Introduction

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 optics3,4,5,6,7,8,9, acoustics10,11,12, elastics13 and thermotics14. Invisibility is a special illusionary effect15,16,17,18,19 of empty space, while from the perspective of wave scattering20,21,22, 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 metamaterials23,24,25,26,27 and metasurfaces28, acoustic invisibility cloaks29,30,31,32,33,34,35,36 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 media37,38,39, 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.
figure 1

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.

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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 \(1\,{{{{\rm{kHz}}}}}\) to \(16\,{{{{\rm{kHz}}}}}\). 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 manner28,45,46,47. 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.
figure 2

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 \(\Delta \phi=0,\,{{{{\rm{\pi }}}}},\,2{{{{\rm{\pi }}}}}\). c Phase shift \(\Delta \phi\) and transmittance \({\left|t\right|}^{2}\) under normal incidence as a function of \(h/w\). Vertical gray dotted lines indicate the cases with \(\Delta \phi={{{{\rm{\pi }}}}},\,2{{{{\rm{\pi }}}}}\). d Equivalent relative refractive index \({n}_{r}\) as a function of \(h/w\) and the frequency \(f\).

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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 \(t=1{{{{\rm{mm}}}}}\), \(d=5{{{{\rm{mm}}}}}\) and \(a=20\,{{{{\rm{cm}}}}}\). The maximum and minimum width of the tunnel are set as \(8.95\,{{{{\rm{mm}}}}}\) and \(3.95\,{{{{\rm{mm}}}}}\), respectively. A frequency of \(f=6860{{{{\rm{Hz}}}}}\) 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 \(\Delta \phi\) and the transmittance \({\left|t\right|}^{2}\) as a function of \(h/w\) at \(f=6860{{{{\rm{Hz}}}}}\). We can observe that the transmitted phase shift can be conveniently modulated to cover the range of \(2{{{{\rm{\pi }}}}}\) 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 \(\phi={k}_{0}L\), where \({k}_{0}\) is the wave number in the background (air) and \(L={n}_{r}a\) is the acoustic path. Here, \({n}_{r}=\phi /({k}_{0}a)\) is the effective refractive index. In Fig. 2d, we plot the calculated \({n}_{r}\) as a function of \(h/w\) and the frequency \(f\) for the tunnel in Fig. 2a. It is clearly seen that \({n}_{r}\) increases with the increase of \(h/w\), and is almost unchanged from \(0.1\,{{{{\rm{kHz}}}}}\) to \(15\,{{{{\rm{kHz}}}}}\). When \(h/w\) is small, the dispersion of \({n}_{r}\) 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 (\(L=22.5\,{{{{\rm{cm}}}}}\)) 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 \(3\,{{{{\rm{cm}}}}}\). A cylindrical wave is emitted by a speaker located at a distance of \({d}_{1}=20\,{{{{\rm{cm}}}}}\) 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 \(\Delta x=10\,{{{{\rm{cm}}}}}\) and \(\Delta y=20\,{{{{\rm{cm}}}}}\) is of \({d}_{2}=5\,{{{{\rm{cm}}}}}\) 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.
figure 3

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 \(a=20\,{{{{\rm{cm}}}}}\). 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.

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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 \(f=4860\,{{{{\rm{Hz}}}}}\) (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 \((-30\,{{{{\rm{cm}}}}},\,0\,{{{{\rm{cm}}}}})\) to a virtual point of \((-10{{{{\rm{cm}}}}},0{{{{\rm{cm}}}}})\). Such a position change of \(20{{{{\rm{cm}}}}}\) 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.
figure 4

a, c, e Simulated acoustic pressure field distributions for the scenario of bare obstacles under a point source radiation at different operating frequencies of \(f=4860\,{{{{\rm{Hz}}}}},\,6860\,{{{{\rm{Hz}}}}},\,8860\,{{{{\rm{Hz}}}}}\), 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 (\({\gamma }_{t}\)) and reflection region (\({\gamma }_{r}\)).

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To demonstrate the ultra-broadband property, the numerical and experimental results obtained for two more frequencies, i.e. \(f=6860\,{{{{\rm{Hz}}}}}\) and \(f=8860\,{{{{\rm{Hz}}}}}\) 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 \({\gamma }_{t,r}={\left|{p}_{s}\right|}^{2}/{\left|{p}_{0}\right|}^{2}\), where \({p}_{s},\,{p}_{0}\) are the scattered field and the empty field, respectively. The subscript \(t\) and \(r\) represent the transmission region \((x\in \left(10\,{{{{\rm{cm}}}}},\,50\,{{{{\rm{cm}}}}}\right))\) and incidence region \((x\in \left(-50\,{{{{\rm{cm}}}}},-10\,{{{{\rm{cm}}}}}\right))\), respectively, and such intensity value is obtained from a spatial average. The scattered field \({p}_{s}\) is obtained by subtracting the empty field without the sample (\({p}_{0}\)) 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 \((-10\,{{{{\rm{cm}}}}},\,0\,{{{{\rm{cm}}}}})\). On the other hand, the empty field for the incidence region is calculated by using the real point source located at \((-30\,{{{{\rm{cm}}}}},\,0\,{{{{\rm{cm}}}}})\). The results are plotted in Fig. 4g and h. It is clearly seen that \({\gamma }_{t,r}\) maintain almost zero with the illusion metamaterial in an ultra-broad spectrum \(1\,{{{{\rm{kHz}}}}} \sim 16\,{{{{\rm{kHz}}}}}\). In contrast, \({\gamma }_{t,r}\) 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 space29,30,31,32,33,34,35,36 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 \((-10\,{{{{\rm{cm}}}}},\,0\,{{{{\rm{cm}}}}})\). 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 \({t}_{{{{{\rm{shift}}}}}}=L/{c}_{0}=0.66\,{{{{\rm{ms}}}}}\), 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.
figure 5

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 \(\Delta {{{{\rm{t}}}}}=0.1{{{{\rm{ms}}}}}\).

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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 \(2.25a\,(a=20\,{{{{\rm{cm}}}}})\), 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 \(L=47.5\,{{{{\rm{cm}}}}}\). 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 \(45{{{{\rm{cm}}}}}\) and a time shift of \(1.38{{{{\rm{ms}}}}}\) 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 \(1\,{{{{\rm{kHz}}}}} \sim 16\,{{{{\rm{kHz}}}}}\).

Fig. 6: Ultra-broadband illusion metamaterial for a random collection of obstacles.
figure 6

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 \(f=6860\,{{{{\rm{Hz}}}}}\). The point source is located at \((-20\,{{{{\rm{cm}}}}},\,0\,{{{{\rm{cm}}}}})\), and the center of three obstacles are located at \((10\,{{{{\rm{cm}}}}},\,10\,{{{{\rm{cm}}}}})\), \((35\,{{{{\rm{cm}}}}},\,0\,{{{{\rm{cm}}}}})\) and \((12.5\,{{{{\rm{cm}}}}},\,-12.5\,{{{{\rm{cm}}}}})\), respectively. c, d Frequency dependence of the normalized scattered pressure intensity for the transmission region (\({\gamma }_{t}\)) and reflection region (\({\gamma }_{r}\)).

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Discussion

It is recognized that when the tunnel width is small, e.g. \( < \,1\,{{{{\rm{mm}}}}}\), 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 \(16{{{{\rm{kHz}}}}}\) 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 \(1.21\,{{{{\rm{kg}}}}}/{{{{{\rm{m}}}}}}^{3}\) and \(343\,{{{{\rm{m}}}}}/{{{{\rm{s}}}}}\), 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, \(0.1\,{{{{\rm{mm}}}}}\) 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 \(1{{{{\rm{cm}}}}}\) and in total 231 points are scanned. The measurement platform is a 2D waveguide with a height of \(30\,{{{{\rm{mm}}}}}\). The sound absorbing foams are set around the system to reduce the reflections.