducing its thickness\nwould integrate a larger portion of the strain field and therefore enhance the output amplitude\nand signal-to-noise ratio (SNR).\nHowever, the central design objective of our work is not to maximize the absolute output but\nto preserve the fidelity of the encoded spectral information. When the disk size becomes\ncomparable to or larger than a significant fraction of the working wavelength, phase variations\nwithin the strain field can cause partial cancellation, leading to distorted frequency responses.\nSimilarly, excessive thinning of the disk can introduce impedance mismatching or unwanted\nresonances, which again risks biasing or distorting the spectral information extracted from the\ncoupled wave dynamics. For this reason, we intentionally selected geometric parameters that\nensure stable piezoelectric output within the linear response regime, where the conversion\nefficiency varies smoothly with frequency and does not introduce spurious resonance effects.\nIn this sense, our design represents a trade-off between output amplitude and spectral fidelity,\nwith the latter being prioritized given the focus of this work. We agree with the reviewer that\nsystematic optimization of the piezoelectric geometry for maximizing SNR, while\nsimultaneously safeguarding spectral fidelity, is a valuable research direction. To reflect this\nconstructive suggestion, we have revised the Discussion section to explicitly state this\nlimitation and to highlight geometry optimization as a promising avenue for future work.\n(Revised manuscript, page 13)\n“In addition, coupling the DEM with optimally designed piezoelectric elements tailored to the\nencoded frequencies can enhance the signal-to-noise ratio, thereby sharpening the retrieved\nspectral information, while integration with electrical circuitry can offer pathways toward more\nreliable signal detection.”\nResponse to Reviewer #3’s comments\nGeneral Comment: The authors have addressed almost all the concerns raised by this reviewer.\nThere is one question about the usage of Taylor series expansion in dispersion analysis. The\nauthors now included higher order terms in the analysis. It is still unclear to this reviewer how\nthe higher order terms entail in the context of ‘design’. Nevertheless, this reviewer agrees that\nsuch analysis could be future work. Overall, this reviewer feels the revised manuscript is\nacceptable.\nResponse: We sincerely thank Reviewer #3 for recognizing our efforts in addressing the earlier\nconcerns and for the positive evaluation of the revised manuscript. Regarding the usage of\nhigher-order terms in the Taylor series expansion, we agree that a more detailed exploration of\ntheir role in practical design would be highly valuable. As the reviewer correctly pointed out,\nthis goes beyond the current scope of our work and is indeed a promising direction for future\nresearch. We are grateful for this insightful suggestion, and we will consider it carefully in our\nfollow-up studies.\nResponse to Reviewer #4’s comments\nGeneral Comment: The authors addressed all our concerns and the comments from other\nreviewers very diligently and provided sufficient changes to grant publication of this\nsignificantly improved manuscript. Thank you!\nResponse: We sincerely thank Reviewer #4 for the careful evaluation of our revised\nmanuscript and for the very encouraging feedback. We are truly grateful that the reviewer\nrecognizes our efforts in addressing all the concerns and is supportive of the publication of the\nimproved version of our work.\nResponse to Reviewer #2’s comments\nGeneral Comment: The authors have addressed all my comments appropriately and, in this\nreviewer opinion, the manuscript deserves now consideration for publication.\nResponse: We sincerely thank Reviewer #2 for carefully evaluating our revised manuscript\nand for expressing satisfaction with the revisions. We greatly appreciate the reviewer’s\nsupportive assessment and are encouraged by their positive feedback.\nResponse to Reviewer #3’s comments\nGeneral Comment: The authors have addressed my concern.\nResponse: We sincerely thank Reviewer #3 for carefully evaluating our revised manuscript\nand for expressing satisfaction with the revisions. We greatly appreciate the reviewer’s\nsupportive assessment and are encouraged by their positive feedback."}]

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