Graded phononic metamaterials based on scalable microfabrication and design

Domain: chemistry
Published: 2026-02-25
Editorial outcome: minor_revision

Abstract
Metamaterials’ engineered internal structures enable customized material properties beyond those found in nature, such as the capability to guide, attenuate, and focus waves at will. Phononic metamaterials aim to manipulate mechanical waves, with broad applications in acoustics, elastodynamics and structural vibrations. A key bottleneck in the advancement of phononic metamaterials is their scalability beyond tens of unit cells per spatial dimension, which equally affects their design, simulation, and fabrication. Here, we present a framework for scalable inverse design of spatially graded phononic metamaterials for elastic waveguiding, together with a scalable microfabrication method. This framework enables the design and realization of complex waveguides including hundreds of thousands of unit cells, potentially extendable to millions with no change in protocol. Scalable designs are optimized with a ray tracing model for waves in spatially graded beam lattices and fabricated by photolithography and etching of silicon wafers, to create free-standing microarchitected films. Wave guiding is demonstrated experimentally by using pulsed laser excitation and interferometric displacement measurements. Broadband wave guiding is demonstrated, indicating the promise of our scalable design and fabrication methods for on-chip elastic wave manipulation.

Introduction
The nature of wave propagation in periodic lattices drives the fundamental behavior of materials, from the electronic band structure to thermal and optical properties. Phononic metamaterials build a bridge from wave phenomena at the atomic [1] to engineering scales, where lattices can be carefully designed to manipulate elastic waves [2]. As rapidly advancing manufacturing technology has enabled the fabrication of intricate geometries, a rich literature on phononic metamaterials has emerged in pursuit of harnessing waves for broad applications including vibration isolation [3, 4], sensing [5], and energy harvesting [6].

A key limitation that plagues the advancement of phononic metamaterials is the scalability of both computation and fabrication [7–9]. Modeling, design, and manufacturing efficiency remains a bottleneck for architectures spanning tens of thousands to millions of unit cells, which aim to dissolve the differences between materials and structures. Solving the problem of scalability promises an enlarged and widely untapped design space beyond periodic architectures. Extension of the design space to non-periodic architectures introduces wave manipulation functionalities out of reach of periodic architectures. Examples include wave guiding along topological interfaces between different unit cell architectures within a bandgap [10] and the introduction of defects for localizing frequencies within a bandgap [11]. Alternatively, spatial grading (introducing smooth variations of the unit cell in space) enables wave guiding and focusing within a pass-band [12] with potential for broadband applicability. Scaling spatial grading to architectures with many unit cells opens a large design space of complicated multiscale spatial grading profiles. This expansion of the design space can enhance applications that spatial graded phononic metamaterials already show promise for (such as energy harvesting [13] and signal processing [14]), as well as uncover new wave manipulation capabilities.

On the computational side, the efficiency of modeling and design methods is a bottleneck to scalability due to the multiscale nature of phononic metamaterials. While efficient multiscale modeling and design methods are available for periodic architectures thanks to Bloch’s theorem [1], periodic architectures barely scratch the surface of the vast metamaterial design space. Looking beyond periodic architectures, spatial grading promises enhanced functionalities such as wave focusing [15], broadband attenuation [16, 17], and signal processing [18]. However, graded architectures are more complicated to model than periodic architectures, since multiple length scales must be resolved, compared to modeling a single unit cell of a periodic architecture. High-fidelity computations (such as transient dynamic finite element analysis) scale poorly to large architectures, where each unit cell must be modeled, which becomes increasingly expensive for inverse problems that require many evaluations of a forward model. Therefore, most existing spatial grading design methods have relied on heavily restricted design spaces (for example, linear gradings [19] and analytical solutions [20]) or restricting assumptions of long wavelengths to enable homogenization [21, 22]. Specialized homogenization methods have been developed to capture dispersive effects at finite frequency [23, 24]. High-frequency homogenization offers means of approximating dispersion relations, which has been used, for example, to design meta-structures consisting of multiple unit cell domains to achieve localize energy [25] (rather than modeling spatial gradings). Thus, there is a rich but largely unexplored design space of spatially graded phononic metamaterials spanning large numbers of unit cells.

The authors’ previous work [26, 27] offers a foundation for the design of complex spatial gradings using ray theory (which is not restricted to the low-frequency homogenization regime and is based on the full local dispersion relations in the neighborhood of each unit cell). Ray tracing provides a computationally efficient tool for modeling wave motion through graded architectures that enables optimal design (see Supplementary Material Section 2.3 for a discussion of computational efficiency). In this work, our design methodology generalizes [27] to capture physically realistic beam-based architectures and broader design objectives that enable scalability though modular design.

On the experimental side, the fabrication of 2D wave-guiding phononic metamaterials using standard machining and 3D-printing techniques are limited to tens to hundreds of unit cells [19, 28, 29], making it difficult to truly fabricate metamaterials rather than structures. Going from tens to hundreds of thousands of unit cells at the meso-scale requires the adaptation of specialized microfabrication techniques that have rarely been exploited for free-standing elastic wave guides [30] – especially spanning tens of micrometers to millimeters. The current state of the art in commercial macroscale 3D printing technology enables architectures with hundreds of unit cells per dimension [31]. These methods, which fall within the broad category of additive manufacturing (AM), have been well understood for polymeric materials that unfortunately are not suitable for wave propagation because of material damping. Although exciting new directions in metal and ceramic AM have emerged [32–34], there is still much work to be done to control printing resolutions and material properties. Microfabrication techniques enabled by multi-photon lithography, traditionally used with polymeric materials at much superior spatial resolution [35], have breached this limitation [36, 37]. Such techniques, while growing rapidly, are still nascent in their ability to achieve large sample sizes close to 100 mm while maintaining sub-micrometer spatial resolution [38, 39], lesser so when spatially-graded non-periodic architectures are involved. Scalable fabrication of low-damping 2D elastic waveguides is especially challenging for nano- and microscale manufacturing, where two-photon lithography [35, 40–44] and its variants [45] can hardly scale beyond samples with tens of hundreds of unit cells per dimension in a tractable build time. The current state of the art in fast fabrication rates is still limited to polymeric materials [46]. Microfabrication of optical metamaterials has achieved better scalability, with two-dimensional architectures reaching over a billion unit cells, using electron beam lithography and atomic layer deposition [47, 48]. Such applications traditionally require smaller sample sizes than those required here to guide elastic waves. At such small length and time scales, the excitation and measurement of waves becomes a challenge, which often requires custom-developed pump-probe optical systems [49–51].

In this work, we introduce a solution to phononic metamaterial scalability both computationally and experimentally. Our results enable the design and realization of elaborate spatially graded metamaterials spanning at least three orders of magnitude in length scales with tens of thousands of unit cells. To achieve scalable computational design, we leverage an optimization framework based on ray tracing for efficient modeling of wave propagation. A modular design approach is developed, where multiple tiles are independently designed to achieve different wave guiding functionalities. The tiles act as building blocks that are assembled like puzzle pieces into large designs with customizable waveguiding capabilities. To realize our designs, microfabrication based on photolithography and etching of silicon wafers creates free-standing architected films. Experimental demonstration of wave guiding is achieved using a nanosecond pulsed laser for excitation and heterodyne interferometry for measurement of particle displacements. The demonstrated design and microfabrication concept opens doors to new applications such as on-chip vibration isolation and signal processing for microscale electromechanical systems (MEMS).

Results
Modular waveguide design

Spatially grading the properties of a material is a powerful tool for manipulating propagating waves, which has long been used, for example, for gradient-index optics [52]. Extending this idea to grading the unit cells of phononic metamaterials introduces a multiscale problem, which is computationally challenging to simulate and optimize. Brute-force transient simulations (for example, using finite elements) are accurate but prohibitively slow for inverse design of large microstructures. Hence, existing work has relied primarily on simple, intuitive designs (such as radially symmetric and linear gradings [12, 15, 19, 53]), leaving a rich untapped design space of complex spatial gradings to be explored.

To circumvent the computational efficiency bottleneck, we leverage ray tracing to model wave motion in graded phononic metamaterials, generalizing well-established ray theories for smooth continua (for example, seismic ray theory [54] and geometric optics [55]) to graded metamaterials [26]. Ray tracing in phononic metamaterials provides an efficient modeling tool, which relies on local Bloch wave analysis in the neighborhood of each unit cell to compute local dispersion relations (assuming smooth spatial gradings and hence locally an approximately periodic medium). Consequently, this method applies both within and above the low-frequency homogenization limit, since the complete dispersion relations are accounted for. The resulting spatially variant local dispersion relations act as a Hamiltonian for tracing rays to determine how waves propagate in graded phononic metamaterials. While ray tracing is appealing due to its efficiency, it is an approximation under the assumption that spatial grading is slow relative to relevant wavelengths, which inherently introduces some error. Rigorous theoretical error analysis given complicated grading profiles and general dispersion relations is an open problem; here we ensure that ray tracing solutions are representative by comparison to transient finite element simulations.

In this work, we adopt ray tracing for efficient forward modeling, which is crucial to achieve scalability to large and complicated graded geometries. An optimization-based inverse design framework is built around ray tracing by extending our previous formulations [27]. Specifically, an optimization problem is posed to design the spatial distribution of unit cells such that the corresponding ray paths are shaped in a prescribed way. Consider the unit cells in Figure 1c, in which each edge represents an elastic beam. The unit cell has one design parameter θ, which is restricted to the range 0.1L ≤ θ ≤ 0.6L for manufacturability (L is the height and width of the square unit cell). The foundation for forward modeling using ray tracing is the local dispersion relation throughout the metamaterial, which is computed using a beam finite element model [28, 56]. Focusing on out-of-plane bending vibrations as in our experiments, the lowest dispersion surface is plotted in Figure 1c for the two extremes of the design space. For this unit cell, the lowest out-of-plane dispersion surface does not intersect other out-of-plane dispersion surfaces, so it can be studied in isolation. We aim to optimally design the spatial distribution of θ in a metamaterial consisting of many unit cells to achieve a prescribed wave guiding objective through the resulting spatial variation of the (local) dispersion relations.

An optimization problem is solved to shape the ray trajectories in two spatially graded square tiles, shown in Figure 1d, each spanning 64×64 unit cells. The first design, Tile 1, considers a point excitation at the center unit cell (from which the rays originate) and designs the spatial distribution of θ to guide the resulting wave to exit the tile along the four tile edges with rays perpendicular to the edges. The second tile, Tile 2, considers an incident plane wave, with horizontal rays entering from the left tile boundary, and redirects the rays to exit perpendicular to the bottom tile edge, thus rotating the incident plane wave by 90◦. Both tiles are designed to have θ = 0.4L around the perimeter, to ensure compatibility between tiles when assembled into the metamaterial (Figure 1a). The distribution of θ and the corresponding ray trajectories are plotted in Figure 1d for both tiles. Details of the optimization setup for tile design are presented in the Supplementary Information Section 1. This optimization setup is generally formulated to handle a wider range of objectives, so additional tiles achieving functionality beyond that of Tiles 1 and 2 could be developed in future work.

Since the two tiles are designed to be compatible, there is no jump in θ at the shared boundary when the two tiles are placed next to each other. Consequently, ray paths of Figure 1d can be continuously connected between adjacent tiles. This allows for solutions to be assembled by placing tiles next to each other such that the rays of each tile are connected to form a desired set of ray paths. Figure 2a provides an example of a tile assembly, which plots the spatial distribution of θ with dotted lines highlighting the boundary between tiles. The corresponding ray trajectories are shown in Figure 2b, which are connected to form a figure-eight shape. Thus, waves emerging from out-of-plane excitation at the origin are guided along the figure-eight. The geometric realization of this assembled design is shown in Figure 2c, where the geometry of each unit cell is determined from the designed distribution of θ in Figure 2a. Note that, since all rays in the figure-eight return to the point of excitation, the design – in the ideal case without dissipative losses – would result in continued traversal of the figure-eight even after the point excitation is removed.

Design by tile assembly is a scalable approach that leads to many possible designs. While brute-force optimization of ray trajectories on a large domain could be formulated, such an approach scales poorly with increasing domain size compared to modular design by tile assembly. Once each tile is designed, connecting them to achieve large-scale designs is trivial and introduces no additional computational cost. A second example of assembled tiles is shown in Figure 2d. Its ray trajectories guide the wave emerging from a point excitation along the outline of a cross, as shown in Figure 2e. The corresponding beam architecture is shown in Figure 2f.

Transient finite element simulations are performed to validate the designs. Supplementary Videos 1 and 2 show animations of the finite element simulation results for the case of harmonic loading at the design frequency. The transient finite element simulations take on the order of 5 hours, highlighting the necessity of ray tracing for inverse design, as it takes on the order of seconds to trace the rays of a tile. Supplementary Information Section 2 provides details about the finite element model setup and analysis of computational efficiency. Additionally, we note that ray trajectory design is performed for a specific target frequency but – in both simulations and experiments – wave guiding is observed over a broad frequency band surrounding this target frequency (see the Supplementary Information Sections 2 and 5 for details).

This modular approach offers a means of designing broadband phononic metamaterial waveguides spanning a large number of unit cells, taking full advantage of spatial grading. The examples of Figure 2a and 2d span approximately 37,000 and 66,000 unit cells, respectively, and this approach directly scales to larger designs simply by assembling more tiles (see Supplementary Material Fig. S2). Thus, through efficient modeling and optimization based on ray tracing together with a modular tile assembly approach, we can circumvent the challenges of computational scalability to compute large and elaborate metamaterial designs. The challenge that remains is scalable fabrication to realize such designs, for which we turn to semiconductor microfabrication, drawing inspiration from chip manufacturing methods.

Microfabrication: Silicon wafers to beam-based phononic metamaterials

Fabrication of spatially graded architectures spanning multiple length scales, such as those in Figure 2, is out of reach of standard manufacturing methods at the macroscale. We adopt silicon microfabrication techniques for a solution.

A schematic of the wafer cross-section after fabrication is shown in Figure 3b and micrographs of the prototype in Figure 3c. Two samples are fabricated on a single wafer to reduce the number of wafers used. The figure-eight design prototype spans a 3×3 cm region with a square unit cell of L = 100µm side length. Individual beams are 5µm wide. To enable excitation via a pulsed laser in the experiments, holes are etched in the substrate layer to allow the pulsed laser to reach the device layer from the backside.

In practice, the entire wafer can be architected with micron-scale features, with the only scalability limitation being the size of the wafer itself. We have demonstrated this scalability by also manufacturing ∼ 600,000 unit cells in a single SOI wafer of 100mm diameter in a separate study [57]. We thus exploit the state of the art in microfabrication technology to manufacture scalable phononic waveguides with micrometer feature resolution of sizes up to 80 mm in diameter. Microfabrication technology has pushed resolution limits down to the nanometer scale [35, 58–61], including advances from the fields of micro- and nano-electromechanical devices (MEMS/NEMS) [30, 62–66]. Most of those studies focused on devices bonded to a substrate or free-standing nano-architectures no larger than hundreds of micrometers. Using the aforementioned well-established techniques, we target a mesoscale, where the functional unit cells span micrometers and the free-standing structure spans almost a full silicon wafer. At these scales, our computational-fabrication-experimental framework achieves robust control of spatially-graded architecture. This admits controlling wave motion in the tens of kHz to MHz regime across centimeters – a regime of long-standing interest to the field [16, 19, 28, 29, 67].

Experimental wave guiding demonstration

Experiments were performed to demonstrate wave guiding in the prototype, using a pump-probe setup. A 1030 nm nanosecond pulsed laser was used for photoacoustic pump excitation, sending broadband out-of-plane elastic wave modes through the sample. The aluminum film on the sample surface enables the transduction of optical to mechanical pulses for wave excitation. The out-of-plane displacement response was measured by a custom-built heterodyne interferometer. A photograph and schematic of the experimental setup are shown in Figure 3d-e. Due to the repeatability of elastic wave propagation, data is collected by successively pumping the excitation pulse at the same spatial location, while the probe laser scans different positions on the wafer. The signal-to-noise ratio was excellent at frequencies up to 2 MHz and measurable up to 4 MHz. This was possible due to low intrinsic damping of single-crystal silicon and the high displacement resolution of the interferometer. Though not necessary here, any dissipation arising from the surrounding air can be mitigated by experiments in vacuum. The experiment hence has a unique potential to probe wave attenuation due to architecture alone.

To capture the metamaterial sample’s ability to guide elastic waves, data is collected along three lines, at x=0 (denoted L1), x=−32L (L2), and x=−64L (L3), see Figure 4a. At 100 measurement points along each line, the time series displacement signal is measured immediately following a pump excitation. Pump excitation occurs at the same point on the sample for all measurement scans. Figure 4c shows the measured response at each position along L1, L2, and L3, comparing the experimental results to simulated data from a finite element model (see Supplementary Information Section 2). The signal-to-noise ratio was excellent for all measurement points except two (marked by the black arrow for line scan L2). For reference, Figure 4a shows the maximum displacement at each spatial location during a transient finite element simulation, indicating each of the line scans with respect to the guided wave path.

The measured response of the wafer clearly captures wave guiding along the designed figure-eight trajectory. The experimental data closely matches the finite element simulation data, with agreement in both the location and timing of large displacement amplitudes. Furthermore, while the waveguide is designed for a specific target frequency of 750 kHz, broadband wave guiding is observed. A detailed frequency analysis is presented in Supplementary Information Section 5, showing that wave guiding is achieved in the window of 250-800 kHz, both in experimental and simulated data. The experiment further shows a different wave guiding response above 800 kHz, which has been validated using finite element analysis (see Figures 4(b,d)). The latter was not explicitly introduced during inverse design. Thus, the proposed methodology is capable of customized manipulation and discovery of phenomena across a frequency range broader than that of the inverse design. The observed broadband wave guiding likely stems from having dispersion relations with approximately self-similar isofrequency contours and angular distributions of group velocity (which drive ray trajectories) over the frequency range 250-800 kHz for all unit cells in the chosen design space.

Methods
Microfabrication

We use cleanroom-based microfabrication processes on commercially-procured 100 mm Silicon-On-Insulator (SOI) wafers to manufacture the figure-eight design prototype of Figure 2a-c. The graded structure of Figure 2c is fabricated in the top (device) layer of a silicon-on-insulator (SOI) wafer. A schematic of the cross-section of an as-received SOI wafer is shown in Figure 3a. Our fabrication process involves photolithography and deep reactive ion etching to architect the device layer, followed by removal of the buried oxide (SiO ) layer, using vapor hydrofluoric acid etching. This results in a free-standing architected film supported by the substrate only around the outer perimeter, similar to a drum with an architected membrane. Additionally, windows are etched in the supporting substrate to enable remote excitation of the acoustic wave from the bottom with a wide range of potential spatio-temporal profiles. To enable optimal excitation and measurement of wave propagation in these films, thin aluminum films (∼30 nm thick) were vapor-deposited on both sides of the architected membrane. A detailed description of the microfabrication procedure is presented in the Supplementary Information Section 3.

Experimental characterization

A photoacoustic pump-probe experiment was developed to resolve propagating elastic waves in our architected films. Excitation of the acoustic waves was achieved by an infrared pulsed laser beam (Coherent FLARE-N X, wavelength 1030 nm, pulse energy 500 µJ, pulse width 1 ns). The mechanism of photoacoustic excitation involves rapid thermal expansion of thin aluminum films deposited on the sample (acting as transducers), resulting in the propagation of an acoustic pulse through the film. The duration of this pulse is determined, in part, by the thickness of the film and its bulk elastic wave speed. Particle displacements were measured by a custom-built heterodyne interferometer. The heterodyne system uses a continuous wave laser (20 mW 633 nm) source with the reference branch frequency shifted by 80 MHz, using an Acousto-Optic Modulator (AOM: EQ Photonics 3080-120) driven by an RF driver (Gooch and Housego 3910). The sample branch of the interferometer was focused on the measurement point on the sample, using a 20X objective lens before being recombined and detected at a balanced photo detector (ThorLabs PDB 230A). Under ambient conditions, this generates an 80 MHz beat signal, which undergoes phase shifts due to displacement of the measurement point. During the propagation of the acoustic wave, the time-resolved phase shift was measured using a lock-in amplifier (Zurich Instruments GHFLI) and readout into a high-speed digital oscilloscope (Tektronix MSO64B). Particle displacements are directly proportional to the measured phase shift (φ ), and are calculated as

δ(t) = λ / 2π * φ (t), (1)

where λ is the wavelength of the laser source. Scanning measurements were performed using two automated stages remotely controlled using Python code. At each measurement point, 50 time series data were averaged to increase the signal-to-noise ratio. Further details regarding data analysis are summarized in Supplementary Information Sections 4 and 5.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This paper deals with the introduction of a framework for the scalable inverse design of spatially graded metamaterials for elastic wave guiding and a scalable microfabrication method. This framework enables the design and realization of complex waveguides with hundreds of thousands of unit cells, with the potential to expand these to millions without changing the protocol. The scalable design is achieved through optimization using a ray tracing model for waves in spatially graded beam lattices.\n\nThe goal of the paper to expand metamaterials’ design with the aim of manufacturing large architectures spanning thousands to millions of unit cells is commendable and extremely timely.\n\nIndeed, as the authors point out, assembling large metamaterials’ structures necessarily minimizes boundary effects that occur when combining different metamaterials together or combining a metamaterial to more « classical » engineering materials. The problem of addressing pertinent boundary conditions between different metamaterials is here addressed by designing different unit cells that are « compatible », i.e. can be connected together without creating material discontinuity.\n\nHowever, the sentence used by the authors « which truly dissolve the differences between materials and structures » seems a bit excessive, because in true applications a metamaterial will always be connected to other materials, so that macroscopic boundary effects cannot be totally « dissolved ». I would rather suggest to replace the word « dissolved » with a more precise sentence in which it is better explained that, increasing the size of the metamaterial’s structure, the effect of the unavoidable boundary effects is indeed minimized and can be neglected in the targeted applications after a certain specimen’s size.\n\nAs said, the goal of looking for scalable solutions is commendable in view of reducing computational time for large-scale structures. However, also other strategies are available to approach the challenge of designing large scale metamaterials’ structures within reasonable computational times. Among them, the most explored ones are those connected to the use of macroscopic (homogenized) models, especially those of micromorphic type (see e.g., https://doi.org/10.1016/j.jmps.2022.104995). Recently, such kind of simplified and computationally affordable methods have also been complemented with the concept of « effective interface forces\" in order to deal with the aforementioned challenge related to boundary effects that are unavoidable in finite-sized metamaterials structures (see e.g., https://doi.org/10.1016/j.jmps.2024.105558).\n\nIndeed coupling the proposed scalability methods with effective (macroscopic) models could further widen the possibility of approaching realistic designs at large scales.\n\nIt would thus be beneficial if the potentialities of the proposed models could be enlightened in this more extended viewpoint (e.g. in the introduction).\n\nThe presentation of an experimental scalable method to truly manufacture large-scale metamaterial structures is also extremely important and meritorious.\n\nIn the light of the previous comments, I highly recommend the publication of the present paper in Nature Communication by implementing only Minor Revisions.", "reviewer_2": "In this manuscript, the authors claim to “present a framework for scalable inverse design of spatially graded metamaterials for elastic wave guiding, together with a scalable microfabrication method”. While the authors’ metamaterial design and experiment are well conducted and documented, in the opinion of the reviewer there is little novelty, and there would thus be minimal impact for the community. Similarly, the introduction is incomplete to the degree that, when combined with the claims of the authors, is misleading.\n\nIn the opinion of the referee, the claimed novelty could be broken down into two sub-claims, as follows.\n\n1. Introducing a scalable microfabrication method for large numbers of metamaterial unit cells in high Q factor suspended membranes. Their method is not remotely new. To pick just one uncited example from ~9 years ago, is Ref. [Sledzinska, Marianna, et al. \"Fabrication of phononic crystals on free-standing silicon membranes.\" Microelectronic engineering 149 (2016): 41-45]. In this reference, the authors use a similar technique to manufacture 12 different patterns of 40,000 to 100,000 unit cells in each pattern on a freestanding silicon membrane for the creation of phononic crystals. Indeed, it is the referee’s understanding that, particularly at the scales the authors conducted their experiments, this sort of microfabrication process is quite standard. Particular communities where this type of fabrication is used regularly is cavity optomechanics, MEMS, and NEMS, in the form of suspended silicon nitride membranes. In addition, it is unclear how this article’s process is any different from their (cited) preprint Ref. [32].\n\n2. A “scalable” phononic crystal design method allowing for gradients. It is unclear how this contribution is meaningfully different than their previous (cited) publication Ref. [28] from 2022.\n\nI do not see that putting these two contributions together, and testing the result with laser ultrasonics, is novel, particularly given the existence of their prior two manuscripts Ref. [28] and [32]. The acoustic response of suspended silicon membranes is so standard at this point, one would not expect any surprises compared to simulation.\n\nFor publication anywhere, an appropriate literature review should be conducted, and the claims revised for accuracy. As will be listed below, it seems every few sentences in the introduction one encounters an erroneous, misleading, or vague statement. Even after this is corrected, it seems there will be minimal novelty, such that a journal like Scientific Reports could be appropriate.\n\nErroneous or misleading statements in the introduction:\n- “It is challenging to model, design, and manufacture large architectures spanning thousands to millions of unit cells.”\n“Challenging” is subjective, and meaningless as concerns novelty.\n- “Solving the problem of scalability promises a significantly enlarged and widely untapped design space for programmable control of (meta-)material functionality to enable emerging engineering applications.” What does “widely untapped” precisely mean? What examples specifically will be enabled? What specific “emerging engineering applications” are envisioned? The key example of this paper is a “figure 8” waveguide. Many other methods have previously shown more complex (and one-way) waveguiding, such as in the field of topological insulators, e.g., Ref. [Ma, Jingwen, Xiang Xi, and Xiankai Sun. \"Experimental demonstration of dual‐band nanoelectromechanical valley-hall topological metamaterials.\" Advanced Materials 33.10 (2021): 2006521], using less unit cells.\n- “However, modeling and design of non-periodic architectures are difficult to scale to large architectures with many unit cells since, unlike for periodic architectures, each unit cell must be modeled.” “Difficult” is subjective, and meaningless as concerns novelty. One could consider a comparison of runtime for related design methods.\n- “Existing spatial grading design methods have relied on heavily restricted design spaces (e.g., linear gradings [10] and analytical solutions [11]) or restricting assumptions of long wavelengths to enable homogenization [12, 13].” The author’s previous publication Ref. [28] is an “Existing spatial grading design method”.\n- “On the experimental side, the fabrication of metamaterials consisting of large numbers of unit cells is challenging. Scaling fabrication beyond tens of unit cells per dimension is extremely challenging, making it difficult to truly fabricate metamaterials rather than structures.” “Challenging” is subjective, and meaningless as concerns novelty.\n- “At the current limits of 3D printing, architectures with hundreds of unit cells per dimension [14] are within reach, but such methods are limited to polymeric materials, which are not ideal for wave guiding due to material damping.” This is not true. Here is one example for non-polymer two-photon 3D printing [Kotz, Frederik, et al. \"Two‐photon polymerization of nanocomposites for the fabrication of transparent fused silica glass microstructures.\" Advanced Materials 33.9 (2021): 2006341.].\n- Scalability is especially challenging for nano- and microscale manufacturing, where two-photon lithography [15–20] and its variants [21] can hardly scale beyond samples with tens of unit cells per dimension in a tractable build time.” This is not true. For instance, Ref. [Zhang, Leran, et al. \"High-throughput two-photon 3D printing enabled by holographic multi-foci high-speed scanning.\" Nano Letters 24.8 (2024): 2671-2679.] shows printing 384 unit cells using two-photon lithography in 4.2 seconds. Scaling this to 100,000 unit cells would result in a print time of 18 minutes.\n- “Microfabrication of optical metamaterials has achieved better scalability, with two-dimensional architectures reaching over a billion unit cells, using electron beam lithography and atomic layer deposition [22, 23]; however, this is achieved by patterning features onto a substrate, rather than creating a free-standing material architecture.” This statement is misleading, as the authors here do not show a billion unit cells. As stated above, there are prior examples with similar demonstrated numbers of unit cells as this work.", "reviewer_3": "This manuscript introduces a scalable design and modeling methodology for gradient-based phononic metamaterials with very large lattices. To overcome computational challenges, dispersion relations of individual unit cell designs are combined with ray tracing of large lattice tiles. Individual tiles are then combined to achieve targeted wave guiding. This approach is demonstrated for 2D lattices consisting of beam structures for guiding of elastic waves and experimentally realized using SOI microfabrication.\n\n**Referee assessment**\nThe paper is well written and comprehensively covers the design methodology, modeling approach, fabrication, and experimental characterization. Extensive supplementary material covers most details missing from the main manuscript. The proposed methodology is interesting, however limitations and performance of the ray-tracing are not discussed. A critical point is an inaccurate claim of novelty regarding SOI microfabrication. A revision is recommended.\nThis and some other shortcomings are elaborated below:\n1. Introduction and terminology\nThe authors call their structures \"phononic metamaterial\" and use both terms \"elastic waves\" and \"mechanical waves\". While usage differs from field to field and group to group, commonly the more general term would be acoustic metamaterial and acoustic wave which implies frequency-dependence and which applies to any wave medium. Phononic MM and phononic crystals have been used for devices covering both air/fluid-borne sound and structure-born sound. Elastic waves and often also mechanical waves refer to structure-borne sound, i.e. waves in solids. The term mechanical metamaterials is often also limited to structures with unusual static/non-wave mechanical behavior, e.g. displaying negative Poisson ratio when stretched/compressed. The authors have mixed references of MM for waves in fluids and solid, but their specific design applies only to elastic waves.\nIt is recommended to use acoustic waves and acoustic MM when referring to the general idea of dynamic metamaterials for sound waves (incl. ultrasound, infrasound, ...) to distinguish from mechanical metamaterials for non-waves. And to specify elastic MM and elastic waves when limited to solids.\n2. Incorrect fabrication claims\nIn the introduction the authors write: \"At the current limits of 3D printing, architectures with hundreds of unit cells per dimension [14] are within reach, but such methods are limited to polymeric materials\". But 3D printing is not limited to polymers. Metal and ceramic printing has been available for over a decade and utilized in various acoustic metamaterials and phononic crystals, e.g.:\n(2017) https://doi.org/10.3390/ma10101125\n(2017) https://doi.org/10.1016/j.ultras.2017.07.017\n(2019) https://doi.org/10.1016/j.jsv.2019.115167\nMore critically, later the authors write about the SOI microfabrication:\n\"The novelty of this microfabrication method is that free-standing architected films (as opposed to a pattern deposited on a substrate) are manufacturable in a scalable fashion.\"\nUsing SOI wafers to fabricate free-standing structured slabs of silicon, back-etching silicon substrates, or using sacrificial layer technology for free-standing elements is nothing new to the world of acoustic metamaterials and phononic crystals. Early examples include, but are not limited to:\n(2008) https://doi.org/10.1016/j.sna.2007.10.081\n(2008) https://doi.org/10.1063/1.2939097\n(2010) https://doi.org/10.1021/nl102918q\n(2015) https://doi.org/10.1016/j.mee.2015.09.004\nThe authors need to temper their claims and acknowledge a long history of CMOS, MEMS, SOI-MEMS, etc. microfabrication in the fields of metamaterials.\nLooking at photonic/optical metamaterials further broadens the existing literature. Incidentally, the introduction statement regarding optical MM of \"rather than creating a free-standing material architecture\" is also not accurate. There are countless examples of photonic crystals and MMs built on suspended membranes, beams, etc.\n3. Ray-tracing vs 2D FE simulation\nOne of the main claims in this manuscript is the much more efficient modelling, where finite element models of the mechanical vibration is only used for eigenmode analysis of individual unit cells and calculation of the dispersion relation/band structure. The long range model uses ray tracing. However, the authors do not fully describe the limits of the ray tracing model. By its nature, ray tracing is a high-frequency/short-wavelength approximation instead of the full wave equation(s), here further limited to out-of-plane bending waves. It neglects any wave coupling or interaction with the structure. The authors need to justify this approach by at least mentioning the relevant wave speeds/wavelengths for their materials and designs. The dispersion relation in Fig. 1 is insufficiently detailed to infer this information.\nWhile the experimental results appear to validate the ray-tracing, a comparison with a FEM wave equation result, maybe for a single tile, would be more convincing.\nMore importantly, the authors need to put the higher efficiency of their modeling approach into quantitative values. Time, memory, CPU/GPU cycles or similar measures should be given at least in the supplementary material. A computational resources comparison of full wave equation model vs ray-tracing for maybe a single tile would elevate the claims.\nThis may also include computational quantification of the inverse design procedure.\n4. Other medium/minor issues\n- All tile examples show only 90° turns. Are other angles not possible? If not, why?\n- Fig. 2: Font size becoming too small and hard to read. In general, font in figures should not be much smaller than in caption.\n- Fabrication: \"thin aluminum transducers (∼ 30 nm thick)\" are mentioned. Aluminum is not an active material and calling it a transducer like a piezoelectric material is strange. It appears to play the role of mirror/reflective surface for the acousto-optical transduction for excitation and detection.\n- Fig. 3: The photo in d) has a scale bar but no mention of the scale in the caption.\n- Fig. 4: Font size too small and hard to read.\n- Fig. 4: order of a),c),b),d) is confusing.\n- Methods section: Several aspects are repeated from previous text, esp. doubling information regarding the microfabrication p. 6-7 and 9-10."}, "author_response": "Revision of NCOMMS-25-56068-T: “Graded phononic metamaterials: scalable design meets scal- able microfabrication” by Charles Dorn, Vignesh Kannan, Ute Drechsler, and Dennis M. Kochmann\nWe thank the reviewers for their thoughtful and constructive comments. We have taken them as basis for the revision of our manuscript.\nIn the following, please allow us to respond to the reviewer’s questions and comments – explaining the changesmadeinourrevisedmanuscript(whicharehighlightedinredintheattachedmanuscript). Werefer to the reference numbering of the revised manuscript, which is copied below.\nKindlynotethatwediscoveredasmallmistakeinourdataanalysiscode, resultinginaminorerrorof∼1% in the displacement magnitude. This has been corrected in the revised manuscript.\nReviewer 1:\nThis paper deals with the introduction of a framework for the scalable inverse design of spatially graded metamaterials for elastic wave guiding and a scalable microfabrication method. This framework enables the design and realization of complex waveguides with hundreds of thousands of unit cells, with the potential to expand these to millions without changing the protocol. The scalable design is achieved through optimization using a ray tracing model for waves in spatially graded beam lattices.\nThe goal of the paper to expand metamaterials’ design with the aim of manufacturing large architectures spanning thousands to millions of unit cells is commendable and extremely timely.\nIndeed, as the authors point out, assembling large metamaterials’ structures necessarily minimizes boundary effects that occur when combining different metamaterials together or combining a metamaterial to more “classical” engineering materials. The problem of addressing pertinent boundary conditions between different metamaterials is here addressed by designing different unit cells that are “compatible”, i.e. can be connected together without creating material discontinuity.\nHowever, the sentence used by the authors “which truly dissolve the differences between materials and struc- tures” seems a bit excessive, because in true applications a metamaterial will always be connected to other materials, so that macroscopic boundary effects cannot be totally “dissolved”. I would rather suggest to re- place the word “dissolved” with a more precise sentence in which it is better explained that, increasing the size of the metamaterial’s structure, the effect of the unavoidable boundary effects is indeed minimized and can be neglected in the targeted applications after a certain specimen’s size.\nResponse: The reviewer makes a valid point, which is why we have modified the quoted statement in the revised manuscript. We only “aim at dissolving the differences between materials and structures”, though this can never be “truly” achieved, unless when dealing with an infinite medium. In the reported system, weapproachascloseaspossibletheseparationofscalesbetweenmicro-andmacroscale(thoughatruescale separation is out of reach).\nReviewer 1: As said, the goal of looking for scalable solutions is commendable in view of reducing computa- tional time for large-scale structures. However, also other strategies are available to approach the challenge of designing large scale metamaterials’ structures within reasonable computational times. Among them, the most explored ones are those connected to the use of macroscopic (homogenized) models, especially those of micromorphic type (see e.g., https://doi.org/10.1016/j.jmps.2022.104995). Recently, such kind of simplified and computationally affordable methods have also been complemented with the concept of “effective interface forces” in order to deal with the aforementioned challenge related to boundary effects that are unavoidable in finite-sized metamaterials structures (see e.g., https://doi.org/10.1016/j.jmps.2024.105558).\nIndeed coupling the proposed scalability methods with effective (macroscopic) models could further widen the possibility of approaching realistic designs at large scales.\nResponse: Thank you for pointing out micromorphic models as a potentially useful concept in pursuit of efficient and scalable metamaterial modeling. We agree that micromorphic models, among other low- and finite-frequency homogenzation methods, could be very useful in certain settings for certain design objectives. We pointouta key distinctionbetween raytracing andtypical homogenizedmodels: raytracing uses the entire dispersion relations (under the assumption that they are “locally” representative), while finite-frequency homogenization methods provide means for dispersion relation approximation. Thus, high- frequency homogenization methods may be particularly useful for cases when the dispersion relation itself is expensive to compute, providing an efficient alternative for, e.g., bandgap design. Alternatively, ray tracing isparticularlysuitableforthecasestudiedinthispaper–givenaunitcelldesignspaceandthecorresponding dispersion relations, design a spatial grading for wave guidance. We are not aware of micromorphic models thataccuratelytakeintoaccountgeneralspatialvariationsoftheunitcellgeometry(thiswouldlikelyrequire homogenization on the fly, which is prohibitively expensive), which is required for the design optimization presented here. Moreover, the type of optimization targets presented here are nontrivial to realize in a classicalcontinuumsetting(e.g., makingwavesturnby90◦ withoutimposingtheexacttimingofthewave), which is why we use ray tracing. We have updated our discussion of homogenization-based methods in the introduction accordingly.\nReviewer 1: The presentation of an experimental scalable method to truly manufacture large-scale meta- material structures is also extremely important and meritorious.\nIn the light of the previous comments, I highly recommend the publication of the present paper in Nature Communication by implementing only Minor Revisions.\nResponse: We thank the reviewer for the positive feedback.\nReviewer 2:\nIn this manuscript, the authors claim to “present a framework for scalable inverse design of spatially graded metamaterialsforelasticwaveguiding,togetherwithascalablemicrofabricationmethod”. Whiletheauthors’ metamaterial design and experiment are well conducted and documented, in the opinion of the reviewer there is little novelty, and there would thus be minimal impact for the community. Similarly, the introduction is incomplete to the degree that, when combined with the claims of the authors, is misleading.\nIn the opinion of the referee, the claimed novelty could be broken down into two sub-claims, as follows.\n1. Introducing a scalable microfabrication method for large numbers of metamaterial unit cells in high Q factor suspended membranes. Their method is not remotely new. To pick just one uncited example from 9 years ago, is Ref. [Sledzinska, Marianna, et al. “Fabrication of phononic crystals on free-standing silicon membranes.” Microelectronic engineering 149 (2016): 41-45]. In this reference, the authors use a similar technique to manufacture 12 different patterns of 40,000 to 100,000 unit cells in each pattern on a freestand- ing silicon membrane for the creation of phononic crystals. Indeed, it is the referee’s understanding that, particularly at the scales the authors conducted their experiments, this sort of microfabrication process is quite standard. Particular communities where this type of fabrication is used regularly is cavity optomechan- ics, MEMS, and NEMS, in the form of suspended silicon nitride membranes. In addition, it is unclear how this article’s process is any different from their (cited) preprint Ref. [32].\nResponse: We thank the reviewer for this critical and very helpful feedback. First, we would like to state\n2\nthat this manuscript does not purely focus on the microfabrication technique, but also on inverse computa- tionaldesignanddetailedfull-fieldspatio-temporalcharacterizationofthewavethroughourfabricatedstruc- ture. In this sense, the main focus of this manuscript is the application of our experimental-computational frameworktodesignofnewelasticwaveguidesacrossmicrometertomillimeterlengthscales,likethefigure-8 case discussed. This is also the main difference between the cited preprint, ref. [32], which focuses on the development of the fabrication and experimental techniques and protocol that form this framework. We note that the two manuscripts are designed to complement each other ([32] does not constitute prior work but was submitted in parallel to this manuscript). With this clarification, allow us to address the specific criticism regarding fabrication.\nIn general, we stress that it was not our intention to claim that we have invented a new technique in mi- crofabrication. We apologize if we inadvertently created that impression. Studies that have breached the nanoscale (like the one that the reviewer correctly pointed out) have achieved free-standing structures of hundreds of micrometers. In certain applications of mechanical wave guiding, however, it may be critical to guide waves over much longer distances; this is non-trivial to achieve. We have used existing microfab- rication techniques to create elastic wave-guiding metamaterials across length scales spanning micrometers to millimeters – a niche range that we refer to as a “meso” scale. Our manuscript presents in detail the advantages and challenges of accessing this “intermediate” range.\nThe reviewer’s statement that the number of unit cells reported in Sledzinska et al. (2016) are similar to ours is indeed true. However, our work distinguishes itself from the afore-cited reference (and others in the MEMS and NEMS literature) through the following aspects:\n1. The cited reference developed techniques to create nanoscale structures (with maximum ranges of 100 × 100 µm) with target frequencies in the GHz regime. Our objective was to create membranes that target the frequency range of 100’s of kHz to tens of MHz. Fabrication of such high densities of unit cells at our “meso” scales in free-standing membranes spanning tens of millimeters are not as common buthavepowerfulapplicationsinmechanicalwave-guidingatthosefrequencyranges. Multiplestudiesin the mechanics literature [2, 64, 12, 41] seek to target such scales (and above), but have not tapped into the microfabrication routes exploited here. Hence, the realization of such large waveguides has not been reported yet to the best of our knowledge.\n2. The development of large free-standing wave guides paves the way for critical technology requiring the propagation of waves across large physical distances (irrespective of the microstructural length scales). For example, if we could fabricate nanoscale structures on a free-standing membrane spanning tens of millimeters (like ours), this would be a momentous step forward in scalable wave-guiding technology, e.g., for controlled design of the “long wavelength” dispersion response of mechanical waves in signal processing. We believe that our microscale structures are a significant step in this direction.\n3. Our “meso” scales also allow measurements of the full spatio-temporally resolved wave propagation with sufficiently high resolution. Such high-dimensional experimental data are crucial to validate compu- tational models, verify and realize complex waveguide designs obtained from optimization schemes, as demonstrated in this study. This would be significantly more difficult to achieve in nanoscale phononic waveguides.\n4. PriorworksintheMEMSandNEMScommunities(whichareclearlyrelevantherefromamicrofabrication perspective),toourknowledgehavenotexploredlarge,free-standing microstructuredthinfilms(spanning micrometers to millimeters)—owing, among others, to the prevalent interest in resonator technology and surface acoustic waves (SAWs), which do not require structures to be free-standing across a large spatial domain. However, such large, free-standing structures naturally enable the propagation of more complex wave modes (e.g., Lamb waves).\n5. As a side note, Sledzinska et al. (2016) focused on stabilizing ultra-thin films, using a polymer layer. While a novel technique, it is unclear how waves can propagate in that configuration over long distances due to damping—especially when aiming for lower frequencies, as discussed here. Discrepancies due to dampingmayaffectquantitativecomparisonsofthefullspatio-temporalevolutionofwavesthroughthose\n3\nwave guides.\n6. Finally,Sledzinskaetal.,(2016)andseveralotherswithinthisdomainfocusedonimprovingthefabrication ofnanoscalewaveguidesratherthanspatialgradingforpre-definedwaveguidingcapabilities. Ashasbeen evident from literature [41, 12, 14, 47], this is a non-trivial problem with recent progress at macroscopic scales (including our own prior work, e.g., [25, 15]). The present manuscript demonstrates significantly improved synergy between scalability in computation, fabrication, and measurement for more robust spatial grading strategies.\nToaddressthereviewer’scommentsandtoclarifytheaboveinourmanuscript, wehavemademodifications intheSectionMicrofabrication: Silicon wafers to beam-based metamaterials,nowstatingthefollowing: “We exploit the state of the art in microfabrication technology to manufacture scalable phononic waveguides with micrometer feature resolution of sizes up to 80 mm in diameter. Microfabrication technology has pushedresolutionlimitsdowntothenanometerscale[32,52,53,54,55],includingsignificantadvancesfrom the fields of micro- and nano-electromechanical devices (MEMS/NEMS) [56, 57, 58, 59, 60, 27]. Most of those studies focused on devices bonded to a substrate or free-standing “nano-architectures” no larger than hundreds of micrometers. Using the aforementioned well-established techniques, we target a “mesoscale”, where the functional unit cells span micrometers and the free-standing structure spans almost a full silicon wafer. At these scales, our computational-fabrication-experimental framework achieves robust control of spatially-graded architecture. This admits controlling wave motion in the tens of kHz to MHz regime across centimeters – a regime of long-standing interest to the field [13, 25, 16, 61, 26].”\nReviewer 2: 2. A “scalable” phononic crystal design method allowing for gradients. It is unclear how this contribution is meaningfully different than their previous (cited) publication Ref. [28] from 2022. I do not see that putting these two contributions together, and testing the result with laser ultrasonics, is novel, particularly given the existence of their prior two manuscripts Ref. [28] and [32]. The acoustic response of suspended silicon membranes is so standard at this point, one would not expect any surprises compared to simulation.\nResponse: We kindly disagree with the reviewer. First, Ref. [51] (Ref. [32] in the original submission) is not prior work but was submitted in parallel to this manuscript (here, focusing on the scalable design and wave-guidance, there on experimental techniques). [51] is being reviewed by other experts in the field (focusing on experimental methods). Therefore, [51] does not infringe on the novelty or originality of this submission.\nSecond,Ref.[24](Ref.[28]intheoriginalsubmission)doesindeedconstituterelevantpriorwork,asitforms the theoretical basis for the optimization framework presented here, but there are many scientific advances made between Ref. [24] and the present manuscript:\n1. The presented large-scale waveguides are made possible by the “tile design” approach, which is intro- duced here for thefirst time(and nontrivial due to, among others, requirementsof matched dispersion relations at tile interfaces, treatment of boundaries, and the design targets required in such tiles). Brute-force optimization of the entire domain based on [32] would not be computationally feasible for the designs considered in this manuscript.\n2. Ref. [24] exclusively optimized rays based on target locations (e.g., wave focusing). Here, we present for the first time how to optimize designs such that rays enter and exit perpendicular to the boundary (which is essential for the tiling to function).\n3. Ref. [24] only implements mass-spring network toy problems with analytical dispersion relations; the current manuscript’s demonstration of beam networks with numerically computed dispersion relations is a major improvement that is key for enabling experimental demonstration.\nReviewer 2: For publication anywhere, an appropriate literature review should be conducted, and the claims revised for accuracy. As will be listed below, it seems every few sentences in the introduction one encounters an erroneous, misleading, or vague statement. Even after this is corrected, it seems there will be minimal novelty, such that a journal like Scientific Reports could be appropriate.\nReviewer 2: Erroneous or misleading statements in the introduction:\n“It is challenging to model, design, and manufacture large architectures spanning thousands to millions of unit cells.” “Challenging” is subjective, and meaningless as concerns novelty.\nResponse: Thank you for pointing out that this wording is not precise. We have updated the wording to “Modeling, design, and manufacturing efficiency remains a bottleneck for architectures spanning tens of thousands to millions of unit cells...” We point out that the Springer-Nature journal family does not appreciate precise novelty claims (“we ask authors to refrain from making priority or novelty claims and to removequalitativeevaluationsoftheirownwork”;“avoidphraseslike‘forthefirsttime’and‘unprecedented’).\nReviewer 2: “Solving the problem of scalability promises a significantly enlarged and widely untapped de- sign space for programmable control of (meta-)material functionality to enable emerging engineering appli- cations.” What does “widely untapped” precisely mean? What examples specifically will be enabled? What specific “emerging engineering applications” are envisioned? The key example of this paper is a “figure 8” waveguide. Many other methods have previously shown more complex (and one-way) waveguiding, such as in the field of topological insulators, e.g., Ref. [Ma, Jingwen, Xiang Xi, and Xiankai Sun. “Experimental demonstration of dual-band nanoelectromechanical valley-hall topological metamaterials.” Advanced Materi- als 33.10 (2021): 2006521], using less unit cells.\nResponse: When referring to a “widely untapped design space”, we are specifically referring to spatially graded architectures with complicated spatial profiles spanning multiple length scales. As mentioned in paragraph3oftheintroduction,existingworkonspatiallygradedphononicmetamaterialshasbeenrestricted tosimplegradingprofiles—–suchasgradingalongonedimensiononly,orradiallysymmetricpatterns. Two- dimensionalspatialgradingswithcomplicatedgradingprofilesspanningmultiplelengthscaleshavenotbeen been previously considered. Scaling to large architectures with many unit cells creates the opportunity to explore gradings with more elaborate spatial structures.\nWe are not tied to a particular application, as we present a fundamental framework for exploring what new capabilities are possible in a previously unexplored design space of complex, multiscale, 2D spatial grad- ings. However, we anticipate that expanding the design space and functionality of graded metamaterials will enhance applications that graded metamaterials (with simple grading profiles) have shown promise for already, such as energy harvesting [10] and mechanical/acoustic signal processing [11], in addition to poten- tially enabling new applications that could benefit from enhanced control over wave guiding, localization, and focusing, such as microfluidics [65].\nWeacknowledgethatmanyapproachestowaveguidinghavebeenexplored. Topologicalmetamaterials,such astheexampleof(Maetal.,2021),areappealinginmanycases,e.g.,duetotheirrobustnesstodefects. One potential drawback is that they are restricted to often narrow bandgaps in the bulk, and to discrete wave guiding angles aligned with lattice interfaces. Similarly, many other approaches to acoustic and phononic metamaterials, which leverage bandgap design, are limited to narrow frequency bands, unlike the results presented here.\nThe introduction has been updated to reflect these points.\nReviewer 2: “However, modeling and design of non-periodic architectures are difficult to scale to large architectures with many unit cells since, unlike for periodic architectures, each unit cell must be modeled.” “Difficult” is subjective, and meaningless as concerns novelty. One could consider a comparison of runtime for related design methods.\nTo model finite-frequency (i.e., above the homogenization limit) wave propagation through spatially graded metamaterials, the options are limited to high-fidelity simulations, where each unit cell is included in the model, or a ray tracing-based model. A computation time comparison between forward modeling via high- fidelity transient dynamic finite element simulations and ray tracing has been added in Supplementary Material Section 2.3, where the finite element simulations took on the order of hours, while the ray tracing simulations took on the order of seconds.\nReviewer 2: “Existing spatial grading design methods have relied on heavily restricted design spaces (e.g., linear gradings [10] and analytical solutions [11]) or restricting assumptions of long wavelengths to enable homogenization [12, 13].” The author’s previous publication Ref. [28] is an “Existing spatial grading design method”.\nResponse: The reviewer makes a fair point that the authors’ previous work, Ref. [24] (Ref. [28] in the original submission), is indeed an existing spatial grading design method. We emphasize that, while Ref. [24]isnotrestrictedtosimplegradingsorlongwavelengths,thecurrentworkisadistinctimprovementwith new capabilities not possible with the formulations of Ref. [24]. The key improvements are as follows:\n• The current work, unlike Ref. [24], includes arbitrary exit curves and the capacity to design the wave vector at which the ray exits. These lead to a different form of the cost function and its gradient. These key differences are discussed in the Supplementary Material Section 1.2.\n• The current work employs a modular design approach by designing a set of compatible tiles that can be assembled, which is crucial for pushing the limits of scalability. Design of compatible tiles is not possibleunlessthewavevectorisprescribedatthetileboundary,hencemodulardesignisnotpossible viaRef.[24]. ApplyingthedesignapproachofRef.[24]totheentiredomainintheexamplespresented here is prohibitively expensive to scale to architectures of the size studied here. Therefore, the tile approach makes such design optimization possible.\n• Ref. [24] only implements mass-spring network toy problems with analytical dispersion relations; the current manuscript’s demonstration of beam networks with numerically computed dispersion relations is a major improvement that is key for enabling experimental demonstration.\nWe have updated the introduction to have more precise wording and mention these key differences.\nReviewer 2: “On the experimental side, the fabrication of metamaterials consisting of large numbers of unit cells is challenging. Scaling fabrication beyond tens of unit cells per dimension is extremely challenging, making it difficult to truly fabricate metamaterials rather than structures.” “Challenging” is subjective, and meaningless as concerns novelty.\nResponse: Thank you for pointing out the lack of precision in this statement. We have modified these sentences to the following: “On the experimental side, the fabrication of 2D wave-guiding metamaterials, using standard machining and 3D-printing techniques, are limited to tens to hundreds of unit cells per dimension [25, 16, 26]. Going from tens to hundreds of thousands of unit cells at the meso-scale requires the adaptation of specialized microfabrication techniques that have rarely been exploited for free-standing elastic wave guides [27] – especially spanning tens of micrometers to millimeters.”\nWe point out again that we have made these changes also keeping in mind that the Springer-Nature journal family does not appreciate precise novelty claims in submitted manuscripts.\nReviewer 2: “At the current limits of 3D printing, architectures with hundreds of unit cells per dimension [14] are within reach, but such methods are limited to polymeric materials, which are not ideal for wave guiding due to material damping.” This is not true. Here is one example for non-polymer two-photon 3D printing [Kotz, Frederik, et al. “Two-photon polymerization of nanocomposites for the fabrication of transparent fused silica glass microstructures.” Advanced Materials 33.9 (2021): 2006341.].\nResponse: Thank you for pointing out this incorrect statement. We were referring to conventional\n6\nmacroscale 3D printing techniques as enabled by commercially available machines, not microfabrication processes. In the context of microfabrication, we have added further references on the fabrication of non- polymeric materials to the revised manuscript, though those have the drawback that the trade-off between resolution and field of view still remains a challenge. In response to similar comments from Reviewer 3, we have also included the broader class of conventional metal and ceramic additive manufacturing in our statement.\nThesentenceinquestionhasbeenmodifiedasfollows: “Thecurrentstateoftheartincommercialmacroscale 3D printing technology enables architectures with hundreds of unit cells per dimension [28]. These meth- ods, which fall within the broad category of additive manufacturing (AM), have been widely exploited for polymericmaterials, whichunfortunatelyarenotsuitableforwavepropagationbecauseofmaterial-intrinsic damping. AlthoughexcitingnewdirectionsinmetalandceramicAMhaveemerged[29,30,31], carefulcon- trolofprintingresolutionandmaterialpropertiesstillposeschallenges. Microfabricationtechniquesenabled bymulti-photonlithography,traditionallyusingpolymericmaterialsatsignificantlyhigherspatialresolution [32], have overcome this limitation [33, 34]. Such techniques, while growing rapidly, are still nascent in their ability to achieve large sample sizes close to 100 mm while maintaining sub-micrometer spatial resolution [35, 36], lesser so when spatially-graded non-periodic architectures are involved.”\nReviewer 2: Scalabilityisespeciallychallengingfornano-andmicroscalemanufacturing, wheretwo-photon lithography [15–20] and its variants [21] can hardly scale beyond samples with tens of unit cells per dimen- sion in a tractable build time.” This is not true. For instance, Ref. [Zhang, Leran, et al. “High-throughput two-photon 3D printing enabled by holographic multi-foci high-speed scanning.” Nano Letters 24.8 (2024): 2671-2679.] showsprinting384unitcellsusingtwo-photonlithographyin4.2seconds. Scalingthisto100,000 unit cells would result in a print time of 18 minutes.\nResponse: While the overall scaling argument is true in principle, we kindly disagree with the reviewer on its applicability to the technique proposed in the the cited reference. It is crucial to take into account the technique itself:\n• The scaling from 384 to 100,000 unit cells (for the same unit cell size) is non-trivial – the sample sizes are limited by the range of the galvanometer mirrors used in their setup (300×300 µm).\n• Toincreasethisrange,thegalvanometerscanningsystemmustbecoupledtoascanningstage. Hence, a simple linear time scaling from 384 to 100,000 unit cells is an unreasonable assumption.\n• Such “stitching” methods have been shown to introduce defects in printed samples [33]. As mentioned before, while attempts at making such scalable samples are improving fast, the technology is still in the nascent/development stage.\n• Finally, we reiterate that this technique has been shown to work for polymeric samples only as yet.\nHowever, we realize that the statement cited by the reviewer may be misleading if read out of context. Hence, we have made a slight modification as follows: “Two-photon lithography [15–20] and its variants [21] demonstrateexcellentaccuracyandrepeatabilityforsampleswithtenstohundredsofunitcellsperdimension in a tractable build time. Recent modifications of the technique have shown significant improvement of fabrication time scales [43], albeit for polymeric base materials. Scaling those techniques to large sample sizes well beyond hundreds of micrometers has remained a challenge.”\nResponse: We are afraid the reviewer may have misread the quoted statement from our manuscript. We write that microfabrication has already achieved over a billion unit cells in the scientific literature (by patterning features onto a substrate), for which we also provide references. We do not claim anywhere in our manuscript (and in particular not in those sentences) that we achieve over a billion unit cells with our samples.\nReviewer 3: This manuscript introduces a scalable design and modeling methodology for gradient-based phononic metamaterials with very large lattices. To overcome computational challenges, dispersion relations of individual unit cell designs are combined with ray tracing of large lattice tiles. Individual tiles are then combined to achieve targeted wave guiding. This approach is demonstrated for 2D lattices consisting of beam structures for guiding of elastic waves and experimentally realized using SOI microfabrication.\nThepaperiswellwrittenandcomprehensivelycoversthedesignmethodology,modelingapproach,fabrication, and experimental characterization. Extensive supplementary material covers most details missing from the main manuscript. The proposed methodology is interesting, however limitations and performance of the ray- tracing are not discussed. A critical point is an inaccurate claim of novelty regarding SOI microfabrication. A revision is recommended. This and some other shortcomings are elaborated below:\n1. Introduction and terminology The authors call their structures “phononic metamaterial” and use both terms “elastic waves” and “mechanical waves”. While usage differs from field to field and group to group, commonly the more general term would be acoustic metamaterial and acoustic wave which implies frequency- dependence and which applies to any wave medium. Phononic MM and phononic crystals have been used for devices covering both air/fluid-borne sound and structure-born sound. Elastic waves and often also me- chanical waves refer to structure-borne sound, i.e. waves in solids. The term mechanical metamaterials is often also limited to structures with unusual static/non-wave mechanical behavior, e.g. displaying negative Poisson ratio when stretched/compressed. The authors have mixed references of MM for waves in fluids and solid, but their specific design applies only to elastic waves. It is recommended to use acoustic waves and acoustic MM when referring to the general idea of dynamic metamaterials for sound waves (incl. ultrasound, infrasound, ...) to distinguish from mechanical metamaterials for non-waves. And to specify elastic MM and elastic waves when limited to solids.\nResponse: Weagreethatterminologyvariesthroughouttheliteratureandisacommonpointofambiguity. Wehaveupdatedthemanuscripttoconsistentlyreferto“elasticwaves”insteadof“mechanicalwaves. How- ever, we believe it is justified to keep our terminology as “phononic metamaterials” rather than “acoustic metamaterials”. Our reasoning is that many (but not all) authors use “acoustic metamaterials” specifically torefertoarchitectureswhoseresponseisdominatedbylocalresonanceeffects,e.g.,in[2,9,63]. Inourcase, our ray tracing formulation is agnostic to whether local resonance and/or Bragg scattering dominates the response. In fact, in graded architectures it is conceivable to have local resonance behavior only in certain spatial regions. We think that the terminology “phononic metamaterials” is the most appropriate to the cases studied here, though we acknowledge that the terminology is not consistent throughout the literature.\nReviewer 3: 2. Incorrectfabricationclaims. Intheintroductiontheauthorswrite: “Atthecurrentlimitsof 3D printing, architectures with hundreds of unit cells per dimension [14] are within reach, but such methods are limited to polymeric materials”. But 3D printing is not limited to polymers. Metal and ceramic printing\n8\nhas been available for over a decade and utilized in various acoustic metamaterials and phononic crystals, e.g.:\n(2017) https://doi.org/10.3390/ma10101125\n(2017) https://doi.org/10.1016/j.ultras.2017.07.017\n(2019) https://doi.org/10.1016/j.jsv.2019.115167\nResponse: We thank the reviewer for correcting us, and apologize for missing the broader class of AM studies. In response to similar comments from Reviewer 2, we have also included the smaller but growing class of microfabrication techniques for metallic and ceramic materials. The sentence has been modified in the revised manuscript as follows: “The current state of the art in commercial macroscale 3D printing technology enables architectures with hundreds of unit cells per dimension [28]. These methods, which fall within the broad category of additive manufacturing (AM), have been widely exploited for polymeric materials, which unfortunately are not suitable for wave propagation because of material-intrinsic damping. Although exciting new directions in metal and ceramic AM have emerged [29, 30, 31], careful control of printing resolution and material properties still poses challenges. Microfabrication techniques enabled by multi-photon lithography, traditionally using polymeric materials at significantly higher spatial resolution [32], have overcome this limitation [33, 34]. Such techniques, while growing rapidly, are still nascent in their ability to achieve large sample sizes close to 100 mm while maintaining sub-micrometer spatial resolution [35, 36].”\nReviewer 3: More critically, later the authors write about the SOI microfabrication: “The novelty of this microfabrication method is that free-standing architected films (as opposed to a pattern deposited on a sub- strate)aremanufacturableinascalablefashion.”UsingSOIwaferstofabricatefree-standingstructuredslabs of silicon, back-etching silicon substrates, or using sacrificial layer technology for free-standing elements is nothing new to the world of acoustic metamaterials and phononic crystals. Early examples include, but are not limited to:\n(2008) https://doi.org/10.1016/j.sna.2007.10.081\n(2008) https://doi.org/10.1063/1.2939097\n(2010) https://doi.org/10.1021/nl102918q\n(2015) https://doi.org/10.1016/j.mee.2015.09.004\nResponse: We thank the reviewer for this critical feedback. This critique is well received (and also in line with some of the points raised by Reviewer 2, which is why our response here is similar). We stress that it was not our intention to claim that we have invented a new microfabrication technique. We apologize if we inadvertently created that impression. Studies that have reached the nanoscale (like the ones that the reviewercorrectlypointedout)haveindeedachievedfree-standingstructuresofhundredsofmicrometers. In certainapplicationsofmechanicalwaveguiding,however,itmaybecriticaltoguidewavesovermuchlonger distances; this is non-trivial to achieve. We have used existing microfabrication techniques to create elastic wave-guidingmetamaterialsacrosslengthscalesspanningmicrometerstomillimeters–anicherangethatwe refertoasa“meso”scale. Moreover,thewaveguidespresentedhererequirenotperiodicbutspatiallygraded designs. Our manuscript presents in detail the advantages and challenges of accessing this “intermediate” range.\nThe revised manuscript now contains the following explanation: “We exploit the state of the art in mi- crofabrication technology to manufacture scalable phononic waveguides with micrometer feature resolution of sizes up to 80 mm in diameter. Microfabrication technology has pushed resolution limits down to the nanometer scale [32, 52, 53, 54, 55], including significant advances from the fields of micro- and nano- electromechanical devices (MEMS/NEMS) [56, 57, 58, 59, 60, 27]. Most of those studies focused on devices bonded to a substrate or free-standing “nano-architectures” no larger than hundreds of micrometers. Us- ing the aforementioned well-established techniques, we target a “mesoscale”, where the functional unit cells span micrometers and the free-standing structure spans almost a full silicon wafer. At these scales, our computational-fabrication-experimental framework achieves robust control of spatially-graded architecture. This admits controlling wave motion in the tens of kHz to MHz regime across centimeters – a regime of long-standing interest to the field [13, 25, 16, 61, 26].”\nResponse: We hope that the aforementioned modifications in the revised manuscript have also addressed this critique from the reviewer. We have also removed the statement “rather than creating free-standing material architecture” in the revised manuscript.\nReviewer 3: 3. Ray-tracing vs 2D FE simulation\nOne of the main claims in this manuscript is the much more efficient modelling, where finite element models of the mechanical vibration is only used for eigenmode analysis of individual unit cells and calculation of the dispersion relation/band structure. The long range model uses ray tracing. However, the authors do not fully describe the limits of the ray tracing model. By its nature, ray tracing is a high-frequency/short-wavelength approximation instead of the full wave equation(s), here further limited to out-of-plane bending waves. It neglects any wave coupling or interaction with the structure. The authors need to justify this approach by at least mentioning the relevant wave speeds/wavelengths for their materials and designs. The dispersion relation in Fig. 1 is insufficiently detailed to infer this information. While the experimental results appear to validate the ray-tracing, a comparison with a FEM wave equation result, maybe for a single tile, would be more convincing.\nResponse: The ray tracing model is indeed an approximation that has limitations, and we agree that the nature of this approximation merits deeper discussion. Specifically, our formulation of ray tracing in the metamaterial setting is based on the assumption that there is a separation of scales between the wavelength (theshortlengthscale)andthelengthscaleassociatedwithspatialgrading(thelonglengthscale);see[23]for the derivation, which follows classical ray tracing derivations assuming the length scale of smooth variation ofmaterialpropertiesissignificantlylongerthanwavelengthsofinterest. Thus,sufficientlyaggressivespatial grading with respect to wavelength may enter a regime not captured by ray theory, which would lead to both error in the ray solutions and its failure to capture physical phenomena like mode coupling.\nToensurethevalidityofraytheory, werelyonnumerical/experimentalvalidation. Ofcourse, rigoroustheo- reticalvalidityconditionswouldbeideal. Whiletheabovequalitativevaliditycondition(gradinglengthscale must be much larger than wavelength) provides rough guidelines, the development of rigorous quantitative validity conditions is an open problem. One approach for developing quantitative validity conditions would be to evaluate the discarded terms in the asymptotic expansion from which ray theory is derived. This has beenexplored,forexample,insimplersettingssuchasanisotropicelasticcontinuumwithconstantgradient properties (see Ref. [S6] in the Supplementary Material); if such an analysis is tractable in our setting of ray theory in dispersive media with anisotropic microstructure and complex grading profiles, it is highly non-trivial and out of scope of this work.\nWehaveaddedadiscussionaboutraytheoryvaliditytoSupplementaryMaterialSection1.2. Additionally,as recommended by the reviewer, we have listed quantitative values for for wavelengths and wave speeds of the designsinSupplementaryMaterialSections1.3.1and1.3.2. However,wecautionthatinterpretationofthese numbers toward ray theory validity is not straightforward, which is why we adopt numerical/experimental validation. Overall, we adopt the approach of using ray theory for design (due to its efficiency and differ- entiability), followed by transient numerical and experimental validation to confirm that the ray solution of the final design is in fact representative.\nFinally, we note that transient dynamic finite element simulations are presented in Supplementary Material Section 2.2 both for assembled design 1 (Fig. S3) and assembled design 2 (Fig. S4), which show that the FE simulation closely follows the predicted ray trajectories. We feel that this is both more impactful and easiertointerpretthanananalogoussimulationofasingletile–ourreasoningisthatintheassembleddesign the wave propagates in a closed loop so the solution can be interpreted with minimal effects of boundary conditions.\nResponse: We agree that quantifying the efficiency is important for emphasizing the value of using a ray tracing model. A discussion has been added to the Supplementary Material Section 2.3, highlighting the numerical efficiency of ray tracing versus transient finite element simulations of the figure-eight example. To summarize this newly added discussion, it takes less than a second to trace one ray (∼5.2 seconds to trace 100 rays to capture wave propagation through Tile 1 when running in parallel), while the parallelized transient dynamic finite element solution takes ∼5 hours.\nAdditionally, we emphasize two additional key advantages of using ray tracing that make it suitable for inverse problems:\n• Ray tracing quantities are differentiable. The (semi-)analytical derivative of a cost function based on ray quantities (specifically the position/wave vector at a give point on a ray) is derived in Eq. (S11) of the Supplementary Material. This is crucial for enabling gradient-based optimization, since finite differences arenotneededforgradientcomputation,whicharecomputationallyexpensivewhenthenumberofdesign variables is large and often introduce error.\n• Ray tracing enables partial solutions to be computed. That is, we have the freedom to choose how many/which rays are traced, so it is possible to only compute the solution in a region of interest. In contrast, transient finite element simulations require computation of a complete full-field solution.\nThese additional points of discussion have been added to the Supplementary Material Section 2.3 to further elevate the value of using ray tracing to enable inverse design problems.\nReviewer 3: 4. Other medium/minor issues:\nAll tile examples show only 90◦ turns. Are other angles not possible? If not, why?\nResponse: It is certainly possible to consider alternative objective functions in the optimal design problem to, for example, produce solutions with other than 90◦ turns. This corresponds to changing the exit wave vector kˆ in the cost function of Supplementary Material Eq. (S3), such that the corresponding group r\nvelocityataray’sexitpointsinthedesireddirection. However,wechooseexampleswith90◦ turnstoenable a straightforward modular design process where square tiles can be placed next to each other since rays are perpendicular to the tile edge. Alternative designs that tesselate the 2D domain are equally possible (such as,e.g.,triangularorhexagonaltiles). WehaveaddedadiscussioninSupplementaryMaterialSection1.3to highlight this point. However, we feel that examples exploring the large design space of potential tile design andassemblyconceptsarebeyondthescopeofthisworkandrequiresignificantnewcomputations(andalso experiments for validation).\nReviewer 3:\nFig. 2: Font size becoming too small and hard to read. In general, font in figures should not be much smaller than in caption.\nResponse: Thank you for pointing this out. We have updated Fig. 2 to have a larger font size.\nReviewer 3:\nFabrication: “thin aluminum transducers (∼30 nm thick)” are mentioned. Aluminum is not an active material and calling it a transducer like a piezoelectric material is strange. It appears to play the role of mirror/reflective surface for the acousto-optical transduction for excitation and detection.\nResponse: We thank the reviewer for pointing this out. The aluminum film serves the transduction of the optical to the mechanical pulse. We have modified this statement in the revised manuscript.\nReviewer 3:\nResponse: A scale bar has been added to the caption.\nReviewer 3:\nFig. 4: Font size too small and hard to read.\nResponse: We have increased the font size in the revised figure.\nReviewer 3:\nFig. 4: order of a),c),b),d) is confusing.\nResponse: The order has been modified.\nReviewer 3:\nMethods section: Several aspects are repeated from previous text, esp. doubling information regarding the microfabrication p. 6-7 and 9-10.\nResponse: The sections on microfabrication have been re-arranged in the revised manuscript to reduce the overlap.\nReviewer 3: I have checked availability and type of code/files. Most calculation and data processing appears to use MATLAB but I have not reviewed the MATLAB code. Dispersion relations calculated from the finite element wave equation/eigenmode model are only provided as results, not the models/FEM codes. Many figure files are provided. However, experimental data is not sorted well and does not appear to include any file description or file naming scheme.\nResponse: We thank the reviewer for carefully checking the code and experimental data.\nRegarding the code, we note that we have included the inverse design codes, which encompasses all novel computational contributions of this work. The finite element-based dispersion relation computation and transient dynamic simulations are standard, well-known numerical procedures (e.g., in Refs. [2, 16]), and the underlying FE code we used, ae108, is open source – [S10] of the Supplementary Material references. Thus, we feel it is appropriate to keep the supplementary codes accompanying this manuscript concise and focused on our new contributions.\nRegarding the experimental data, we have re-named the files and added a readme.txt file in the master directory, with short descriptions of the directory structure and data available in sub-directories. 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Proceedings of the Royal Society A 481, 20240429 (2025).\n15"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The comments have been addressed satisfactorily.\nI thus recommend publication of the paper in its present form.", "reviewer_2": "While the authors have made substantial replies in the rebuttal letter, I do not feel substantive improvements were made to the manuscript regarding modifying their arguments to accurately place their contribution in the context of existing and prior capabilities. Key references that would conflict with their arguments of novelty are still missing.\n\nI do not believe this is appropriate to be published anywhere in its current form. Even if their description of the context of their contribution is improved, this work does not have sufficient novelty for Nature Communications. Scientific Reports or NJP Metamaterials could be alternate suitable venues.\n\nHere is just one other, again uncited, example of using laser ultrasonics to measure phononic responses of patterned plates [Zhao, Jinfeng, et al. \"On-chip valley phononic crystal plates with graded topological interface.\" International Journal of Mechanical Sciences 227 (2022): 107460.]. In that reference they study ~1.7-1.9 MHz waves in a patterned silicon plate with patterned area stretching over 31x37 cm within a 4 inch diameter wafer.\n\nThe authors’ focus in the rebuttal letter on subtle difference from articles brought to their attention by the Reviewers misses the point. Reviewers should not have to do the authors’ literature review for them.\n\nBelow, I will also respond to the subset of their replies that I feel warrant particular response.\n\nAuthor: “the main focus of this manuscript is the application of our experimental-computational framework to design of new elastic wave guides across micrometer to millimeter length scales, like the figure-8 case discussed.”\nI do not understand what the experimental part is in “experimental-computational framework”. It seems the design is purely computational, then there is experimental validation. Neither the fabrication method nor the characterization method is substantially new.\n\nIn addition, I am unclear from the manuscript as to why someone would want to make these large gradient based structures, when the same thing can be done in a much smaller footprint. For instance, conventional (not even topological or phononic-crystal-based) elastic waveguides have been around for many decades. A broadband, figure 8 waveguide could be easily done in this conventional context via design of impedance mismatch and total internal reflection.\n\nAuthor: In general, we stress that it was not our intention to claim that we have invented a new technique in microfabrication. We apologize if we inadvertently created that impression. Studies that have breached the nanoscale (like the one that the reviewer correctly pointed out) have achieved free-standing structures of hundreds of micrometers. In certain applications of mechanical wave guiding, however, it may be critical to guide waves over much longer distances; this is non-trivial to achieve.\nWhat applications, specifically, are these critical for? This has still been left as nebulous in the manuscript.\n\nAuthor: We have used existing microfabrication techniques to create elastic wave-guiding metamaterials across length scales spanning micrometers to millimeters – a niche range that we refer to as a “meso” scale. Our manuscript presents in detail the advantages and challenges of accessing this “intermediate” range.\nWhat advantages and where are they stated? Do you mean “This admits controlling wave motion in the tens of kHz to MHz regime across centimeters – a regime of long-standing interest to the field”. If so: i) this is still nebulous; ii) Someone controlling kHz to MHz waves would just use larger structures there is no need to use few micron beams (see paper cited above).\n\nAuthor: For example, if we could fabricate nanoscale structures on a free-standing membrane spanning tens of millimeters (like ours), this would be a momentous step forward in scalable wave-guiding technology, e.g., for controlled design of the “long wavelength” dispersion response of mechanical waves in signal processing.\nHow is this “momentous” exactly? What would this enable in terms of new phenomenology or functionality? Why would one just not use larger unit cells (see paper cited above)?\n\nAuthor: Our “meso” scales also allow measurements of the full spatio-temporally resolved wave propagation with sufficiently high resolution. Such high-dimensional experimental data are crucial to validate computational models, verify and realize complex waveguide designs obtained from optimization schemes, as demonstrated in this study.\nI do not understand this comment. Laser ultrasonic characterization, including full field imaging has been around for decades (e.g. [Tachizaki, Takehiro, et al. \"Scanning ultrafast Sagnac interferometry for imaging two-dimensional surface wave propagation.\" Review of Scientific Instruments 77.4 (2006)], or the paper cited above)\n\nAuthor: Prior works in the MEMS and NEMS communities (which are clearly relevant here from a microfabrication perspective), to our knowledge have not explored large, free-standing microstructured thin films (spanning micrometers to millimeters)—owing, among others, to the prevalent interest in resonator technology and surface acoustic waves (SAWs), which do not require structures to be free-standing across a large spatial domain. However, such large, free-standing structures naturally enable the propagation of more complex wave modes (e.g., Lamb waves).\nI do not understand this comment. Lamb waves are routinely considered in phononic crystal design – see, e.g., another, uncited, example [Hyun, Jaeyub, Wonjae Choi, and Miso Kim. \"Gradient-index phononic crystals for highly dense flexural energy harvesting.\" Applied Physics Letters 115.17 (2019)].\n\nAuthor: As a side note, Sledzinska et al. (2016) focused on stabilizing ultra-thin films, using a polymer layer. While a novel technique, it is unclear how waves can propagate in that configuration over long distances due to damping—especially when aiming for lower frequencies, as discussed here. Discrepancies due to damping may affect quantitative comparisons of the full spatio-temporal evolution of waves through those wave guides.\nThere are many suspended membrane studies that do not use polymer layers. Here is another, uncited, example [Graczykowski, Bartłomiej, et al. \"Phonon dispersion in hypersonic two-dimensional phononic crystal membranes.\" Physical Review B 91.7 (2015): 075414].\n\nAuthor: Finally, Sledzinska et al., (2016) and several others within this domain focused on improving the fabrication of nanoscale waveguides rather than spatial grading for pre-defined wave guiding capabilities. As has been evident from literature [41, 12, 14, 47], this is a non-trivial problem with recent progress at macroscopic scales (including our own prior work, e.g., [25, 15]). The present manuscript demonstrates significantly improved synergy between scalability in computation, fabrication, and measurement for more robust spatial grading strategies.\nI do not understand this comment. Most other papers just treat this “synergy” as experimental validation of a computationally designed metamaterial.\n\nAuthor: The presented large-scale waveguides are made possible by the “tile design” approach, which is introduced here for the first time (and nontrivial due to, among others, requirements of matched dispersion relations at tile interfaces, treatment of boundaries, and the design targets required in such tiles). Brute-force optimization of the entire domain based on [32] would not be computationally feasible for the designs considered in this manuscript.\nThere is no proof by the authors that this brute force optimization is not feasible, particularly given the used (and prior demonstrated in Ref. [24]) Ray tracing algorithm. Why is the tiling needed? There is no evidence.\n\nAuthor: Ref. [24] exclusively optimized rays based on target locations (e.g., wave focusing). Here, we present for the first time how to optimize designs such that rays enter and exit perpendicular to the boundary (which is essential for the tiling to function). Ref. [24] only implements mass-spring network toy problems with analytical dispersion relations; the current manuscript’s demonstration of beam networks with numerically computed dispersion relations is a major improvement that is key for enabling experimental demonstration.\nThese both seem like relatively incremental advances in the opinion of the reviewer.\n\nAuthor: Thank you for pointing out that this wording is not precise. We have updated the wording to “Modeling, design, and manufacturing efficiency remains a bottleneck for architectures spanning tens of thousands to millions of unit cells...” We point out that the Springer-Nature journal family does not appreciate precise novelty claims (“we ask authors to refrain from making priority or novelty claims and to remove qualitative evaluations of their own work”; “avoid phrases like ‘for the first time’ and ‘unprecedented’).\nThe statements the authors make must still be accurate. In the Reviewers opinion, given the prior references and related work in other fields, this is not a bottleneck as the authors have claimed.\n\nAuthor: When referring to a “widely untapped design space”, we are specifically referring to spatially graded architectures with complicated spatial profiles spanning multiple length scales. As mentioned in paragraph 3 of the introduction, existing work on spatially graded phononic metamaterials has been restricted to simple grading profiles—–such as grading along one dimension only, or radially symmetric patterns. Two-dimensional spatial gradings with complicated grading profiles spanning multiple length scales have not been been previously considered. Scaling to large architectures with many unit cells creates the opportunity to explore gradings with more elaborate spatial structures. We are not tied to a particular application, as we present a fundamental framework for exploring what new capabilities are possible in a previously unexplored design space of complex, multiscale, 2D spatial gradings. However, we anticipate that expanding the design space and functionality of graded metamaterials will enhance applications that graded metamaterials (with simple grading profiles) have shown promise for already, such as energy harvesting [10] and mechanical/acoustic signal processing [11], in addition to potentially enabling new applications that could benefit from enhanced control over wave guiding, localization, and focusing, such as microfluidics [65].\nThe authors still have not clearly stated how this would be beneficial over simpler solutions. They have not demonstrated how energy harvesting, acoustic signal processing, focusing, or microfluidics would be improved via this design, compared to simpler, existing approaches.\n\nAuthor: We acknowledge that many approaches to wave guiding have been explored. Topological metamaterials, such as the example of (Ma et al., 2021), are appealing in many cases, e.g., due to their robustness to defects. One potential drawback is that they are restricted to often narrow bandgaps in the bulk, and to discrete wave guiding angles aligned with lattice interfaces. Similarly, many other approaches to acoustic and phononic metamaterials, which leverage bandgap design, are limited to narrow frequency bands, unlike the results presented here. The introduction has been updated to reflect these points.\nTopological waveguides were just one example. There are many more examples of topologically trivial waveguides simply by virtue of effective elastic property mismatches, which do not have a bandwidth limitation.\n\nAuthor: “Thank you for pointing out the lack of precision in this statement. We have modified these sentences to the following: “… Going from tens to hundreds of thousands of unit cells at the meso-scale requires the adaptation of specialized microfabrication techniques that have rarely been exploited for free-standing elastic wave guides [27] – especially spanning tens of micrometers to millimeters.”\n“Rarely” does not contribute to the scientific novelty of the work.\n\nAuthor: Thank you for pointing out this incorrect statement. We were referring to conventional macroscale 3D printing techniques as enabled by commercially available machines, not microfabrication processes. In the context of microfabrication, we have added further references on the fabrication of non-polymeric materials to the revised manuscript, though those have the drawback that the trade-off between resolution and field of view still remains a challenge. In response to similar comments from Reviewer 3, we have also included the broader class of conventional metal and ceramic additive manufacturing in our statement. The sentence in question has been modified as follows: “The current state of the art in commercial macroscale 3D printing technology enables architectures with hundreds of unit cells per dimension [28]. These methods, which fall within the broad category of additive manufacturing (AM), have been widely exploited for polymeric materials, which unfortunately are not suitable for wave propagation because of material-intrinsic damping. Although exciting new directions in metal and ceramic AM have emerged [29, 30, 31], careful control of printing resolution and material properties still poses challenges. Microfabrication techniques enabled by multi-photon lithography, traditionally using polymeric materials at significantly higher spatial resolution [32], have overcome this limitation [33, 34]. Such techniques, while growing rapidly, are still nascent in their ability to achieve large sample sizes close to 100 mm while maintaining sub-micrometer spatial resolution [35, 36], lesser so when spatially-graded non-periodic architectures are involved.”\nI do not think the arguments added to the introduction here are accurate. Even commercial entities are now advertising “wafer scale” production through 3D printing, including the availability of silica resins [https://www.nanoscribe.com/en/applications/wafer-scale-production-through-3d-direct-printing/,\nhttps://www.nanoscribe.com/en/products/gp-silica/].\n\nAuthor: While the overall scaling argument is true in principle, we kindly disagree with the reviewer on its applicability to the technique proposed in the the cited reference. It is crucial to take into account the technique itself. The scaling from 384 to 100,000 unit cells (for the same unit cell size) is non-trivial – the sample sizes are limited by the range of the galvanometer mirrors used in their setup (300 × 300 µm). To increase this range, the galvanometer scanning system must be coupled to a scanning stage. Hence, a simple linear time scaling from 384 to 100,000 unit cells is an unreasonable assumption. Such “stitching” methods have been shown to introduce defects in printed samples [33]. As mentioned before, while attempts at making such scalable samples are improving fast, the technology is still in the nascent/development stage. Finally, we reiterate that this technique has been shown to work for polymeric samples only as yet.\nI assert that this is incorrect, see the reply to the prior statement.", "reviewer_3": "The authors have revised their manuscript in a thorough manner and addressed most reviewer comments in an adequate fashion. The argumentation in regards to sufficient novelty is mostly convincing and the authors have tempered and modified incorrect/inaccurate claims.\n\nHowever, in several responses the authors purely point to changes/additions in the supplementary materials. They should carefully consider whether statements/claims are needed in the main manuscript, considering that supplementary material is only viewed by a fraction of readers. One example would be the computational cost and gains of the ray tracing approach, which is not reflected in the main manuscript.\n\nA note regarding the PDF manuscript, it is quite large, especially due to the figures. When viewing the manuscript in a browser, Fig. 4 does not load completely (it does load when viewing in Acrobat). In general, all figures load slowly and may even impede scrolling. The authors should make sure that figures (esp. pixel graphics) only use the resolution necessary for normal viewing and check their PDF in different viewers."}, "author_response": "Second Revision of NCOMMS-25-56068-T: “Graded phononic metamaterials: scalable design meets scalable microfabrication” by Charles Dorn, Vignesh Kannan, Ute Drechsler, and Dennis M. Kochmann\nWe thank the reviewers for their thoughtful and constructive comments. We have taken them as basis for the revision of our manuscript.\nIn the following, please allow us to respond to the reviewer’s questions and comments – explaining the changesmadeinourrevisedmanuscript(whicharehighlightedinredintheattachedmanuscript). Werefer tothereferencenumberingofthesecondrevisionofthemanuscript, whichiscopiedbelow. Pleasenotethat we have also taken the chance to clarify the labeling in the schematic of Fig. 3(e).\nAdditionally, please note that the affiliation of author Vignesh Kannan has been revised from “Laboratoire de M´ecanique des Solides, E´cole Polytechnique, Palaiseau, 91128, France” to “Laboratoire de M´ecanique des Solides, CNRS, E´cole Polytechnique, Institut Polytechnique de Paris, Palaiseau, 91128, France”. This does not reflect a change in the author’s institutional affiliation but is merely a change in the institution’s recommended nomenclature to include parent organizations as per the French academic system.\nReviewer 1:\nThe comments have been addressed satisfactorily. I thus recommend publication of the paper in its present form.\nResponse: Thank you for the positive feedback.\nReviewer 2: While the authors have made substantial replies in the rebuttal letter, I do not feel substantive improvements were made to the manuscript regarding modifying their arguments to accurately place their contribution in the context of existing and prior capabilities. Key references that would conflict with their arguments of novelty are still missing.\nI do not believe this is appropriate to be published anywhere in its current form. Even if their descrip- tion of the context of their contribution is improved, this work does not have sufficient novelty for Nature Communications. Scientific Reports or NJP Metamaterials could be alternate suitable venues.\nResponse: We strongly disagree with the reviewer’s assessment, and we are glad that Reviewers 1 and 3 also do not agree. Before we respond in detail to each point raised by Reviewer 2, we would like to make a general comment.\nReviewer 2 does not criticize the appropriateness of our methods nor the validity of our results, but they merely make a lack of “sufficient novelty” the basis of their negative assessment of our manuscript. We previously agreed with the reviewer that our manuscript leverages well-established techniques of microfab- rication and of wave modeling, so the novelty does not lie in the general techniques being used. However, we combine and extend existing experimental and numerical techniques to design, fabricate, and characterize a smoothly spatially-graded phononic metamaterial with the largest number of unit cells ever reported in the literature to the best of our knowledge (and over an extremely wide frequency range that is unusual for phononic metamaterials). Spatial gradings are of significant interest in the context of wave physics and metamaterials(fromtransformationelastodynamicstodispersivewaveguidance),yettodateallapproaches have relied on significantly smaller systems (in terms of the number of unit cells) and/or on considerably simplerdesigns(e.g.,periodicstructureswithorwithoutsharpinterfaces). Thenumberofunitcellsmatters, as theories based on homogenization require a separation of scales. From a modeling perspective, having a playgroundtoexperimentallyexploresmoothlyspatially-gradeddesignsiskeytoexploringtheopportunities and limitations of dynamic homogenization – which is exactly what our approach offers (while also exhibit- ing minimal damping, minimal geometric defects, and excellent scalability). In addition, our manuscript introduces a new tiling design strategy for metamaterials, a new spatially graded design parameterization, 1\na new optimization formulation, and a new multiscale setting of micro- (beam and unit cell), meso- (tile) and macro-scales (boundary value problem) – to mention but a few of the novel aspects of our manuscript.\nAs we elaborate in our responses below, several (very recent) review papers in the metamaterials literature [7, 8, 9] have highlighted the need for scalable design and fabrication approaches, which is what we are ad- dressing. Reviewer 2 further criticizes that we do not present practical applications, yet this disregards the fundamental scientific relevance: the presented setup can be used to explore many interesting wave physics questions,beyondtechnologicaldevices. Forallofthesereasons,wedobelievethatourmanuscriptwarrants consideration for publication in Nature Communications.\nIn the following, we will respond to each point of criticism raised by the reviewer.\nReviewer 2: Here is just one other, again uncited, example of using laser ultrasonics to measure phononic responses of patterned plates [Zhao, Jinfeng, et al. “On-chip valley phononic crystal plates with graded topological interface.” International Journal of Mechanical Sciences 227 (2022): 107460.]. In that reference they study 1.7-1.9 MHz waves in a patterned silicon plate with patterned area stretching over 31x37 cm within a 4 inch diameter wafer.\nResponse: We,ofcourse,acknowledgethatthereisarichbodyofliteratureonphononicwaveguidesaswell as on micro- and nanoscale structures fabricated by MEMS/NEMS technology. We appreciate the reviewer pointing out references that are related to our manuscript, including this one (there is likely a long list of references that are tangentially related but cannot be included for practical reasons). As also mentioned before, we agree with the reviewer that “using laser ultrasonics to measure phononic responses of patterned plates” is not a new invention, as stated in our previous response letter (where we cited [30]). However, the ability to develop microarchitecture through predictive computational design for arbitrary wave guiding is still a challenge. MEMS/NEMS technology can fabricate millions of small-scale unit cells on a wafer, but computational design has been lacking the tools to efficiently optimize for wave motion in such complex, spatially graded architectures.\nWith reference to the suggested paper (Zhao et al., 2022), the authors there engineered interfaces with a specific periodic lattice geometry, which is much simpler than the spatially varying designs presented here. That manuscript focuses on the engineering of interfaces, which is an important albeit considerably smaller subset of microarchitecture designs for wave motion. Furthermore, the excitation of waves in that paper is achieved by PZT plates (hence, not optical pump-probe laser ultrasonics), which exhibit limited control in frequency range and spatial distribution for excitation. Completely optical pump-probe setups (such as our system)deliversignificantlybettermodularityandcontrolofexcitation(see,e.g.,[50,51]). Laserultrasonics is a relatively vast topic applied across various material systems and length scales, hence a fully detailed literature review was not within the scope of this manuscript.\nIn light of this critique from the reviewer, these citations (i.e., [50, 51]) have been added to the revised manuscript.\nReviewer 2: The authors’ focus in the rebuttal letter on subtle difference from articles brought to their attention by the Reviewers misses the point. Reviewers should not have to do the authors’ literature review for them.\nBelow, I will also respond to the subset of their replies that I feel warrant particular response.\nAuthor: “the main focus of this manuscript is the application of our experimental-computational framework to design of new elastic wave guides across micrometer to millimeter length scales, like the figure-8 case discussed.”\nI do not understand what the experimental part is in “experimental-computational framework”. It seems the design is purely computational, then there is experimental validation. Neither the fabrication method nor the characterization method is substantially new.\nResponse: To us, this is a matter of semantics (and we refer to the comments made above about sufficient vs. insufficient novelty). When speaking of a new “experimental-computational framework”, this does not necessarily imply that (a) all experimental or computational details are new and (b) that there is two-way 2\nfeedback between experiments and simulations. We introduce a new computational approach to designing large spatially graded waveguides, which would have little merit without an experimental route to fabri- cate such structures (and which additionally benefits from an experimental validation). At the same time, MEMS/NEMS technology to make such wafer-scale structures has been available for a long time – as the reviewerrightlypointsout. Yet,ithasnotbeenusedtoproducethetypeofgradedwaveguidesreportedhere, among others exactly because appropriate design tools were previously lacking. In our revised manuscript, we do not claim that the wafer fabrication route is novel nor that we are the first to use interferometry to characterize wave motion. Yet, we are the first to combine our novel design and optimization tools with a suitable fabrication approach that can produce hundreds of thousands of spatially graded unit cells in a single, smoothly graded structure and characterize the wave motion in situ for a successful validation. This is what we refer to as our “experimental-computational framework”.\nReviewer 2: In addition, I am unclear from the manuscript as to why someone would want to make these large gradient based structures, when the same thing can be done in a much smaller footprint. For instance, conventional (not even topological or phononic-crystal-based) elastic waveguides have been around for many decades. A broadband, figure 8 waveguide could be easily done in this conventional context via design of impedance mismatch and total internal reflection.\nResponse: We would like to re-emphasize that, as the reviewer points out, there are many wave guiding methodologies,eachwiththeirbenefitsandlimitations. Theaimofthisresearchisnottoproduceasuperior figure-eightwaveguidewithrespecttoallotherpotentialwaveguidingapproaches. Thefigure-eightexample provides a platform for numerical and experimental validation, as we explore the design space of spatially graded architectures spanning multiple length scales. This expansion of the design space, specifically to include gradings on multiple length scales, may unlock new capabilities not possible with simpler designs (the design of Fig. S2 as one example). Using reflections (as a consequence of impedance mismatch, as pointedoutbythereviewer)isadifferentphysicalmechanismthatcanbeexploited. Usingsmoothgradings insteadhasbenefitssuchasslowvariationsinotherphysicalproperties(e.g.,mechanicalstiffnessandstrength and therefore lower stress concentrations). Interfaces may also excite other Bloch branches, be frequency- selective,orleadtoparasiticscattering. Ouraimistodemonstrateanapproachthatenablessmoothspatial gradings to accommodate new opportunities for wave guidance.\nReviewer 2: Author: In general, we stress that it was not our intention to claim that we have invented a new technique in microfabrication. We apologize if we inadvertently created that impression. Studies that have breached the nanoscale (like the one that the reviewer correctly pointed out) have achieved free-standing structures of hundreds of micrometers. In certain applications of mechanical wave guiding, however, it may be critical to guide waves over much longer distances; this is non-trivial to achieve.\nWhat applications, specifically, are these critical for? This has still been left as nebulous in the manuscript.\nResponse: We acknowledge that, as the reviewer correctly points out, this research is not tied to a specific application. This is curiosity-driven research in pursuit of exploring the design space of complex, multiscale spatial gradings. We strongly believe there is scientific merit in exploring phononic metamaterial dynamics within this largely unexplored design space. While our interest is of scientific nature and we do not develop a specific application, we are confident that the tools and knowledge presented in this manuscript will capture the interest of the metamaterials and broader wave physics communities and may lead to novel applications and devices. Our broad intent is that expanding the metamaterial design space expands the achievable functionalities. This has been evident in the past, for example, when the expansion of the design space from periodic architectures to simple spatial gradings increased functionalities (e.g., metamaterial Luneburg lenses) driving interest in applications such as energy harvesting and signal processing. Our work further expands the grading design space to complex, multiscale architectures on a fundamental level. The precise future application domains that benefit from our fundamental scientific contributions are yet to be determined. Complex wave manipulation functionalities over long distances with respect to the unit cell scale and wavelength is made possible. Moreover, the fact that we can guide and characterize waves in spatiallygradedstructureswithhundredsofthousandstomillionsofunitcellspresentsanew,versatile,and powerful experimental platform for research in wave motion (going far beyond the specific unit cell design and optimization objectives presented here).\nWhat advantages and where are they stated? Do you mean “This admits controlling wave motion in the tens of kHz to MHz regime across centimeters – a regime of long-standing interest to the field”. If so: i) this is still nebulous; ii) Someone controlling kHz to MHz waves would just use larger structures there is no need to use few micron beams (see paper cited above).\nResponse: We appreciate that reviewer acknowledging that “controlling wave motion in the tens of kHz to MHz regime across centimeters” is novel and at present possible only with significantly “larger structures”. First,weagainstressthatscientificnoveltywithoutanimmediatetechnologicalapplicationinourviewdoes warrant publication in a scientific journal. Second, the small-scale design presented in our work admits easy integrationintodeviceswithlimitedoperatingvolume. Forexample,on-chipenergyharvestersormechanical signal processors cannot include significantly larger systems. Finally, to “just use larger structures” would limittherangeoffrequenciesandlengthscaleswithinwhichwaveguidingispossible. Therefore, thecontrol of architecture across length scales is essential to controlling a “broadband” response.\nReviewer 2: Author: For example, if we could fabricate nanoscale structures on a free-standing membrane spanning tens of millimeters (like ours), this would be a momentous step forward in scalable wave-guiding technology, e.g., for controlled design of the “long wavelength” dispersion response of mechanical waves in signal processing.\nHow is this “momentous” exactly? What would this enable in terms of new phenomenology or functionality? Why would one just not use larger unit cells (see paper cited above)?\nResponse: With regards to enabling of new functionality, we refer to our previous response. Using larger unit cells limits the length scales over which wave guiding can be achieved (and requires a larger footprint). Working on the simple assumption that the unit cell size imposes the smallest length scale over which a wave path in physical space can be guided, achieving the physically smallest possible unit cell size while controllingthelengthscaleoverwhichtheunitcellarchitectureisvariedtakesusonestepclosertoachieving independent control of elastic waves in the real and Fourier spaces.\nAn extreme example would be achieving billions of nano- to micro-scale unit cells over an 8-inch free- standing film (as opposed to our 4-inch ones). With the right computational design and experimental tools, we can guide kHz to GHz waves across distances over a hundred millimeters, with wave-paths achieving curvatures on the hundreds-of-nanometer length scales (while our paper does not reach billions of unit cells and GHz frequencies, we take a significant step toward scalable broadband wave control). While still a fundamental study, the applications in high-throughput cell sorting, materials characterization, broadband signal processing, encoding and decoding, to name a few, could well be imagined. This requires true multi- scale capabilities in fabrication, and more importantly computational design, that cannot be achieved just through simply changing the size of unit cells.\nReviewer2: Author: Our“meso”scalesalsoallowmeasurementsofthefullspatio-temporallyresolvedwave propagation with sufficiently high resolution. Such high-dimensional experimental data are crucial to validate computational models, verify and realize complex waveguide designs obtained from optimization schemes, as demonstrated in this study.\nI do not understand this comment. Laser ultrasonic characterization, including full field imaging has been around for decades (e.g. [Tachizaki, Takehiro, et al. “Scanning ultrafast Sagnac interferometry for imaging two-dimensionalsurfacewavepropagation.”ReviewofScientificInstruments77.4(2006)], orthepapercited above)\nResponse: Thismaybeamatterofsemantics,butwedonotseewherethereviewer’sstatementinvalidates ours. We agree that laser ultrasonic characterization has existed for a long time. We previously referred to [30], where laser ultrasonic characterization was used. Laser ultrasonics is a relatively vast topic applied across various material systems and length scales, hence a fully detailed literature review was not within the scope of this manuscript, though some of those references have been added to the revised manuscript\n4\n[49, 50, 51]. As pointed out in previous responses, a key novelty of our work lies in the design and fab- rication of smoothly spatially-graded structures containing hundreds of thousands of unit cells, which we achieve via microfabrication. This in turn requires a suitable characterization technique for measurements of the full spatio-temporally resolved wave propagation. It is for this purpose that we used laser ultrasonic characterization.\nReviewer 2: Author: Prior works in the MEMS and NEMS communities (which are clearly relevant here from a microfabrication perspective), to our knowledge have not explored large, free-standing microstructured thinfilms(spanningmicrometerstomillimeters)—owing,amongothers,totheprevalentinterestinresonator technology and surface acoustic waves (SAWs), which do not require structures to be free-standing across a large spatial domain. However, such large, free-standing structures naturally enable the propagation of more complex wave modes (e.g., Lamb waves).\nI do not understand this comment. Lamb waves are routinely considered in phononic crystal design – see, e.g., another, uncited, example [Hyun, Jaeyub, Wonjae Choi, and Miso Kim. “Gradient-index phononic crystals for highly dense flexural energy harvesting.” Applied Physics Letters 115.17 (2019)].\nResponse: Thank you for pointing out this mistake in our wording. We meant to say Bloch waves instead of Lamb waves in this statement.\nTherevieweriscorrectthatLambwavesincontinuum(orhomogenized)free-standingplateshavebeenwell understood and utilized. Bloch waves, which interact with the microstructure above the homogenization limit, have been much less utilized – especially in the context of spatial grading. For example, the reference pointedout(Hyunetal., 2019)considersaneffectivemediumapproximation, restrictingtheapplicabilityof their approach to the low-frequency homogenization limit. By leveraging ray tracing, our work captures the complete local dispersion relations thus capturing the Bloch modes above the homogenization limit.\nReviewer 2: Author: As a side note, Sledzinska et al. (2016) focused on stabilizing ultra-thin films, using a polymer layer. While a novel technique, it is unclear how waves can propagate in that configuration over long distances due to damping—especially when aiming for lower frequencies, as discussed here. Discrepancies due to dampingmayaffectquantitativecomparisonsofthefullspatio-temporalevolutionofwavesthroughthose wave guides.\nThere are many suspended membrane studies that do not use polymer layers. Here is another, uncited, example [Graczykowski, Bart(cid:32)lomiej, et al. “Phonon dispersion in hypersonic two-dimensional phononic crystal membranes.” Physical Review B 91.7 (2015): 075414].\nResponse: As before, we agree with the reviewer that nanoscale device fabrication is a well-established field. Given this fact, it is impractical to cite all studies on nano-fabrication, especially since our targeted rangeoflengthandtimescales, aswellasourfocusonspatially graded architectures, areverydifferentfrom previous work in this vast field.\nReviewer 2: Author: Finally, Sledzinska et al., (2016) and several others within this domain focused on improving the fabrication of nanoscale waveguides rather than spatial grading for pre-defined wave guiding capabilities. As has been evident from literature [41, 12, 14, 47], this is a non-trivial problem with recent progress at macroscopic scales (including our own prior work, e.g., [25, 15]). The present manuscript demonstratessignificantlyimprovedsynergybetweenscalabilityincomputation,fabrication,andmeasurement for more robust spatial grading strategies.\nI do not understand this comment. Most other papers just treat this “synergy” as experimental validation of a computationally designed metamaterial.\nResponse: We refer to our previous responses: this is a matter of semantics to us. By “synergy” we refer to the benefits of being able to simulate, optimize, fabricate, and test complex, spatially graded phononic structures covering three length scales (from unit cell to mesoscale to structure). Simply put, our use of the term “synergy” refers to the added benefits of combining experimental fabrication and characterization\n5\nanddesignoptimization. Microfabricationhaslongenabledsmall-scalestructures,butsuitableoptimization techniques for the graded designs presented here have been lacking. Ray tracing-based optimization of metamaterials was previously introduced (by us), but without the tiling strategy introduced here could not reach the relevant scales. Combining the two yields synergy, especially when complemented by laser ultrasonic characterization for validation. This is what we refer to by “synergy”.\nReviewer 2: Author: The presented large-scale waveguides are made possible by the “tile design” approach, which is introduced here for the first time (and nontrivial due to, among others, requirements of matched dispersion relations at tile interfaces, treatment of boundaries, and the design targets required in such tiles). Brute-force optimization of the entire domain based on [32] would not be computationally feasible for the designs considered in this manuscript.\nThere is no proof by the authors that this brute force optimization is not feasible, particularly given the used (andpriordemonstratedinRef. [24])Raytracingalgorithm. Whyisthetilingneeded? Thereisnoevidence.\nResponse: Thankyouforpointingthisout,andweagreethattheadvantageofamodular,tile-baseddesign over brute-force optimization merits further discussion. We cannot, in absolute terms, prove the negative – that brute-force optimization on a large domain is not feasible with arbitrary computational resources. However,wecanprovideastrongargumentthatbrute-forceoptimizationbecomessignificantlylesseffective (to the point of being practically infeasible) for larger design domains and more complex cost functions.\nFirst, as the design domain grows, brute-force optimization would see an increase in design variables. For example, design of a single 64×64 unit cell tile, which has one design parameter per unit cell, involves op- timization over 3844 design variables (after removing design variables on the boundary). This optimization takes on the order of hours (depending on parallelization and specific computing hardware). Thus, each independently designed tile involves a 3844 variable design problem. Modular design by tile assembly intro- duces no additional computation time once individual tiles are designed, with no limit on scalability (e.g., Fig. S2 presents a 448×320 unit cell design). Alternatively, brute-force optimization scales poorly with increasing domain size. As an example, optimization of a 448×320 unit cell domain would require optimization over ∼143,000 variables.\nA second barrier to brute-force optimization is non-convexity. While the optimization of a single tile is non-convex, we demonstrate that it is within reach to find global optima (i.e., we found designs that send the cost function to zero, which is the case for Tiles 1 and 2 of the manuscript). In our experience, as the number of design variables grows and as the complexity of cost function increases, it is increasingly difficult to navigate the nonconvex design space, to the point that brute-force optimization becomes practically infeasible. However, we prefer to refrain from presenting a detailed study of failed brute-force optimization attempts.\nA discussion clarifying the challenges of brute-force optimization and the advantages of modular design has been added to the Modular Waveguide Design section of the main text.\nReviewer 2: Author: Ref. [24] exclusively optimized rays based on target locations (e.g., wave focusing). Here, we present for the first time how to optimize designs such that rays enter and exit perpendicular to the boundary (which is essential for the tiling to function). Ref. [24] only implements mass-spring network toy problems with analytical dispersion relations; the current manuscript’s demonstration of beam networks with numerically computed dispersion relations is a major improvement that is key for enabling experimental demonstration.\nThese both seem like relatively incremental advances in the opinion of the reviewer.\nResponse: Wekindlydisagree. Ofcourse,thejudgmentofaresearchcontributionastransformativeversus incremental is subjective. However, we argue that our manuscript takes a significant and nontrivial step to transformatheoreticalframework(i.e.,theauthors’previousworkof[27]–ref.[24]ofthepreviousrevision) previously demonstrated only on a simple, non-realistic toy problem (a mass-spring network with analytical dispersion relations) to physically realistic system that can be implemented and rigorously validated experi- mentally. Inaddition,wemakesignificantgeneralizationtoref.[27],makingamuchricherandmoregeneral\n6\nfamily of design objectives within reach. Considering these advances with respect to previous work, we feel that the innovation of this work compared to ref. [27] is substantial and will be of notable interest to the metamaterials, mechanics, and wave physics communities.\nReviewer 2: Author: Thank you for pointing out that this wording is not precise. We have updated the wording to “Modeling, design, and manufacturing efficiency remains a bottleneck for architectures spanning tens of thousands to millions of unit cells...” We point out that the Springer-Nature journal family does not appreciate precise novelty claims (“we ask authors to refrain from making priority or novelty claims and to removequalitativeevaluationsoftheirownwork”;“avoidphraseslike‘forthefirsttime’and‘unprecedented’). Thestatementstheauthorsmakemuststillbeaccurate. IntheReviewer’sopinion, giventhepriorreferences and related work in other fields, this is not a bottleneck as the authors have claimed.\nResponse: We respectfully disagree with the reviewer, and we would like to point out that the challenge of scalability is frequently acknowledged in contemporary review papers that discuss the state of the art in phononic metamaterials. Examples of phononic metamaterial reviews from 2024-2025 commenting on the scalability bottleneck include the following:\n• Theauthorsof[7]statethat“Advancesinnumericalmodellingtechniqueswillallowresearcherstoex- ploreavastdesignspaceefficiently. Ontheotherhand,scalableproductionmethodsandassessingthe long-term durability and stability of metamaterials in various environments are crucial considerations not only for metamaterials designed for acoustic absorption but also for any material or technology.”\n• The authors of [8] state that “On the practical side, developing innovative fabrication techniques is essential for constructing the most challenging and advanced metamaterial structures. Current limitations in manufacturing, such as achieving the necessary precision and scalability, hinder the practical deployment of metamaterials in real-world devices.”\n• The authors of [9] state that “Other notable challenges are, for example, the need to develop scalable manufacturing techniques.”\nTheauthorsoftheabovereviewpaperscollectivelyincludeasignificantlistofprominentcontributorstothe field of phononic metamaterials, indicating that there is notable consensus that scalability is an important bottleneck faced by current metamaterials research, and that there is significant interest in research efforts towardscalability(andwethereforefeelthatourmanuscriptwillbeofsignificantinteresttothecommunity). Thus, we believe that our wording of the sentence pointed out by the reviewer accurately represents the consensus of much of the research community while avoiding absolute statements.\nReviewer 2: Author: When referring to a “widely untapped design space”, we are specifically referring to spatially graded architectures with complicated spatial profiles spanning multiple length scales. As mentioned in paragraph 3 of the introduction, existing work on spatially graded phononic metamaterials has been restricted to simple grading profiles—–such as grading along one dimension only, or radially symmetric patterns. Two-dimensional spatial gradings with complicated grading profiles spanning multiple length scales have not been been previously considered. Scaling to large architectures with many unit cells creates the opportunity to explore gradings with more elaborate spatial structures. We are not tied to a particular application, as we present a fundamental framework for exploring what new capabilities are possible in a previously unexplored design space of complex, multiscale, 2D spatial gradings. However, we anticipate that expanding the design space and functionality of graded metamaterials will enhance applications that graded metamaterials (with simple grading profiles) have shown promise for already, such as energy harvesting [10] and mechanical/acoustic signal processing [11], in addition to potentially enabling new applications that could benefit from enhanced control over wave guiding, localization, and focusing, such as microfluidics [65].\nThe authors still have not clearly stated how this would be beneficial over simpler solutions. They have not demonstrated how energy harvesting, acoustic signal processing, focusing, or microfluidics would be improved via this design, compared to simpler, existing approaches.\nResponse: The objective of this work is not to directly advance applications such as energy harvesting\n7\nor acoustic signal processing. The objective of this work is to provide knowledge and tools to expand the metamaterial design space such that complex, multiscale spatially graded designs are within reach. We argue that all expansions of the metamaterial design space thus far have proven fruitful. For example, simple gradings (e.g., to create Luneburg lenses) are more complex than periodic architectures; this added complexity increases metamaterial functionality to include wave focusing capabilities. While we concentrate onspatialgradings,therearemanysimilarexamplesinotherregionsofthemetamaterialdesignspacewhere adding non-periodic complexity increases functionality (e.g., interfaces and defects). Thus, we are confident thattheexpansionofthemetamaterialdesignspacetomultiscalespatialgradingspresentedinthisworkwill inspire new metamaterial functionalities and applications. Due to the success of simple spatial gradings for applications such as energy harvesting and signal processing, we explicitly mention those as potential future applicationdomains. Wespecificallyanticipatethatourresultswillprovideafoundationforon-chipsensing and energy harvesting applications, where larger-scale metamaterial systems are not feasible. However, we do not claim to specifically advance device design in those application domains.\nReviewer2: Author: Weacknowledgethatmanyapproachestowaveguidinghavebeenexplored. Topological metamaterials, such as the example of (Ma et al., 2021), are appealing in many cases, e.g., due to their robustness to defects. One potential drawback is that they are restricted to often narrow bandgaps in the bulk, and to discrete wave guiding angles aligned with lattice interfaces. Similarly, many other approaches to acoustic and phononic metamaterials, which leverage bandgap design, are limited to narrow frequency bands, unlike the results presented here. The introduction has been updated to reflect these points.\nTopological waveguides were just one example. There are many more examples of topologically trivial waveg- uides simply by virtue of effective elastic property mismatches, which do not have a bandwidth limitation.\nResponse: Followingourpreviousresponses,weagainacknowledgethattherearemanyapproachestowave guidinginmetamaterials,eachwithcertainbenefitsandlimitations. Here,weweresimplypointingoutthat the topological waveguide example mentioned by the reviewer is typically band-limited. Other approaches (i.e., different concepts for topologically trivial waveguides) have different benefits and limitations. We do not believe this conflicts with our contributions, which explore multiscale spatial gradings spanning many unit cells – a setting that has not previously been explored.\nReviewer 2: Author: “Thank you for pointing out the lack of precision in this statement. We have modified these sentences to the following: “... Going from tens to hundreds of thousands of unit cells at the meso- scale requires the adaptation of specialized microfabrication techniques that have rarely been exploited for free-standing elastic wave guides [27] – especially spanning tens of micrometers to millimeters.”\n“Rarely” does not contribute to the scientific novelty of the work.\nResponse: This is again a matter of semantics. Within the rich literature of phononic metamaterials and MEMS/NEMS devices we cannot cite every possible publication in this sentence. Those “rare” cases we referred to did not investigate spatially graded, free-standing elastic waveguides across the current length and frequency ranges. Thus, we believe that this does constitute novelty.\nReviewer 2: Author: Thank you for pointing out this incorrect statement. We were referring to conven- tionalmacroscale3Dprintingtechniquesasenabledbycommerciallyavailablemachines,notmicrofabrication processes. In the context of microfabrication, we have added further references on the fabrication of non- polymeric materials to the revised manuscript, though those have the drawback that the trade-off between resolution and field of view still remains a challenge. In response to similar comments from Reviewer 3, we have also included the broader class of conventional metal and ceramic additive manufacturing in our state- ment. The sentence in question has been modified as follows: “The current state of the art in commercial macroscale3Dprintingtechnologyenablesarchitectureswithhundredsofunitcellsperdimension[28]. These methods, which fall within the broad category of additive manufacturing (AM), have been widely exploited for polymeric materials, which unfortunately are not suitable for wave propagation because of material-intrinsic damping. Although exciting new directions in metal and ceramic AM have emerged [29, 30, 31], careful con- trolofprintingresolutionandmaterialpropertiesstillposeschallenges. Microfabricationtechniquesenabled\n8\nbymulti-photonlithography,traditionallyusingpolymericmaterialsatsignificantlyhigherspatialresolution [32], have overcome this limitation [33, 34]. Such techniques, while growing rapidly, are still nascent in their ability to achieve large sample sizes close to 100 mm while maintaining sub-micrometer spatial resolution [35, 36], lesser so when spatially-graded non-periodic architectures are involved.”\nI do not think the arguments added to the introduction here are accurate. Even commercial entities are now advertising“waferscale”productionthrough3Dprinting,includingtheavailabilityofsilicaresins[https:// www.nanoscribe.com/en/applications/wafer-scale-production-through-3d-direct-printing/,\nhttps://www.nanoscribe.com/en/products/gp-silica/].\nResponse: We acknowledge the reviewer’s comment on existence of commercial solutions for “wafer scale” production. First,thestatementinquestion(intherevisedmanuscript)wasaimedatreinforcingthestrength ofsiliconmicrofabricationtechniquesoverotheradditivemanufacturingtechniques(inresponsetocomments from this and one other reviewer); hence, it is not conflicting with the reviewer’s critique.\nSecond, the commercial solution cited by the reviewer is much more advanced technology than commonly available systems (in large-scale cleanroom facilities). We expect, based on our experience, that the ac- cessibility of academic groups (or even industries) to such specialized systems is limited. Consequently, we believethatthecommercializationofcertaintechnologiesdoesnotnecessarilyimplythattheyarecommonly available for all fundamental or applied research related to the topic. This is evidenced by the relatively limitedstudiesonwaveguideswithsuchlargescaleseparationintheliterature,andthepreviouslypresented comments from reviews in the literature on the limited scalability of current fabrication routes. We, of course, do not claim that these technologies cannot be used for similar purposes; on the contrary, these may prove powerful for even more advanced wave guide designs – yet those require design approaches like the one presented here.\nReviewer 2: Author: While the overall scaling argument is true in principle, we kindly disagree with the reviewer on its applicability to the technique proposed in the the cited reference. It is crucial to take into account the technique itself. The scaling from 384 to 100,000 unit cells (for the same unit cell size) is non-trivial – the sample sizes are limited by the range of the galvanometer mirrors used in their setup (300 × 300 µm). To increase this range, the galvanometer scanning system must be coupled to a scanning stage. Hence, a simple linear time scaling from 384 to 100,000 unit cells is an unreasonable assumption. Such “stitching” methods have been shown to introduce defects in printed samples [33]. As mentioned before, while attempts at making such scalable samples are improving fast, the technology is still in the nascent/development stage. Finally, we reiterate that this technique has been shown to work for polymeric samples only as yet.\nI assert that this is incorrect, see the reply to the prior statement.\nResponse: As mentioned in our earlier response, we acknowledge the existence of commercial technology cited by the reviewer but have seen almost no use of this technology in fundamental studies of multiscale wave guide design and development.\nFurthermore, we stand by our (contested) argument in our first response letter, where we claim that the two-photon lithography method of Zhang et al., Nano Letters 24.8 (2024) does not reasonably/accurately scale to a build volume over 200 times larger than that presented in their original paper to achieve 100,000 unit cells. To back up our claim, we point to the following recent paper: Gu et al., Nature, 2025, https: //doi.org/10.1038/s41586-025-09842-x (see their Figure 5), which thoroughly reviews micro/nanoscale additive manufacturing methods including Zhang et al.’s method, and also claims that it is restricted to a <1 mm2 write field without error-prone stitching; this bottleneck has also been highlighted in a previous review by Saccone et al., Nature, 2022, http://dx.doi.org/10.1038/s41586-022-05433-2, referenced in our earlier response [36].\nWe acknowledge that micro-/nanoscale additive manufacturing is a rapidly advancing field, exemplified by theGuetal.paperthatwepreviouslymentioned(whichwasrecentlypublishedafterourpreviousresponse), presenting advances in parallelized two-photon lithography. We believe such advances complement rather thanconflictwiththeimpactofourwork, sinceweemphasizemultiscalemodelinganddesignadvancesthat must parallel manufacturing advances to achieve technological progress.\nThe authors have revised their manuscript in a thorough manner and addressed most reviewer comments in an adequate fashion. The argumentation in regards to sufficient novelty is mostly convincing and the authors have tempered and modified incorrect/inaccurate claims.\nResponse: Thank you for the positive feedback.\nHowever, in several responses the authors purely point to changes/additions in the supplementary materials. They should carefully consider whether statements/claims are needed in the main manuscript, considering that supplementary material is only viewed by a fraction of readers. One example would be the computational cost and gains of the ray tracing approach, which is not reflected in the main manuscript.\nResponse: Goodpoint. Weagreethatsomeofourpreviousrevisionsareworthdiscussinginthemaintext toensuretheyreachthemajorityofreaders. WehavemodifiedtheModularWaveguideDesign sectionofthe maintexttoincludebriefdiscussionsaboutraytracingvalidity, applicabilityofouroptimizationframework to other objectives, and computational efficiency of transient finite element simulations versus ray tracing (while honoring the strict word count limit imposed by the journal).\nReviewer 3: A note regarding the PDF manuscript, it is quite large, especially due to the figures. When viewing the manuscript in a browser, Fig. 4 does not load completely (it does load when viewing in Acrobat). In general, all figures load slowly and may even impede scrolling. The authors should make sure that figures (esp. pixel graphics) only use the resolution necessary for normal viewing and check their PDF in different viewers.\nResponse: We apologize for the excessive file size and have reduced the figure resolution.\nReferences\n[1] Brillouin, L. 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