
==== Front
Nat Commun
Nat Commun
Nature Communications
2041-1723
Nature Publishing Group UK London

49856
10.1038/s41467-024-49856-z
Article
Ultra-broadband illusion acoustics for space and time camouflages
http://orcid.org/0000-0002-5686-3257
Liu Chenkai 12
http://orcid.org/0000-0001-7944-7646
Ma Chu 3
http://orcid.org/0000-0002-0040-9274
Lai Yun laiyun@nju.edu.cn

1
http://orcid.org/0000-0001-5713-629X
Fang Nicholas X. nicxfang@hku.hk

24
1 grid.41156.37 0000 0001 2314 964X MOE Key Laboratory of Modern Acoustics, National Laboratory of Solid State Microstructures, School of Physics, and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093 China
2 https://ror.org/042nb2s44 grid.116068.8 0000 0001 2341 2786 Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 USA
3 https://ror.org/01y2jtd41 grid.14003.36 0000 0001 2167 3675 Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI, 53706 USA
4 https://ror.org/02zhqgq86 grid.194645.b 0000 0001 2174 2757 Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong, China
14 9 2024
14 9 2024
2024
15 804615 5 2023
20 6 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Invisibility cloaks that can suppress wave scattering by objects have attracted a tremendous amount of interest in the past two decades. In comparison to prior methods that were severely limited by narrow bandwidths, here we present a practical strategy to suppress sound scattering across an ultra-broad spectrum by leveraging illusion metamaterials. Consisting of a collection of subwavelength tunnels with precisely crafted internal structures, this illusion metamaterial has the ability to guide acoustic waves around the obstacles and accurately recreate the incoming wavefront on the exit surface. Remarkably, two ultra-broadband illusionary effects are produced, disappearing space and time shift. Sound scatterings are removed at all frequencies below a limit determined by the tunnel width, as confirmed by full-wave simulations and acoustic experiments. Our strategy represents a universal approach to solve the key bottleneck of bandwidth limitation in the field of cloaking in transmission, and establishes a metamaterial platform that enables the long-desired ultra-broadband sound manipulation such as acoustic camouflage and reverberation control, opening up exciting new possibilities in practical applications.

Invisibility cloaks have attracted a tremendous amount of interest. Here, the authors present a strategy to realize ultra-broadband disappearing-space and time-shift camouflages by leveraging illusion metamaterials, thereby removing the longstanding bottleneck of limited bandwidth in invisibility.

Subject terms

Acoustics
Characterization and analytical techniques
https://doi.org/10.13039/501100001809 National Natural Science Foundation of China (National Science Foundation of China) 11974176 Lai Yun the National Key Research and Development Program of China (Grant No. 2020YFA0211400)issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

Invisibility cloaks1,2, the long-cherished desire of human being, have become possible with the advent of artificial metamaterials and transformation optics theory, among the fields of optics3–9, acoustics10–12, elastics13 and thermotics14. Invisibility is a special illusionary effect15–19 of empty space, while from the perspective of wave scattering20–22, such cloaks can significantly suppress the scattering by any objects, which is widespread in the wave fields, as illustrated in Fig. 1a. By manipulating acoustic waves with acoustic metamaterials23–27 and metasurfaces28, acoustic invisibility cloaks29–36 have been demonstrated, which have wide implications like acoustic stealth and reverberation engineering. Besides invisibility cloaks, many other approaches for scattering suppression have been proposed, such as zero-index media37–39, destructive interference40, topological effects41, and spatial dispersion parameters42, etc. Nevertheless, so far, all these approaches have a severe limitation, i.e., narrow operating bandwidth. This limitation originates in the causality principle and resonant nature of general metamaterials and structures, which seriously hindered its real-world applications.Fig. 1 Illustration of the illusion.

Upon the incidence of a sound wave, a multiple scattering is generated by a random collection of obstacles, however, b when the obstacles are embedded in the illusion metamaterial, the backward reflection is removed and the wave can propagate around the obstacles with its wavefront undistorted. The arrows and the alternating solid and dashed lines represent the sound wave and its wavefront, respectively.

In this work, we reveal that this long-standing and seemly unsurmountable bandwidth limitation throughout various disciplines can be completely removed by leveraging a new class of illusion metamaterials that exhibit two ultra-broadband illusionary effects: disappearing space and time shift, instead of the empty space, as illustrated in Fig. 1b. Such illusion metamaterials consist of a collection of subwavelength acoustic tunnels arranged around the obstacles. Attributing to the equal acoustic path design of tunnels and the uniform geometric shape of the entry and exit surfaces, the wavefront of incident wave can be accurately reproduced on the exit surface over an ultra-broad spectrum from the quasistatic limit to an upper frequency limit determined by the tunnel width. Consequently, the space occupied by the metamaterial and obstacles seems to be completely disappeared, while at the same time, the time is shifted backward by a corresponding amount. With such extraordinary illusionary effects, the original sound scattering effects of obstacles are also eliminated in this unprecedented broad spectrum. Through full-wave simulations and acoustic experiments of a typical example, we demonstrate that the bandwidth of this ultra-broadband illusionary effects covers the entire regime from 1kHz to 16kHz. Our work replaces the traditional route of creating empty space by a new paradigm of creating disappearing space and time shift, enabling sound scattering suppression with the long-coveted extreme bandwidth and frequency-independent feature for the first time.

Results

Acoustic tunnels with internal protrusion structures

To show the generality of our approach, the tunnel is designed to possess an arbitrary shape, as shown in Fig. 2a. Suppose the tunnel has a varying cross-section-width w, and a horizontal length a. To achieve a tunable acoustic path, protrusion structures with a height of h and a lattice constant of d are added to the interior of the tunnel. The thickness of the tunnel shell and the protrusion is set as t. When the width of the tunnel is much smaller than the wavelength, there is only one mode in the tunnel and the acoustic tunnel can be described by a transmission-line model. A collection of such tunnels leads to anisotropic metamaterials43,44 that guide sound along certain directions. The introduction of slow-wave structures into the tunnels allows for the manipulation of sound phases and absorption in a more flexible manner28,45–47. Here the internal protrusion structures play a key role in modulating the total acoustic path of each tunnel.Fig. 2 Design of acoustic tunnels with tunable acoustic paths.

a Schematic diagram of a curved tunnel with a varying cross-section. The gray portion represents the solid structure. A magnified view is displayed in the top-right corner. Arrows indicate the incidence and transmission of waves. b Simulated acoustic pressure field distributions that vary with the ratio of h/w, corresponding to the phase shift Δϕ=0,π,2π. c Phase shift Δϕ and transmittance t2 under normal incidence as a function of h/w. Vertical gray dotted lines indicate the cases with Δϕ=π,2π. d Equivalent relative refractive index nr as a function of h/w and the frequency f.

To illustrate this mechanism, we have performed the full-wave simulations for the tunnel plotted in Fig. 2b and c. The geometric parameters are set as t=1mm, d=5mm and a=20cm. The maximum and minimum width of the tunnel are set as 8.95mm and 3.95mm, respectively. A frequency of f=6860Hz is chosen for demonstration. Figure 2b shows the acoustic pressure field distributions inside the tunnels with different geometric parameters of h/w. It is clearly seen that the phase shift of the transmitted wave increases substantially when h/w increases. We note that w and h simultaneously change along the propagation path, but the ratio of h/w is fixed in one tunnel. Figure 2c presents the phase shift Δϕ and the transmittance t2 as a function of h/w at f=6860Hz. We can observe that the transmitted phase shift can be conveniently modulated to cover the range of 2π by changing h/w from 0 to 0.32, while the average transmittance is kept above 97%, indicating that such type of tunnels can achieve a tunable acoustic path with a high efficiency. Investigation on some minor factors to the transmittance, including the incident angle of sound, the bending angle and the variation width of the tunnel, are discussed in the Supplementary Fig. 1. The results demonstrate that the functionality of the tunnel is quite robust against these factors. We note that comparing with other designs based on phase manipulation, such as the zigzag structures37, the protrusion structures exhibits a better performance in impedance matching.

Interestingly, such a type of tunnels with a curved path can be approximated as straight tunnels with a slower and dispersion-free effective sound speed and this works in an ultra-broad spectrum. The transmission phase of the tunnel is described as ϕ=k0L, where k0 is the wave number in the background (air) and L=nra is the acoustic path. Here, nr=ϕ/(k0a) is the effective refractive index. In Fig. 2d, we plot the calculated nr as a function of h/w and the frequency f for the tunnel in Fig. 2a. It is clearly seen that nr increases with the increase of h/w, and is almost unchanged from 0.1kHz to 15kHz. When h/w is small, the dispersion of nr is negligible in the frequency regime. We emphasize that this frequency-independent refractive index holds the key to the realization of ultra-broadband illusion.

Design of the ultra-broadband illusion metamaterial and the experimental setup

This illusion metamaterial is constructed by assembling an array of such tunnels to route around the obstacles and guide sound waves from the entry surface to the exit other. To create the ultra-broadband illusionary effect, there are two criteria. The first criterion is that the entry and exit surfaces of the illusion metamaterial should have exactly the same shape (dotted lines in Fig. 1b). The other criterion is that all the tunnels should have exactly the same acoustic path. If the two criteria are met by the illusion metamaterial, the wavefront of the incident acoustic field can be copied from the entry to the exit surface, therefore realizing the two illusionary effects of disappearing space and time shift.

Figure 3a shows the schematic diagram of a typical example of the illusion metamaterial and the corresponding experimental setup. There are three rhombic hard obstacles arranged along the y direction in air. The entry and exit surfaces of the illusion metamaterial are both flat surfaces in this case. A total number of twenty tunnels are applied for each rhombic obstacle, as shown by the zoom-in figure in Fig. 3b. Because of the mirror symmetry, ten different sets of h/w are designed, as shown in Fig. 3c. Such parameters can guarantee that the acoustic path is almost the same (L=22.5cm) for all twenty tunnels, despite that they have obviously different lengths and bending angles. In experiment, the illusion metamaterial as well as the obstacles are both fabricated by using 3D printing techniques. A photo of the sample and the experimental setup is shown in Fig. 3d. The whole experiment is performed in a plate waveguide with a height of 3cm. A cylindrical wave is emitted by a speaker located at a distance of d1=20cm away from the sample. A microphone is mounted on a horizontal translation stage to scan the transmitted signals located at the black dashed area shown in Fig. 3a in the xy plane. The measured area with the size of Δx=10cm and Δy=20cm is of d2=5cm away from the sample. Sound absorbing foams are placed around the platform to reduce the reflected waves and noise from the environment.Fig. 3 Design of the ultra-broadband illusion metamaterial and the experimental setup.

a Schematic diagram of the illusion metamaterial and the experimental setup. A magnified view of the metamaterial is shown in (b). The side length of the square-shaped unit is a=20cm. The diagonal lengths of the rhombic-shaped obstacle are represented by a and a/2, respectively. c Geometric parameter h/w for various tunnels in the metamaterial. d Picture of the experimental setup.

Ultra-broadband illusionary effect of disappearing space and sound scattering suppression

In Fig. 4, we demonstrate the full-wave simulation and acoustic experimental results obtained by using the illusion metamaterial designed in Fig. 3. For the case of bare obstacles at f=4860Hz (Fig. 4a), large scattering clearly occurs due to the huge impedance mismatch between obstacles and air, as confirmed by the shadows behind the obstacles. However, when the obstacles are embedded in the designed metamaterial (Fig. 4b), the impinging cylindrical waves are guided around the obstacles, and then reproduce the cylindrical wavefront of the incident waves in the transmission region. From the wavefront in the transmission region, the point source seems to have moved forward from the real point of (−30cm,0cm) to a virtual point of (−10cm,0cm). Such a position change of 20cm corresponds to the side length of the illusion metamaterial. Therefore, an illusion of disappearing space, i.e., the whole space of the metamaterial with the embedded obstacles has completely disappeared, is proved. Moreover, the field on the incidence region is almost undisturbed, indicating that the reflection is negligibly small. The measured results, as shown in the right inset graphs of Fig. 4a and b, agree well with the full-wave simulations.Fig. 4 Ultra-broadband illusionary effect of disappearing space and sound scattering suppression.

a, c, e Simulated acoustic pressure field distributions for the scenario of bare obstacles under a point source radiation at different operating frequencies of f=4860Hz,6860Hz,8860Hz, and the corresponding experimental measurements (located in the black dashed area) are on the right. b, d, f Results for the scenario of obstacles covered by the illusion metamaterial. g, h Frequency dependence of the normalized scattered pressure intensity for the transmission region (γt) and reflection region (γr).

To demonstrate the ultra-broadband property, the numerical and experimental results obtained for two more frequencies, i.e. f=6860Hz and f=8860Hz are shown in Fig. 4c–f. We can observe that the illusionary effects of disappearing space maintain perfectly under these frequencies with the illusion metamaterial, proving the ultra-broadband property.

To quantify the performance of illusion metamaterial in a broad spectrum, we calculate the normalized scattered pressure intensity, which is defined as γt,r=ps2/p02, where ps,p0 are the scattered field and the empty field, respectively. The subscript t and r represent the transmission region (x∈10cm,50cm) and incidence region (x∈−50cm,−10cm), respectively, and such intensity value is obtained from a spatial average. The scattered field ps is obtained by subtracting the empty field without the sample (p0) from the total field with the sample (p). It should be noted that, due to the illusionary effects, the empty field in the transmission region is calculated by using the virtual point source located at (−10cm,0cm). On the other hand, the empty field for the incidence region is calculated by using the real point source located at (−30cm,0cm). The results are plotted in Fig. 4g and h. It is clearly seen that γt,r maintain almost zero with the illusion metamaterial in an ultra-broad spectrum 1kHz~16kHz. In contrast, γt,r are significantly larger in the case of bare obstacles.

We emphasize that the wide bandwidth in this approach is far beyond those of any other approaches. Previous methods as empty space29–36 always have narrow bandwidths, except for ray optics cloaks48,49 that have ignored the phase difference and thus cannot be described as a scattering suppression device. Here, the illusion metamaterial works for all frequencies below an upper limit where more than one propagating mode occur inside the tunnels, which is determined by w. We note that this functionality is independent of the wavefront of the incident waves. More examples such as the Gaussian wave incidence and multiple point source incidence are also plotted in the Supplementary Information, which clearly verify the universality of this principle. Such approach is not limited to flat surfaces, and is also valid for curved geometries, as shown in the Supplementary Fig. 4.

Illusionary effect of time shift

We emphasize that besides the illusionary effect of disappearing space, there is, simultaneously, another illusionary effect of time shift. This illusionary effect can be demonstrated by considering a sound pulse propagating through a collection of obstacles with and without the illusion metamaterial, as shown in Fig. 5. A number of rhombic obstacles are arranged along the y direction. The pulse is a time-domain Gaussian signal with a broad bandwidth. Clearly, the scattering by the obstacles has produced intense reflected waves and significantly changed the wavefront in transmission in the case of bare obstacles, as shown in Fig. 5a. While with the illusion metamaterial (Fig. 5b), the reflected waves are much smaller, because of the good impedance matching between the metamaterial and background medium (free space). In the transmission region, the circular wavefront is perfectly maintained, as if emitted from a shifted virtual point source located at (−10cm,0cm). Furthermore, by comparing the wavefronts in the incidence and transmission regions, it is seen that there is a time shift in the transmission region. This is because that although the equal acoustic path in the tunnels leads to the illusionary effect of disappearing space, but such a disappeared space still takes a finite time for the waves to pass. The value of the extra time shift is obtained as tshift=L/c0=0.66ms, which is consistent with the results obtained from Fig. 5b. The zoom-in inset graphs on the right of Fig. 5b exhibit the details of how the wavefront of the sound waves are routed around the obstacles inside the metamaterial, thereby significantly reducing the scattering effect in Fig. 5a. More details are shown in Supplementary Information. The length of time shift can be modulated by controlling the acoustic path.Fig. 5 Pulse radiation and illusionary effect of time shift.

a Snapshots of a sound pulse propagating through a collection of obstacles with significant scattering. b Snapshots of a sound pulse propagating through the obstacles embedded in the illusion metamaterial. The inset graphs display magnified details of the wavefront inside the metamaterial. The numbers indicate the order of the snapshots, where Δt=0.1ms.

Illusion metamaterial for a random collection of obstacles

As a further demonstration of the robustness of this approach, we design a particular illusion metamaterial to suppress the sound scattering from three obstacles in a random arrangement. As shown in Fig. 6a, the illusion metamaterial is set in a square shape with a side length of 2.25a(a=20cm), and is composed of 46 tunnels, whose geometric parameters are presented in the Supplementary Fig. 7. Here, the uniform acoustic path in these tunnels is set as L=47.5cm. The size of the rhombic obstacles is the same as above. From the field distributions plotted in Fig. 6b, it is clearly seen that when the obstacles are covered by the illusion metamaterial, the wavefronts in both the incidence and transmission regions become undistorted, as if the obstacles disappear. The illusionary effects of disappearing space of length 45cm and a time shift of 1.38ms are both observed in the transmission region. On the contrary, mussy scattering occurs in the case of bare obstacles. More field distributions are shown in the Supplementary Information. Furthermore, the normalized scattered pressure intensity is calculated and shown in Fig. 6c and d, which turns out to be enormously reduced in the regime of 1kHz~16kHz.Fig. 6 Ultra-broadband illusion metamaterial for a random collection of obstacles.

a Schematic of a specific illusion metamaterial composed of three rhombic obstacles arranged randomly. b Simulated acoustic pressure field distributions for the scenarios of obstacles with or without the metamaterial under a point source radiation at an operating frequency of f=6860Hz. The point source is located at (−20cm,0cm), and the center of three obstacles are located at (10cm,10cm), (35cm,0cm) and (12.5cm,−12.5cm), respectively. c, d Frequency dependence of the normalized scattered pressure intensity for the transmission region (γt) and reflection region (γr).

Discussion

It is recognized that when the tunnel width is small, e.g. <1mm, dissipation effect cannot be ignored due to the viscous friction between the air and the hard boundaries. The influence of loss in principle can be reduced by applying relatively large tunnel widths as well as shorter lengths. On the other hand, the illusionary effects are impaired when the tunnels support more than one propagating mode, or when the wavelength is small enough to cause the diffraction effects20,21. Therefore, the operating bandwidth has an upper limit, which is around 16kHz in our design. Interestingly, there is no lower limit for our design.

Causality is one of the essential reasons why previous illusions of empty space have a narrow bandwidth48. However, in this work, there is no similar constraint due to the existence of the time shift, which, remarkably, enables ultra-broad operating bandwidth. It’s worth mentioning that the illusion of disappearing space is also fundamentally different from that of empty space extensively studied in invisibility cloaks. Although we are not realizing an invisibility cloak here, nevertheless, the original wave scattering effects of obstacles are completely removed. The only tradeoff is the ultra-broadband illusionary effect of disappearing space and time shift. It is therefore an ideal way to extend the bandwidth of scattering suppression to the extreme limit.

Thanks to its remarkable ultra-broad bandwidth and frequency-independent capabilities, our approach holds significant promise for a diverse range of practical applications in the field of acoustics. For example, this technique could effectively reduce sound scattering in deep-sea environments, especially around seamounts and other substantial features, thereby making their detection more challenging. In the realm of architectural acoustics, the camouflage effect could be used to tailor the acoustic characteristics within a room, leading to the enhancement of sound quality, speech intelligibility, and the reduction of unwanted echoes or reverberation. More applications include acoustic bending. Our work thus enables a powerful platform for the long-desired ultra-broadband sound manipulation.

Methods

Numerical simulations

The finite element software COMSOL Multiphysics is performed for the full-wave simulations with Pressure-Acoustics-Frequency-Domain and Pressure-Acoustics-Transient modules. The mass density and sound velocity of air are set as 1.21kg/m3 and 343m/s, respectively. The resin structures are treated as acoustically rigid materials. The plane wave radiation is used in Fig. 2, and the monopole point source is used in Figs. 4–6. Perfectly matched layers are adopted to reduce the reflections.

Experimental measurements

All samples are fabricated with resin by using stereolithography 3D printing techniques (SLA, 0.1mm in precision). A sound speaker radiates eight periods of sound waves with single frequency we want. A microphone (PCB 130F20) scans the sound field located at the black dashed area shown in Fig. 3a. The scanning has resolution of 1cm and in total 231 points are scanned. The measurement platform is a 2D waveguide with a height of 30mm. The sound absorbing foams are set around the system to reduce the reflections.

Supplementary information

Supplementary Information

Peer Review File

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-49856-z.

Acknowledgements

C.L. thanks J. Luo for helpful discussions. Y.L. acknowledges the support of the National Key Research and Development Program of China (Grant No. 2020YFA0211400) and the National Natural Science Foundation of China (Grants No. 12174188, 11974176) for this work.

Author contributions

C.L. and C.M. contributed equally to this work. Y.L. and N.X.F. supervised the project. C.L. and Y.L. conducted the analysis, simulations and sample fabrication. C.M. and N.X.F. conducted the experiment design and measurements. All the authors contributed to the data analysis and manuscript preparation.

Peer review

Peer review information

Nature Communications thanks Johan Robertsson, Marc Serra-Garcia, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The data during the current study are available from the corresponding author upon request.

Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

These authors contributed equally: Chenkai Liu, Chu Ma.
==== Refs
References

1. Leonhardt U Optical conformal mapping Science 2006 312 1777 1781 10.1126/science.1126493 16728596
Leonhardt, U. Optical conformal mapping. Science 312, 1777–1781 (2006).16728596 10.1126/science.1126493
2. Pendry JB Schurig D Smith DR Controlling electromagnetic fields Science 2006 312 1780 1782 10.1126/science.1125907 16728597
Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic fields. Science 312, 1780–1782 (2006).16728597 10.1126/science.1125907
3. Smith DR Pendry JB Wiltshire MCK Metamaterials and negative refractive index Science 2004 305 788 792 10.1126/science.1096796 15297655
Smith, D. R., Pendry, J. B. & Wiltshire, M. C. K. Metamaterials and negative refractive index. Science 305, 788–792 (2004).15297655 10.1126/science.1096796
4. Li J Pendry JB Hiding under the carpet: A new strategy for cloaking Phys. Rev. Lett. 2008 101 203901 10.1103/PhysRevLett.101.203901 19113341
Li, J. & Pendry, J. B. Hiding under the carpet: A new strategy for cloaking. Phys. Rev. Lett. 101, 203901 (2008).19113341 10.1103/PhysRevLett.101.203901
5. Lai Y Chen H Zhang ZQ Chan CT Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell Phys. Rev. Lett. 2009 102 093901 10.1103/PhysRevLett.102.093901 19392518
Lai, Y., Chen, H., Zhang, Z. Q. & Chan, C. T. Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell. Phys. Rev. Lett. 102, 093901 (2009).19392518 10.1103/PhysRevLett.102.093901
6. Yu N Light propagation with phase reflection and refraction Science 2011 334 333 337 10.1126/science.1210713 21885733
Yu, N. et al. Light propagation with phase reflection and refraction. Science 334, 333–337 (2011).21885733 10.1126/science.1210713
7. Liu Y Zhang X Metamaterials: A new frontier of science and technology Chem. Soc. Rev. 2011 40 2494 2507 10.1039/c0cs00184h 21234491
Liu, Y. & Zhang, X. Metamaterials: A new frontier of science and technology. Chem. Soc. Rev. 40, 2494–2507 (2011).21234491 10.1039/c0cs00184h
8. Kadic M Bückmann T Schittny R Wegener M Metamaterials beyond electromagnetism Rep. Prog. Phys. 2013 76 126501 10.1088/0034-4885/76/12/126501 24190877
Kadic, M., Bückmann, T., Schittny, R. & Wegener, M. Metamaterials beyond electromagnetism. Rep. Prog. Phys. 76, 126501 (2013).24190877 10.1088/0034-4885/76/12/126501
9. Alù A Engheta N Achieving transparency with plasmonic and metamaterial coatings Phys. Rev. E 2005 72 016623 10.1103/PhysRevE.72.016623
Alù, A. & Engheta, N. Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72, 016623 (2005).10.1103/PhysRevE.72.016623
10. Cummer SA Schurig D One path to acoustic cloaking N. J. Phys. 2007 9 45 10.1088/1367-2630/9/3/045
Cummer, S. A. & Schurig, D. One path to acoustic cloaking. N. J. Phys. 9, 45 (2007).10.1088/1367-2630/9/3/045
11. Chen H Chan CT Acoustic cloaking in three dimensions using acoustic metamaterials Appl. Phys. Lett. 2007 91 183518 10.1063/1.2803315
Chen, H. & Chan, C. T. Acoustic cloaking in three dimensions using acoustic metamaterials. Appl. Phys. Lett. 91, 183518 (2007).10.1063/1.2803315
12. Norris AN Acoustic cloaking Acoust. Today 2015 11 38 46
Norris, A. N. Acoustic cloaking. Acoust. Today 11, 38–46 (2015).
13. Farhat M Guenneau S Enoch S Ultrabroadband elastic cloaking in thin plates Phys. Rev. Lett. 2009 103 024301 10.1103/PhysRevLett.103.024301 19659209
Farhat, M., Guenneau, S. & Enoch, S. Ultrabroadband elastic cloaking in thin plates. Phys. Rev. Lett. 103, 024301 (2009).19659209 10.1103/PhysRevLett.103.024301
14. Schittny R Kadic M Guenneau S Wegener M Experiments on transformation thermodynamics: Molding the flow of heat Phys. Rev. Lett. 2013 110 195901 10.1103/PhysRevLett.110.195901 23705719
Schittny, R., Kadic, M., Guenneau, S. & Wegener, M. Experiments on transformation thermodynamics: Molding the flow of heat. Phys. Rev. Lett. 110, 195901 (2013).23705719 10.1103/PhysRevLett.110.195901
15. Lai Y Illusion optics: The optical transformation of an object into another object Phys. Rev. Lett. 2009 102 253902 10.1103/PhysRevLett.102.253902 19659076
Lai, Y. et al. Illusion optics: The optical transformation of an object into another object. Phys. Rev. Lett. 102, 253902 (2009).19659076 10.1103/PhysRevLett.102.253902
16. Pendry J All smoke and metamaterials Nature 2009 460 579 580 10.1038/460579a
Pendry, J. All smoke and metamaterials. Nature 460, 579–580 (2009).10.1038/460579a
17. Chen H Chan CT Sheng P Transformation optics and metamaterials Nat. Mater. 2010 9 387 396 10.1038/nmat2743 20414221
Chen, H., Chan, C. T. & Sheng, P. Transformation optics and metamaterials. Nat. Mater. 9, 387–396 (2010).20414221 10.1038/nmat2743
18. Kan W Acoustic Illusion near boundaries of arbitrary curved geometry Sci. Rep. 2013 3 1427 10.1038/srep01427 23478430
Kan, W. et al. Acoustic Illusion near boundaries of arbitrary curved geometry. Sci. Rep. 3, 1427 (2013).23478430 10.1038/srep01427
19. Zhang Y Luo Y Pendry JB Zhang B Transformation-Invariant Metamaterials Phys. Rev. Lett. 2019 123 67701 10.1103/PhysRevLett.123.067701
Zhang, Y., Luo, Y., Pendry, J. B. & Zhang, B. Transformation-Invariant Metamaterials. Phys. Rev. Lett. 123, 67701 (2019).10.1103/PhysRevLett.123.067701
20. KELLER JB Geometrical theory of diffraction J. Opt. Soc. Am. 1962 52 116 10.1364/JOSA.52.000116 14454948
KELLER, J. B. Geometrical theory of diffraction. J. Opt. Soc. Am. 52, 116 (1962).14454948 10.1364/JOSA.52.000116
21. Ufimtsev, P. Y. Fundamentals of the Physical Theory of Diffraction. (John Wiley & Sons, Inc., Hoboken, New Jersey, 2007).
22. P. Sheng. Introduction to Wave Scattering, Localization and Mesoscopic Phenomena. (Springer, New York, 2006).
23. Liu Z Locally resonant sonic materials Science 2000 289 1734 10.1126/science.289.5485.1734 10976063
Liu, Z. et al. Locally resonant sonic materials. Science 289, 1734 (2000).10976063 10.1126/science.289.5485.1734
24. Fang N Ultrasonic metamaterials with negative modulus Nat. Mater. 2006 5 452 456 10.1038/nmat1644 16648856
Fang, N. et al. Ultrasonic metamaterials with negative modulus. Nat. Mater. 5, 452–456 (2006).16648856 10.1038/nmat1644
25. Ma G Sheng P Acoustic metamaterials: From local resonances to broad horizons Sci. Adv. 2016 2 e1501595 10.1126/sciadv.1501595 26933692
Ma, G. & Sheng, P. Acoustic metamaterials: From local resonances to broad horizons. Sci. Adv. 2, e1501595 (2016).26933692 10.1126/sciadv.1501595
26. Cummer SA Christensen J Alù A Controlling sound with acoustic metamaterials Nat. Rev. Mater. 2016 1 16001 10.1038/natrevmats.2016.1
Cummer, S. A., Christensen, J. & Alù, A. Controlling sound with acoustic metamaterials. Nat. Rev. Mater. 1, 16001 (2016).10.1038/natrevmats.2016.1
27. Ge H Breaking the barriers: Advances in acoustic functional materials Natl Sci. Rev. 2018 5 159 182 10.1093/nsr/nwx154
Ge, H. et al. Breaking the barriers: Advances in acoustic functional materials. Natl Sci. Rev. 5, 159–182 (2018).10.1093/nsr/nwx154
28. Assouar B Acoustic metasurfaces Nat. Rev. Mater. 2018 3 460 472 10.1038/s41578-018-0061-4
Assouar, B. et al. Acoustic metasurfaces. Nat. Rev. Mater. 3, 460–472 (2018).10.1038/s41578-018-0061-4
29. Zhang S Xia C Fang N Broadband acoustic cloak for ultrasound waves Phys. Rev. Lett. 2011 106 024301 10.1103/PhysRevLett.106.024301 21405230
Zhang, S., Xia, C. & Fang, N. Broadband acoustic cloak for ultrasound waves. Phys. Rev. Lett. 106, 024301 (2011).21405230 10.1103/PhysRevLett.106.024301
30. Zhu X Liang B Kan W Zou X Cheng J Acoustic cloaking by a superlens with single-negative materials Phys. Rev. Lett. 2011 106 014301 10.1103/PhysRevLett.106.014301 21231745
Zhu, X., Liang, B., Kan, W., Zou, X. & Cheng, J. Acoustic cloaking by a superlens with single-negative materials. Phys. Rev. Lett. 106, 014301 (2011).21231745 10.1103/PhysRevLett.106.014301
31. Chen Y Broadband solid cloak for underwater acoustics Phys. Rev. B 2017 95 180104 10.1103/PhysRevB.95.180104
Chen, Y. et al. Broadband solid cloak for underwater acoustics. Phys. Rev. B 95, 180104 (2017).10.1103/PhysRevB.95.180104
32. Zhao W Chu H Tao Z Hang ZH Acoustic transmissive cloaking using zero-index materials and metasurfaces Appl. Phys. Express 2019 12 054004 10.7567/1882-0786/ab14ad
Zhao, W., Chu, H., Tao, Z. & Hang, Z. H. Acoustic transmissive cloaking using zero-index materials and metasurfaces. Appl. Phys. Express 12, 054004 (2019).10.7567/1882-0786/ab14ad
33. Li H Ultrathin acoustic parity-time symmetric metasurface cloak Research 2019 2019 8345683 10.34133/2019/8345683 31549089
Li, H. et al. Ultrathin acoustic parity-time symmetric metasurface cloak. Research 2019, 8345683 (2019).31549089 10.34133/2019/8345683
34. Popa BI Zigoneanu L Cummer SA Experimental acoustic ground cloak in air Phys. Rev. Lett. 2011 106 253901 10.1103/PhysRevLett.106.253901 21770640
Popa, B. I., Zigoneanu, L. & Cummer, S. A. Experimental acoustic ground cloak in air. Phys. Rev. Lett. 106, 253901 (2011).21770640 10.1103/PhysRevLett.106.253901
35. Zigoneanu L Popa BI Cummer SA Three-dimensional broadband omnidirectional acoustic ground cloak Nat. Mater. 2014 13 352 355 10.1038/nmat3901 24608143
Zigoneanu, L., Popa, B. I. & Cummer, S. A. Three-dimensional broadband omnidirectional acoustic ground cloak. Nat. Mater. 13, 352–355 (2014).24608143 10.1038/nmat3901
36. Yang Y Wang H Yu F Xu Z Chen H A metasurface carpet cloak for electromagnetic, acoustic and water waves Sci. Rep. 2016 6 20219 10.1038/srep20219 26822429
Yang, Y., Wang, H., Yu, F., Xu, Z. & Chen, H. A metasurface carpet cloak for electromagnetic, acoustic and water waves. Sci. Rep. 6, 20219 (2016).26822429 10.1038/srep20219
37. Liang Z Li J Extreme acoustic metamaterial by coiling up space Phys. Rev. Lett. 2012 108 114301 10.1103/PhysRevLett.108.114301 22540476
Liang, Z. & Li, J. Extreme acoustic metamaterial by coiling up space. Phys. Rev. Lett. 108, 114301 (2012).22540476 10.1103/PhysRevLett.108.114301
38. Dubois M Shi C Zhu X Wang Y Zhang X Observation of acoustic Dirac-like cone and double zero refractive index Nat. Commun. 2017 8 14871 10.1038/ncomms14871 28317927
Dubois, M., Shi, C., Zhu, X., Wang, Y. & Zhang, X. Observation of acoustic Dirac-like cone and double zero refractive index. Nat. Commun. 8, 14871 (2017).28317927 10.1038/ncomms14871
39. Xu C Three-dimensional acoustic double-zero-index medium with a fourfold degenerate Dirac-like point Phys. Rev. Lett. 2020 124 074501 10.1103/PhysRevLett.124.074501 32142328
Xu, C. et al. Three-dimensional acoustic double-zero-index medium with a fourfold degenerate Dirac-like point. Phys. Rev. Lett. 124, 074501 (2020).32142328 10.1103/PhysRevLett.124.074501
40. Sanchis L Three-dimensional axisymmetric cloak based on the cancellation of acoustic scattering from a sphere Phys. Rev. Lett. 2013 110 124301 10.1103/PhysRevLett.110.124301 25166808
Sanchis, L. et al. Three-dimensional axisymmetric cloak based on the cancellation of acoustic scattering from a sphere. Phys. Rev. Lett. 110, 124301 (2013).25166808 10.1103/PhysRevLett.110.124301
41. Zhang X Xiao M Cheng Y Lu MH Christensen J Topological sound Commun. Phys. 2018 1 97 10.1038/s42005-018-0094-4
Zhang, X., Xiao, M., Cheng, Y., Lu, M. H. & Christensen, J. Topological sound. Commun. Phys. 1, 97 (2018).10.1038/s42005-018-0094-4
42. Liu C Wide-angle broadband nonreflecting acoustic metamaterial fence Phys. Rev. Appl. 2020 13 054012 10.1103/PhysRevApplied.13.054012
Liu, C. et al. Wide-angle broadband nonreflecting acoustic metamaterial fence. Phys. Rev. Appl. 13, 054012 (2020).10.1103/PhysRevApplied.13.054012
43. Li J Fok L Yin X Bartal G Zhang X Experimental demonstration of an acoustic magnifying hyperlens Nat. Mater. 2009 8 931 934 10.1038/nmat2561 19855382
Li, J., Fok, L., Yin, X., Bartal, G. & Zhang, X. Experimental demonstration of an acoustic magnifying hyperlens. Nat. Mater. 8, 931–934 (2009).19855382 10.1038/nmat2561
44. Zhu J A holey-structured metamaterial for acoustic deep-subwavelength imaging Nat. Phys. 2011 7 52 55 10.1038/nphys1804
Zhu, J. et al. A holey-structured metamaterial for acoustic deep-subwavelength imaging. Nat. Phys. 7, 52–55 (2011).10.1038/nphys1804
45. Mironov MA Pislyakov VV One-dimensional acoustic waves in retarding structures with propagation velocity tending to zero Acoust. Phys. 2002 48 347 352 10.1134/1.1478121
Mironov, M. A. & Pislyakov, V. V. One-dimensional acoustic waves in retarding structures with propagation velocity tending to zero. Acoust. Phys. 48, 347–352 (2002).10.1134/1.1478121
46. Xia JP Sun HX Yuan SQ Modulating sound with acoustic metafiber bundles Sci. Rep. 2017 7 8151 10.1038/s41598-017-07232-6 28811586
Xia, J. P., Sun, H. X. & Yuan, S. Q. Modulating sound with acoustic metafiber bundles. Sci. Rep. 7, 8151 (2017).28811586 10.1038/s41598-017-07232-6
47. Zhang X Cheng L Broadband and low frequency sound absorption by Sonic black holes with Micro-perforated boundaries J. Sound Vib. 2021 512 116401 10.1016/j.jsv.2021.116401
Zhang, X. & Cheng, L. Broadband and low frequency sound absorption by Sonic black holes with Micro-perforated boundaries. J. Sound Vib. 512, 116401 (2021).10.1016/j.jsv.2021.116401
48. Chen H Ray-optics cloaking devices for large objects in incoherent natural light Nat. Commun. 2013 4 2652 10.1038/ncomms3652 24153410
Chen, H. et al. Ray-optics cloaking devices for large objects in incoherent natural light. Nat. Commun. 4, 2652 (2013).24153410 10.1038/ncomms3652
49. Chen X Macroscopic invisibility cloaking of visible light Nat. Commun. 2011 2 176 10.1038/ncomms1176 21285954
Chen, X. et al. Macroscopic invisibility cloaking of visible light. Nat. Commun. 2, 176 (2011).21285954 10.1038/ncomms1176
