Observation of multiple rotons and multidirectional roton-like dispersion relations in acoustic metamaterials

All metamaterial full-wave simulations are performed in the pressure acoustic module of COMSOL Multiphysics. In all simulations, the background medium air is modelled with a density of 1.29 kg m⁻³ and a speed of sound of 340 m s⁻¹. The plastic stereolithography can be considered as a hard boundary during the simulations due to the substantial acoustic impedance contrast compared with air. For calculating the dispersion relations, periodic boundary conditions are set in the x-direction (1D) or both x- and y-directions (2D), and other boundaries are considered as hard acoustic boundaries. The dispersion slope (i.e. group velocity) is calculated by v_g = dω/dk, where ω is the angular frequency and k is wave vector in the momentum space.

The samples are fabricated with an additive manufacturing technique (that is, stereolithography) with a rein of a thickness of 5 mm and 1 $\times$ 50-unit cells stacked along the x-direction (1D) or 30 $\times$ 30-unit cells stacked along both x- and y-direction (2D). The materials are photosensitive resin with a modulus of 2880 MPa and a density of 1.10 g cm⁻¹. The structures can be viewed as rigid walls for the acoustic waves that propagate in the acoustic metamaterials due to the large impedance mismatch between air and the material. A small hole (r = 2 mm) is located on the centre of each cube resonator for excitation and detection. When not in use, they are blocked by plugs. In the experiment, a broadband sound signal is launched by a balanced armature speaker guided into the sample through a small hole on the centre of cube resonator to excite the acoustic modes. Another microphone (Brüel & Kjær Type 4961, of about 3.2 mm radius) in a sealed sleeve with a tube of 1 mm radius and 250 mm length is inserted into each cube resonator through the small hole to collect the acoustic signals one by one. The measured data is processed by a Brüel & Kjær 3160-A-022 module to extract the frequency spectrum with 2 Hz resolution. Spatial Fourier transforms are applied to the complex acoustic field distributions to obtain the measured dispersion relations.

The three-dimensional (3D) experimental sample of the acoustic metamaterial with the roton-like dispersion relation is fabricated with stereolithography 3D printing technology, as shown in figure 1(a). It consists of 50 unit cells along x-direction. Each unit cell has a cube resonator with side length d₁ (see the red region of figure 1(b)). For the convenience of experimental measurement, each cube resonator has a small hole for detecting the acoustic waves in the resonator. These holes are sealed when not in use and have a negligible impact on the dispersion relation of the acoustic metamaterial. The vertical cylindrical tubes with a radius of r₁ connecting two nearest-neighbour resonators serve as the nearest-neighbour interactions. The oblique cylindrical tubes with a radius of r₂ connecting two third-nearest-neighbour resonators serve as the beyond-nearest-neighbour interactions with an order of N= 3. The hollow frame with height d₂ serves as an auxiliary structure to mediate the third-nearest-neighbour interactions. The oblique cylindrical tubes connect to the auxiliary hollow frame, from which another set of oblique cylindrical tubes is connected to the third-nearest-neighbour of the starting cube resonator to trigger the long-range interaction. The cube resonators, cylindrical oblique tubes, and the auxiliary frames are filled with air and made of hard plastics to ensure that the inner surfaces meet the hard boundary condition. Besides, the acoustic metamaterial structure is achiral, which exhibits mirror symmetry and inversion symmetry.

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In the experiments, the sound wave generated by a broadband speaker is guided into the waveguide through a small hole at the leftmost cube resonator. A microphone probe is inserted into the other holes at each cube resonator to measure the acoustic pressure signal. This measurement is repeated for all cube resonators of the sample one by one (see section 2). Note that all cylindrical holes are plugged except for the one that opens to measure the acoustic signal. After performing a 1D Fourier transform to the measured acoustic field distributions, we obtain the dispersion relations of the nonlocal acoustic metamaterials (see the colour plotted in figure 1(d)). One can clearly observe the pronounced roton-like dispersion relation—the dispersion band starts from zero frequency and zero momentum with linear growth at low wavevectors, then evolves continuously to reach a maximum, and the dispersion slope (i.e. group velocity v_g= dω/dk) reverses from positive to negative (see figure 1(e)), displaying a pronounced 'roton' minimum (white dot in figure 1(d)) before increasing again. The dispersion relation closely resembles the dispersion curve of the elementary excitations of a Bose superfluid predicted by Landau [1]. The experimental results agree with the simulated counterpart (see the colour curve in figure 1(d)) very well.

The roton-like dispersion relation has several intriguing features, which may find potential applications to manipulate acoustic waves in uncommon ways. First, the 'maxon' maximum and 'roton' minimum with ZGV produce singularities with an extremely high density of states that can be utilized to significantly enhance the wave-matter interactions. Second, the negative slope of roton-like dispersion relation gives rise to broadband backward waves that could lead to acoustic negative refraction and superlensing [14–18] based on a new mechanism of nonlocal effect. Third, at a given frequency, the dispersion curve can support multiple coexisting Bloch modes with different wavevectors, phase velocities, and group velocities. Note that Lamb waves in natural materials also show ZGV modes [30, 31] and multiple coexisting modes at a given frequency. However, the dispersion curves of Lamb waves are entirely different from the roton-like dispersion relation. The unique dispersion curve of superfluid comes from the interplay between phonons and rotons. Therefore, for the roton-like dispersion, the dispersion relation starts from a linear growth (corresponding to low-energy phonons); at a certain large momentum, the dispersion relation reaches a maximum, then decreases to a minimum (corresponding to rotons), before increasing again [27, 28]. Consequently, the roton-like dispersion curve usually reverts twice with one local minimum (roton) and one local maximum (maxon), allowing for three modes (two forward modes and one backward mode) with different wavenumbers coexisting at the same frequency [27, 28]. However, the dispersion curve of the ZGV modes does not show a linear growth from zero frequency, and it reverts only once, allowing for only one backward and one forward mode with different wave numbers coexisting at the same frequency.

To understand the origin of the roton-like behaviour in acoustic metamaterials, we plot the simulated mean energy flux along the x-direction in a unit cell for three different Bloch modes at the same frequency of 1.35 kHz in figure 1(f), including two forward modes (A and C) and one backward mode (B) with different wavevectors and group velocities. For two eigenmodes with positive group velocity (A and C), the mean energy flux in both the vertical (nearest-neighbour interaction) and oblique connecting tubes (third-nearest-neighbour interaction) is positive. In sharp contrast, for eigenmode with negative group velocity (B), the mean energy flux is positive in the vertical connecting tube but is negative in the oblique connecting tubes, which supports a backward propagating wave. It is clear that the sum of the two opposite energy fluxes is negative, consistent with the negative group velocity of the dispersion relations shown in figures 1(d) and (e). The two energy fluxes moving forward and backward can give rise to a vortex-like behaviour of the energy flux, very similar to Feynman's intuitive appealing picture of rotons which correspond to some sort of stirring motions due to a 'return flow' [2].

To further elaborate the experimental results, the practical acoustic metamaterial structure can be translated to a one-dimensional (1D) toy spring-mass model with the acoustic resonator and tubes playing roles of mass and spring (see figure 1(c)), respectively. Here, the black straight lines (green arc lines) represent the nearest-neighbour interactions K₁ (the third-nearest-neighbour interactions K₃). Newton's equation of motion in this model can be written as

$$\begin{equation}m\ddot u = {K_1}({u_{n + 1}} - 2{u_n} + {u_{n - 1}}) + {K_N}({u_{n + N}} - 2{u_n} + {u_{n - N}}).\end{equation} \tag{ 1 }$$

where m is the mass, $\ddot u$ is the acceleration of the mass displacement u_n at lattice site n, and K_N is the spring constant representing Nth-nearest-neighbour interactions, with N = 3 in our case. The corresponding dispersion relation is

$$\begin{equation}\omega (k) = 2\sqrt {\frac{{{K_1}}}{m}{{\sin }^2}\left( {\frac{{ka}}{2}} \right) + \frac{{{K_N}}}{m}{{\sin }^2}\left( {\frac{{Nka}}{2}} \right)} .\end{equation} \tag{ 2 }$$

By fitting our measured dispersion relation to equation (2), we obtain k₁= 298 N m⁻¹, k₃ = 454 N m⁻¹, and m= 2.59*10⁻⁵ kg for our experimental sample. The fitted curve is represented by the blue dashed line in figure 1(d), which is consistent with our measured result.

The displacement of the mass at lattice site n can be given by ${u_n}(t) = \tilde u\exp (i(kna - \omega t))$, where $\tilde u$, k and ω represent amplitude, wavenumber and angular frequency. The mean energy flow in the 1D toy model is contributed by two parts [27], is given by

$$\begin{equation}\left\langle {\,p(k)} \right\rangle = \frac{1}{2}{\tilde u^2}\omega ({K_1}\sin (ka) + N{K_N}\sin (Nka)).\end{equation} \tag{ 3 }$$

The first term is always positive for 0 < k < π/a, which stems from the nearest-neighbour interactions. The second term originating from the beyond-nearest-neighbour interactions is negative in the interval $k \in [\pi /(3a),2\pi /(3a)]$, leading to a net negative energy flow if K_N /K₁ > 1/N.

Having established the exotic features of the roton-like dispersion in nonlocal acoustic metamaterials with long-range interactions, we discuss the dispersion engineering of the nonlocal acoustic metamaterials by tuning the long-range interaction orders. It has been theoretically predicted [27] that a roton-like minimum occurs if and only if the interactions order is sufficiently high (i.e. N⩾ 3) and the strength of the beyond-nearest-neighbour interactions is sufficiently large (i.e. K_N /K₁ > 1/N). More remarkably, the number of roton-minimum and the negative slope region of the roton-like dispersion can be controlled by tailoring the long-range interaction order. To verify the influence of the order of beyond-nearest-neighbour interactions on the dispersion band, we design and fabricate three acoustic metamaterials with second- (N = 2), fourth- (N = 4), and fifth-nearest-neighbour (N = 5) interactions, respectively. Their unit cells are schematically shown in figures 2(a), (e) and (i) respectively, where the vertical cylindrical tubes (blue region) provide the nearest-neighbour interaction (N = 1) and the oblique cylindrical connecting tubes with the auxiliary hollow frames (beige regions) provide the second- (N= 2), fourth- (N= 4) or fifth-nearest-neighbour (N = 5) interactions independently.

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For the case of N = 2, the 1D toy model with second-nearest-neighbour interactions K₂ (red arc line) is shown in figure 2(b). The measured (red colour) and calculated (colour line) band dispersions only exhibit a maxon-maximum, and its group velocity switches sign for only once, as shown in figures 2(c) and (d), respectively. In this situation, there are only two coexisting eigenmodes with different wavevectors and group velocities at the same frequency. This dispersion relation has no roton minimum and is similar to traditional metamaterial dispersion with a single negative slope region. We then study the case of N = 4, whose 1D toy model with the fourth-nearest-neighbour interactions K₄ (blue arc line) is shown in figure 2(f). For the case of N = 4, the measured (red colour) and simulated (colour line) dispersion relations are more complex with the sign of group velocity switches three times and support four different coexisting acoustic eigenmodes at one single frequency, displaying two maxon-maximums and one roton-minimum (white dot), as shown in figures 2(g) and (h), respectively. Now we continue to increase the long-range interaction order and explore another case with N = 5, whose 1D toy model with the fifth-nearest-neighbour interactions K₅ (purple arc line) is shown in figure 2(j). The measured (red colour) and simulated (colour line) dispersion relations reverse four times and support five different coexisting acoustic eigenmodes at one single frequency, displaying two maxon-maximums and two roton-minimums (white dots), as shown in figures 2(k) and (l), respectively. Note that the bright regions above the roton-like dispersion in figures 2(g) and (k) come from a higher band. The mean energy fluxes of different coexisting eigenmodes at the same frequency of 1.35 kHz for N = 2, 4 and 5 are shown in figure 3. Both the measured and simulated results reveal, as expected, roton behaviour occurs only for beyond-nearest-neighbour interaction order N ⩾ 3 and more slope inversions, and roton minimums can be brought into the first Brillouin zone by increasing the long-range interaction order. Note that multiple classical rotons on a single dispersion curve achieved when N ⩾ 5 have never been theoretically proposed and experimentally demonstrated in both the quantum and classical-wave systems before. Theoretically derived roton-like dispersion relations for four 1D toy models with different orders of long-range interactions are shown in figure 4.

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Recently, it has been theoretically reported that the roton-like dispersion relations for transverse acoustical elastic waves can be realized in noncentrosymmetric micropolar crystal based on chiral micropolar elasticity theory, where chirality is a necessary condition for roton-like behaviours [32]. Hence, it is natural to ask whether chirality plays an important role in the roton-like behaviours in our airborne acoustic system. Here, we introduce chirality into our acoustic metamaterials with N =3 by twisting and doubling the oblique connecting tubes to construct an acoustic metamaterial with four-fold rotational symmetry around the x-axis, whose unit cell is schematically shown in figure 5(a). As both the inversion and mirror symmetries are absent, the newly designed acoustic metamaterial is structurally chiral. Interestingly, both the measured (red colour) and simulated (colour line) dispersion relations of the chiral acoustic metamaterial (figure 5(b)) display a roton-like behaviour which is very similar to that of the achiral acoustic metamaterials in figure 1(d). The energy flux of the three coexisting modes shown in figure 5(c) further confirmed the similarity between the chiral and achiral acoustic metamaterials with beyond-nearest-neighbour interactions. Therefore, we have experimentally verified that both the chiral and achiral acoustic metamaterials with beyond-nearest-neighbour interactions can support roton-like dispersion relations with similar properties.

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We have also theoretically proposed and experimentally realized multidirectional roton-like dispersion relations in a 2D nonlocal acoustic metamaterial with third-nearest-neighbour interactions in both horizontal (x) and vertical (y) directions. Since the long-range interactions in the 2D system are far more complex than the 1D case, we propose a completely different architecture to construct the 2D nonlocal acoustic metamaterial with long-range interactions. The unit cell of the 2D acoustic metamaterial is schematically shown in figure 6(a); the original ancillary hollow frame mediating the third-nearest-neighbour interactions in 1D acoustic metamaterials is reduced to four hollow rectangular tubes to construct the third-nearest-neighbour interactions in two orthogonal directions. A perspective view of 4 × 4 unit cells out of a corresponding 2D nonlocal acoustic metamaterial is shown in figure 6(b) for clarity. The part highlighted in beige illustrates the third-nearest-neighbour interactions in two orthogonal directions. In the x (y) direction, one circular coupling tube connects the first cube (n,l) to the upper (lower) ancillary rectangular box and the other circular coupling tube connects this ancillary rectangular box to the fourth cube (n+ 3,l) (n,l+ 3), which has a distance of 3a with respect to the first cube in both directions. The 2D toy model with equal third-nearest-neighbour interactions along x- and y-directions (green and red lines) is depicted in figure 6(c). Equation of motion in this model can be written as

$$\begin{align} m{{\ddot u}_{xy}} & = {K_1}\left[\Big(u_{(n + 1,l)}^x - 2u_{(n,l)}^x + u_{(n - 1,l)}^x\Big) + \Big(u_{(n,l + 1)}^y - 2u_{(n,l)}^y + u_{(n,l - 1)}^y\Big)\right]\nonumber \\ &\quad +{\text{ }}{K_N}\left[\Big(u_{(n + N,l)}^x - 2u_{(n,l)}^x + u_{(n - N,l)}^x\Big) + \Big(u_{(n,l + N)}^y - 2u_{(n,l)}^y + u_{(n,l - N)}^y\Big)\right]. \end{align} \tag{ 4 }$$

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where m is the mass, $\ddot{u}_{xy}$ is the acceleration of the mass displacement $u_{(n,l)}^x$ and $u_{(n,l)}^y$ at lattice site (n,l), and K_N is the spring constant representing Nth-nearest-neighbour interactions, with N= 3 in our case. The solution of this equation of motion are Bloch waves in both x- and y-directions, following the theoretical dispersion relation of this 2D toy model:

$$\begin{align} &\omega ({k_x},{k_y})\nonumber \\ &\quad {\text{ }} = 2\sqrt {\frac{{{K_1}}}{m}\Bigg[{{\sin }^2}\Big(\frac{{{k_x}a}}{2}\Big) + {{\sin }^2}\Big(\frac{{{k_y}a}}{2}\Big)\Bigg] + \frac{{{K_N}}}{m}\Bigg[{{\sin }^2}\Big(\frac{{N{k_x}a}}{2}\Big) + {{\sin }^2}\Big(\frac{{N{k_y}a}}{2}\Big)\Bigg]} .\end{align} \tag{ 5 }$$

The theoretically derived dispersion relations of the 2D toy model with third-nearest-neighbour interactions are depicted in figure 7. We then fabricate the 2D acoustic metamaterial (see the photograph of the experimental sample in figure 6(d)) and measure its dispersion relations. The measured (red colour) and simulated (grey line) dispersion relations are shown in figure 6(e), exhibiting multidirectional roton-like dispersion relation at high-symmetry lines. In figure 6(f), we plot the sum of mean energy flux along x and y directions corresponding to the three eigenmodes marked as A, B, and C of the dispersion bands with negative group velocity in figure 6(e). For modes A and B, the sum of mean energy flux in two directions is positive in the nearest-neighbour connecting tube along x- or y-direction, but is negative in the upper or lower third-nearest-neighbour oblique connecting tubes. For mode C, the sum of mean energy flux in two directions is positive both in the nearest-neighbour connecting tube along x- and y-directions, but is negative both in the upper and lower third-nearest-neighbour oblique connecting tubes. This unconventional feature of energy fluxes flow gives rise to a vortex-like behaviour of acoustic waves in multiple directions.

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