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1 Ultra-sparse metasurface for high reflection of low-frequency sound based on artificial Mie resonances Y. Cheng, 1,2 C. Zhou, 1 B.G. Yuan, 1 D.J. Wu, 3 Q. Wei, 1 X.J. Liu 1,2* 1 Key Laboratory of Modern Acoustics, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing , China 2 State Key Laboratory of Acoustics, Chinese Academy of Sciences, Beijing , China 3 School of Physics and Technology, Nanjing Normal University, Nanjing , China * liuxiaojun@nju.edu.cn Note 1. Describing the system with the rigorous scattering theory...2 Note 2. Validity of rigorous scattering derivation...5 Note 3. Effect of losses...6 Note 4. Achieving high transmission loss and broad band width...8 Note 5. System robustness...10 Note 6. Comparison with Helmholtz resonators...13 Note 7. Multipolar Mie-resonance modes for sound-blocking...15 Note 8. Sound tunneling by using the array of Mie resonators...16 References...17 NATURE MATERIALS 1

2 Note 1. Describing the system with the rigorous scattering theory Assume that a harmonic incident plane wave radiates vertically on a single unit cell in an infinite space. It has been demonstrated that the solid plate with a periodic array of subwavelength slits can serve as a homogeneous material (Ref. S1). Hence the physical model of the unit cell in the subwavelength scale can be characterized as a layered structure shown in Fig. S1. The coefficients in each order can be solved by matching the continuous scalar pressure p and radial velocity v at interfaces, as shown by Eq. 1. Here we emphasize that the approximation of long wavelength is not applicable to kk rr and kk rr 2. The unit cell s scattering coefficient rewritten in a physically revealing form (Ref. S2) as Dm can be D = U ( U + iv ), (S1) m m m m where the functions of U m and V m are given by the determinants U m V m J ( k r) J ( kr ) H ( kr ) 0 m 0 2 m 2 m 2 k0 k k J m( k0r2) J m( kr 2 ) H m( kr 2 ) 0 r0 r r = 0 J ( kr ) H ( kr ) J ( k R) m m m k k k 0 0 Jm( kr) Hm( kr) Jm( k0r) r r r0 J ( k r) J ( kr ) H ( kr ) 0 m 0 2 m 2 m 2 k0 k k J m( k0r2) J m( kr 2 ) H m( kr 2 ) 0 r0 r r = 0 J ( kr ) H ( kr ) H ( k R) m m m k k k J kr H kr H k R 0 0 m( ) m( ) m( 0 ) r r r0 0 0, (S2). (S3) From Eq. S1, it can be found that Dm may approach 1 when V m 0, leading to the excitation of resonance scattering in specific modes. In order to observe the resonances, the acoustic total scattering cross-section (SCS) spectrum is investigated. The SCS is defined as the ratio of the total scattered power to the incident power, and the SCS spectrum is obtained by integrating the 2 NATURE MATERIALS

3 far-field scattering pattern over the entire azimuthal angle, as shown in Fig. S2a. The solid and dashed-dotted curves stand for the unit cell of metasurface and the rigid cylinder, respectively. The intensive resonance scattering (denoted by the arrows in Fig. S2a) could drastically increase the SCS in the resonant regions compared with that of the rigid cylinder with the same size. Figure S1 Schematic set-up of the derivation. Left: incident plane wave radiates vertically on a single unit cell in subwavelength scale. Middle: zoom-in view of the physical model. Right: In order to perform rigorous scattering analysis, the physical model is characterized as an interconnection core (rr < rr 2 ), an equivalent uniform coating shell (rr 2 < rr < rr ), and a virtual layer (rr < rr < RR). In order to provide a physical insight into the origin of the resonances, the unit cell s scattering coefficients Dm are presented in Figs. S2b-S2c. Due to the identical incident wave coefficients i n for each order, D m should identify each order wave s contribution to the scattered field. Notice that the peak of D0 appears at the monopole frequency, while the peaks of D1 and D2 appear at the dipole and quadrupole modes. Comparing Figs. S2a-S2c, it can be clearly observed that the monopole, dipole and quadrupole resonances are excited by the 0 th -order, 1 st -order, and 2 nd -order cylindrical waves, respectively. NATURE MATERIALS 3

4 Figure S2 Scattering character of a single unit cell. Frequency dependence of (a) acoustic total scattering cross-section, (b) amplitudes and (c) phases of 0 th -order, 1 st -order, and 2 nd order scattering coefficients. 4 NATURE MATERIALS

5 Note 2. Validity of rigorous scattering derivation In order to check the validity of the above derivations, we apply the FEM full-wave simulations to both the realistic structured model (Fig. 1b) and the physical model (Fig. 1c). Then the monopole frequency of a single unit cell for varied curling number N is illustrated in Fig. S3. All three results are in good agreements. The validity and accuracy of the analytical derivations are verified. Figure S3 Monopole frequency of a single unit cell. Theoretical derivation is convincingly confirmed by numerical full-wave simulations of both the structured and physical models. NATURE MATERIALS 5

6 Note 3. Effect of losses Due to the thermo-viscous losses, the air channel size of the cells is an important factor that affects the practical magnitude of Mie resonance. For illustration, we perform both numerical simulations and experimental measurements to characterize the practical response of a structure with N=22, as shown in Figs. S4a and S4b. It is observed that the Mie resonance can be excited with the peak magnification of 4.9. In order to further illustrate the effect of losses, the pressure-magnification spectra at the central interconnection core with respect to various N are shown in Fig. S4c. Note a trade-off effect on the artificial Mie resonance. First, narrowing the channel size (increasing N) leads to more significant dissipation due to the enhanced viscous and thermal losses in the smaller channels. In air at 20 and 1 atm, the thickness of viscous boundary layer is 0.16 mm at 200 Hz, and the corresponding thickness of thermal boundary layer is 0.19 mm. Thus the llllllll value (0.032) for the structure with N=22 is much higher than that (0.0093) with N=8. However, decreasing the channel size also results in an increase in the effective refractive index of the Mie resonator, which can significantly increase the Mie resonance in the lossless case (assuming llllllll=0, peak magnification = 48.9 for N=22). Therefore, efficient Mie resonance can still be achieved with a high N-value, provided that a good compromise is reached between the magnification and dissipation loss by choosing a moderate channel size. 6 NATURE MATERIALS

7 Figure S4 Effect of losses. a, Photograph of the sample with N=22. b, Pressure magnification in the region of central interconnection core. The red open circles (black solid line) represent the experimental (simulation with llllllll=0.032) results. The measured basic monopole mode at the peak is shown by the inset. c, Maximum pressure magnification and llllllll factor with respect to various N. NATURE MATERIALS 7

8 Note 4. Achieving high transmission loss and broad band width The standard metasurface sample (Fig. S5a) only has single layer cells with identical dip frequency, and the Mie resonators are arranged in ultra-sparse pattern with large intervals of dd = 10RR. We would like to emphasize that the high transmission loss and broad band width can be easily achieved by simply rearranging the Mie-resonant based metamaterial, owing to its unique coupling characters. As illustration, sample S1 (Fig. S5b) is identical in composition as single-layer standard metasurface (N=8), but the interval between cells is dd = 2.4RR in this case. The power transmission spectrum of sample S1 is shown with the red dashed-dotted curve in Fig. S5d. Following the methodology of membrane-type metamaterials (MTM) shield in Ref. 50, we also construct sample S2 (Fig. S5c) by stacking four single-layer compact cells of N=8, 12, 16 and 20 together, as shown by the blue solid curve in Fig. S5d. The result for standard sample is shown as green dotted curve for comparison. Below, we show how the sound blocking power is improved in the rearranged samples. Band width: Interestingly, sample S1 covers the range of Hz using single layer cells with identical dip frequency. This significant enlargement of bandwidth originates from the new hybrid monopolar and dipolar modes in identical cells, which is distinct from MTM shield without interference. Furthermore, sample S2 demonstrates an even broader range from 50 Hz to 1250 Hz. Insertion loss: Sample S2 demonstrates an average STL of > 40 db over a broad range from 50 Hz to 1250 Hz. Here the STL is determined by STL (db) = 40 20log (tt) and tt is the percentage of transmission amplitude, as that in Ref NATURE MATERIALS

9 Thickness: Due to the different resonant mechanisms, a MTM cell is indeed thinner than the current version of Mie cells. It will be able to achieve thinner Mie cells by dramatically downsizing the wall of each cell and optimizing the geometric design. Here we would like to emphasize that an important advantage of a Mie metasurface is its lower weight (< 3.5 kg/m 2 for the standard sample, < 14.4 kg/m 2 for sample S1) and wide interval. Therefore, the material consumption and construction cost should be significantly reduced, and the intervals unblock scenic vistas which reduce the visual impact for users. Therefore, owing to the performance presented above, the Mie-resonant metasurfaces can be very promising to perspective in applications. Figure S5 High transmission loss and broad band width. a, standard metasurface (as reference) with single-layer ultra-sparse cells of N=8 and d=10r; b, sample S1 with single-layer compact cells of N=8 and d=2.4r; c, sample S2 with four single-layer compact cells of N=8, 12, 16 and 20; d, power transmission spectra. NATURE MATERIALS 9

10 Note 5. System robustness System robustness is highly desirable for the practical realization of relevant functional devices. We have tested the sound blocking power taking sample fabrication irregularities into account. The results show good robustness against various variations and objects. First, we demonstrate the robustness dependence of energy transmission under various variations. In Fig. S6a, we show the sensitivity of transmission on the horizontal translations of the cells. Here the cells are shifted randomly with horizontal displacement δδ 3 cm. We test 20 groups of δδ-value and find that the transmission remains considerably high. As illustration, three randomly chosen cases are shown. We also examine the robustness against vertical translations of ξξ 3 cm (shown in Fig. S6b), rotational translations of θθ 22.5 (shown in Fig. S6c), and combined multi-factors of δδ 3 cm, ξξ 3 cm and θθ 22.5 (shown in Fig. S6d). No substantial difference is observed. Note that δδ 3 cm and ξξ 3 cm are 120% of the cell s radius, while θθ 22.5 is the maximum angular irregularity of the cell. In other words, excellent sound blocking power can still be achieved in the presence of high fabrication irregularities. Furthermore, we confirm system robustness against sizable objects in the gap, as shown in Figure S7. Very low transmission (<0.05) is still observed around Mie resonance frequency, while the blocking power can be further improved in the frequency range above the Mie resonance. Further investigations also confirm good robustness against objects with different sizes, shapes, positions, numbers, and material properties (not shown). 10 NATURE MATERIALS

11 Figure S6 Robustness against variations in the equal-spacing of the cells. a, horizontal translations of δδ 3 cm; b, vertical translations of ξξ 3 cm; c, rotational translations of θθ 22.5 ; d, multi-factors of δδ 3 cm, ξξ 3 cm and θθ δδ, ξξ and θθ are pseudorandom values drawn from the standard uniform distribution on given intervals by using MATLAB code. In general, owing to the robust radial inflow features of the field arrangements, the Mie-resonance based system has a large tolerance in sample fabrication irregularities. For example, when a defect object is present between the Mie resonators, then the velocity field will find another route to pass around this defect object and maintain the radial inflow pattern. As a result, the system can promise unique, robust new device functionalities by providing immunity to performance degradation induced by fabrication imperfections or environmental changes. NATURE MATERIALS 11

12 Figure S7 Robustness against sizable objects present in the gap. Here two solid cylindrical objects are inserted between the cells. Their positions are determined randomly by using MATLAB code. 12 NATURE MATERIALS

13 Note 6. Comparison with Helmholtz resonators The ultra-slow fluid-like unit cells and Helmholtz resonators are two distinct methods of creating local resonance. (1) One of the key differences between Helmholtz resonators and ultra-slow fluid-like unit cells is the distinct inertial and intrinsic nature (see Ref. 1). As classic acoustic elements, Helmholtz resonators are basic constitutive units of inertial acoustic metamaterials. They create local resonance through inclusions of two components that function as mass-spring oscillator in each unit cell (see Fig. S8a). The short neck and cavity respectively acts as acoustic mass (counterpart of inductance L in electric circuits) and capacitor (counterparts of capacitance C). The moving fluid in the neck excited by external sound produces vertical oscillating up-and-down motion, which radiates sound into the surrounding medium in the same manner as an open-ended pipe. Thus, Helmholtz resonators radiate in a hemispherical pattern and can simulate monopoles in effect in narrow one- and two-dimensional waveguides at low-frequency limit, leading to negative bulk modulus to block sound. From the dynamic point of view, Helmholtz resonators can be the acoustic counterpart of the well-known metal split-ring resonator (SRR) in electromagnetic metamaterials, since both of them act as LC resonance circuits. Instead of requiring two components that function as mass-spring oscillator, the ultra-slow fluid-like unit cells provide an alternative approach to achieve strong monopolar response, which are promising candidates for constitutive units of intrinsic acoustic metamaterials. As it follows from the Mie solution of wave scattering by high-refractive index particles, one can understand the resonance phenomenon by considering the relative wavelengths in particle and background air at resonance. The first order resonance takes place when the effective wavelength inside the particles NATURE MATERIALS 13

14 reaches the diameter. Under this condition, collective in-phase propagation inside the eight channels can produce the monopolar Mie resonance, which exhibits symmetric radial oscillating pattern (see Fig. S8b). In general, the artificial ultra-slow fluid-like unit cells can be the acoustic counterpart of the new, rapidly developing dielectric nanoparticles in electromagnetic metamaterials, since both of them act as Mie resonators. Figure S8 Schematic representation of fluid motion at monopolar resonance. a, Helmholtz resonators; b, ultra-slow fluid-like unit cell. (2) Another important difference is the supportable eigenmode. The size of the Helmholtz resonator is completely decoupled from the wavelength, and only one simulated monopolar resonance exists. In contrast, it is worth noted that the resonances of ultra-slow fluid-like unit cells are not fixed. They can support rich monopolar and multipolar resonance modes (see Fig. 2). Moreover, the resonant frequency can be tuned by changing the size and high refractive index of the unit cells as well as the ambient conditions. The monopolar and dipolar Mie resonances can be overlapped in spectral range, bringing negative density and modulus. Consequently, the ultra-slow fluid-like unit cells open a practical route to implement versatile important concepts of metamaterials and metadevices with high functionalities. 14 NATURE MATERIALS

15 Note 7. Multipolar Mie-resonance modes for sound-blocking In order to discuss the usage of multipolar Mie-resonance modes for sound-blocking application, we further studied the transmission around dipole and quadrupole Mie-resonance modes for illustration. Figure S9 shows the transmission around dipole and quadrupole Mie-resonance modes at d=5r. Transmission dips are clearly seen in the frequency region of dipole and quadrupole Mie resonance modes, which can be confirmed by the inset pressure fields (see Supplementary Movie 2 and Movie 3 for 3D dynamic view). Figure S9 Sound-blocking around dipole and quadrupole Mie-resonance modes. The insets show pressure fields. NATURE MATERIALS 15

16 Note 8. Sound tunneling by using the array of Mie resonators It should be noted that the proposed Mie resonators may enable flexible control of acoustic waves for realizing versatile functional devices with superior performance. As an example, Fig. S10 shows preliminary illustrations of acoustic cloaking/tunnelling effects by using just a line of these Mie resonators. Figure S10 Acoustic cloaking/tunnelling by using artificial Mie resonators. a, Upper-left: pressure map showing the sound scattering by a rigid object (shaded region). The white arrows indicate the directions of wave propagation. Lower-left: resulting map when the object is embedded in a ρ eff -near-zero metamaterial slab made of artificial Mie resonators. Note that the cloak is thick, like conventional designs. Right: efficient cloaking by ultrathin cloak with only two layers of Mie resonators. b, Acoustic tunnelling through a narrow channel with 120 bent angle. Note that the energy transmittance is 87.5% (right) by using the line of the Mie resonators while it becomes to be 3% (left) for the case without the Mie resonators. 16 NATURE MATERIALS

17 References: S1. Cai, F., Liu, F., He, Z. & Liu, Z. High refractive-index sonic material based on periodic subwavelength structure. Appl. Phys. Lett. 91, (2007). S2. Cheng, Y. & Liu, X. J. Extraordinary resonant scattering in imperfect acoustic cloak. Chin. Phys. Lett. 26, (2009). NATURE MATERIALS 17