Ultra-Thin Metasurface-Based Absorber of Low-Frequency Sound With Bandwidth Optimization

We report, both theoretically and experimentally, a type of ultra-thin metasurface-based low-frequency sound absorber with bandwidth optimization. Such a metasurface unit consists of an ultrathin resonator (thickness∼1/90 wavelength) with a circular hole on the upper panel and four narrow slits inside a multiple-cavity structure. Eigenmode simulations of the unit show rich artificial Mie resonances, in which a type of monopolar Mie resonance mode can be obtained at 238.4 Hz. Based on the excitation of the monopolar mode, we can realize the near-perfect low-frequency sound absorption with the maximum absorption coefficient and fractional bandwidth of 0.97 and 12.9%, respectively, which mainly arises from the high thermal-viscous loss around the circular hole and four narrow slits of the unit. More interestingly, by combining 4 units with different diameters of the circular hole, we further enhance the fractional bandwidth of the compound unit to 18.7%. Our work provides a route to design ultra-thin broadband sound absorbers by artificial Mie resonances, showing great potential in practical applications of low-frequency noise control and architectural acoustics.


INTRODUCTION
Studies on low-frequency sound absorption have attracted great scientific and engineering fascination due to its extensive practical applications in noise control, architectural acoustics, and environmental protection. Traditionally, the realization of sound absorption is mainly based on porous and fibrous materials (Biot, 1956;Zarek, 1978) and micro-perforated plate structures with cavities at the back (Maa, 1998;Arenas and Crocker, 2010). However, these absorbing structures usually have imperfect impedance matching with free space and relatively large sizes comparable to working wavelengths.
Recently, a type of maze-like unit consisting of eight zigzag channels has become a hot topic due to its rich artificial Mie resonances and subwavelength size (Cheng et al., 2015;Landi et al., 2018). Based on different types of Mie resonance modes created by the maze-like units, a variety of application designs of lowfrequency sound have been realized, including rainbow trapping (Zhou et al., 2016), extraordinary transmission (Xia et al., 2015;Zhang et al., 2017), sound filtering , energy harvesting (Gao et al., 2019) and directional propagation . Additionally, a multi-band near-perfect sound absorber based on the multi-orders monopolar and dipolar Mie resonances has been designed (Long et al., 2018). However, this system is composed of a Mie resonator array backed by a rigid wall, and broadband sound absorbers designed by a single layer of Mie resonator array with deep subwavelength thickness still pose a challenge.
In this work, we propose a metasurface unit which consists of an upper surface panel with a central circular hole and a multiple-cavity structure. By applying eigenmode simulations to the unit, a series of artificial Mie resonance modes can be observed, such as a monopolar Mie resonance (MMR) mode at 238.4 Hz and a second MMR mode at 1,145.4 Hz. Based on the thermal-viscous loss created by the circular hole and four narrow slits of the unit under the excitation of the MMR mode, the near-perfect low-frequency sound absorption is observed at 239 Hz, and the maximum absorption coefficient and fractional bandwidth can reach about 0.97 and 12.9%, respectively. Additionally, we discuss the influences of structure parameters on the sound absorption performance, and design two types of broadband compound units by combining 4 units with different central circular holes. The fractional bandwidth of the compound unit can be further enhanced to 18.7%. The measured sound absorption spectra agree well with the simulated ones.

Design of Unit
As schematically shown in Figure 1A, we propose an acoustic metasurface-based absorber consisting of periodic square units with a length a and a thickness h. A central circular hole with a diameter d is located at the upper surface of the unit. Each unit is composed of an upper surface panel (with a thickness t 3 ) and a multiple-cavity structure (with a thickness t 4 ) on the bottom ( Figure 1B). As shown in Figure 1C, the multiple-cavity structure consists of a central square cavity (with a length b) surrounded by four interconnected identical cavities which are divided by four narrow slits (with a width t 1 ), showing a high structure symmetry. The distance between the slits and the outer frame is t 2 , and the frames (with a thickness t) are made of epoxy resin based on 3D-printing technology. Here, the COMSOL Multiphysics software is used to numerically simulate sound absorption characteristics, and the structure parameters are selected as a 100 mm, b 42 mm, d 5 mm, t t 1 2 mm, t 2 10 mm, t 3 1 mm, and t 4 15 mm. In our work, the sound absorption is created by the thermoviscous loss of the unit structure, and we use the module of Thermoviscous Acoustic-Solid Interaction inside the unit, and the module of Acoustic Pressure outside the unit due to the huge computation load. In the simulations, the thermoviscous acoustic boundary is used for all the surfaces inside the unit (include the inner surface of the hole), and the acoustic-thermoviscous acoustic boundary is adopted for the interface between the hole and the external space. The parameters of epoxy resin are the density ρ e 1,180 kg/m 3 , the longitudinal wave velocity c l 2,720 m/s, and the transversal wave velocity c t 1,460 m/s, and those of air are calculated as ρ a p 0 M/RT and c a cRT/M , in which the ratio of the molar heat capacities c, the molar mass M, and the temperature of air are 1.4, 28.97 × 10 -3 kg/mol, and 293 K, respectively, the molar gas constant R 8.31 J/(mol/K), and p 0 101.325 kPa. The paragraph of the unit is shown in Figure 1D. Figure 2 shows the simulated pressure amplitude and phase eigenfunctions of the proposed unit. We can see that two types of eigenmodes present typical characteristics of the MMR, which are denoted as the monopole and second monopole. Additionally, due to high symmetry of the multiple-cavity structure, the Mie resonance of the dipole and quadrupole can also be observed (see Supplementary Material), showing rich Mie resonant modes of the unit. As shown in Figure 2A, for the MMR mode at 238.4 Hz, the sound energy is mainly concentrated into the surrounding four cavities, and the whole structure exhibits a collective in-phase characteristic ( Figure 2B). But for the second MMR mode at 1,145.4 Hz, the sound energy is mainly in the central square cavity (Figure 2C), and an out-of-phase feature ( Figure 2D) is observed between the internal and external cavities. Here, to further demonstrate the mechanism of both MMR modes, we simulate the pressure amplitude and phase eigenfunctions of the units with different number of surrounding cavities (see Supplementary Material). The results show that the eigenfrequencies of both MMR modes change greatly with different number of cavities, but their mode characteristics are almost the same.

Low-Frequency Sound Absorption Created by the MMR Mode
Next, we experimentally measure the absorption performance of low-frequency sound created by the MMR mode in Figure 2A. As shown in Figure 3A, in the experiment, the sample (shown in Figure 1D) is placed at the right side in the straight waveguide which is made of acrylic plates to satisfy sound hard boundary condition. The experimental set-up is presented in the Supplementary Material. Figure 3B shows the measured and simulated sound absorption spectra created by the unit. We find that there exists a sound absorption peak at 239 Hz for both results, and the absorption coefficient can reach about 0.97, showing a near-perfect low-frequency sound absorption. Moreover, the bandwidth of sound absorption (black shaded region) is about 31 Hz, and its corresponding fractional bandwidth (the ratio of the bandwidth to the center frequency) can reach about 12.9%. The measured and simulated sound absorption spectra match well with each other. Beyond that, the thickness h of the unit is only 16 mm, which is equal to λ/90, exhibiting a deep subwavelength thickness of the proposed low-frequency sound absorber.
To explain the existence of the sound absorption peak, we introduce the relative acoustic impedance of the unit defined as Z r 〈p〉 Za〈v⊥〉 , where Z a ρ a c a is the acoustic impedance of air, p and v ⊥ are the total acoustic pressure and the sound velocity normal to the surface, respectively, and 〈.〉 represents averaging over the surface of the unit. The simulated real and imaginary parts of Z r are shown in Figure 3C. We observe that, at the frequency of absorption peak, the real and imaginary parts of Z r are about 1.35 and 0, respectively, indicating better impedance match between the proposed structure and air at 239 Hz. Therefore, the near-perfect sound absorption can be created by the unit structure.
Furthermore, we find that the frequency of sound absorption peak is almost the same as that of the MMR mode, and thus the sound absorption may arise from the MMR mode of the unit. To make a further insight into it, we simulate the distributions of the pressure amplitude and total thermal-viscous power loss density in the unit created by a normal incidence of sound at 239 Hz, which are shown in Figures 3D,E, respectively. Note that the excited pressure amplitude distribution of the unit ( Figure 3D) agrees well with that of the MMR mode (Figure 2A), indicating that the low-frequency sound absorption is created by the MMR mode of the unit. Moreover, as shown in Figure 3E, there exist an obvious thermal-viscous sound loss around the central circular hole and four narrow slits, especially the central circular hole. Therefore, we deduce that the sound absorption of the unit arises from the thermoviscous loss around the central circular hole and four narrow slits under the excitation of the MMR mode. Beyond that, we also simulate the sound absorption spectra created by the MMR mode of the unit with different incident angles (θ), and the absorption spectra are relatively stable below θ 60°. (see Supplementary Material).
Besides the sound absorption created by the MMR mode, we simulate the performances of sound absorption created by the second MMR mode of the unit. The results show that the sound absorption can also be created by the second MMR mode, but its absorption performance is reduced greatly due

BANDWIDTH OPTIMIZATION OF SOUND ABSORBER
Finally, we discuss the influences of the parameters b and d on the sound absorption and further optimize the working bandwidth of the sound absorber. Figures 4A,B show the simulated sound absorption spectra created by the MMR mode as a function of the parameters b and d, respectively, in which other parameters remain unchanged. It is found that, with the decrease of both parameters, the working bandwidth moves to the low-frequency region with a high sound absorption coefficient. The corresponding measured results for the parameters b and d are displayed in Figures 4C,D, which agree well with the simulation ones. Thus, we can reduce the working frequency of the sound absorption by simply decreasing the values of b and d.
To further optimize the working bandwidth, we design two types of compound units A and B consisting of 4 units (2 × 2 array) with different values of d (d 8, 10, and 12 mm for the units I, II and III), and experimentally measure sound absorption of both compound units. The experiment set-up is shown in Figure 5A, in which the width and height of the waveguide double those in Figure 3A, and the other parameters are the same. As shown in Figure 5B, the compound unit A consists of two types of units (I and II), and the arrangement of 4 units is shown in the sample photograph (shown in bottom inset). Note that, by combining the units I and II, the fractional bandwidth of the compound unit A can reach about 16.4%, in which the working frequency range (266-313.5 Hz, black shaded region) can cover those of a single unit I or II. Compared with the result in Figure 3B, the maximum sound absorption coefficient decreases slightly, but the absorption peak becomes wide and flat due to their coupling effect of both types of units. Additionally, as shown in Figure 5C, the compound unit B is composed of three types of units (I, II and III

CONCLUSION
In conclusions, we have demonstrated a metasurface-based unit with near-perfect low-frequency sound absorption based on artificial Mie resonances. The results show that a series of artificial Mie resonance modes can be observed in the unit, including the MMR mode at 238.4 Hz and the second MMR mode at 1,145.4 Hz. Based on the excited MMR mode and the thermal-viscous loss around the circular hole and four narrow slits of the unit, the near-perfect low-frequency sound absorption is achieved at 239 Hz, the maximum absorption coefficient and fractional bandwidth of the proposed unit can reach 0.97 and 12.9%. It is noted that the thickness of the unit is only about λ/90, showing a deep subwavelength thickness of the proposed metasurface-based sound absorber. In addition, we discuss the influences of structure parameters b and d on the sound absorption in detail, and find that the working bandwidth moves to the low-frequency region with a high absorption coefficient by decreasing both parameters. Finally, we improve the working bandwidth of the sound absorption by combining 4 units with different values of b, and the fractional bandwidth of the compound unit B can be further enhanced to 18.7%. The measured and simulated sound absorption spectra match well with each other. The proposed multiple-cavity units with the near-perfect sound absorption and broadband feature provide diverse routes to design advanced sound absorption structures with great potential applications in low-frequency noise control, architectural acoustics and environmental protection.

AUTHOR CONTRIBUTIONS
Y-JG and YG contributed equally to this work.

FUNDING
This work was supported by the National Natural Science Foundation of China (11774137, 51779107, 11834008, 61671314, 11974176, and 12174159).