Bilayer ventilated labyrinthine metasurfaces with high sound absorption and tunable bandwidth

The recent advent of acoustic metamaterials offers unprecedented opportunities for sound controlling in various occasions, whereas it remains a challenge to attain broadband high sound absorption and free air flow simultaneously. Here, we demonstrated, both theoretically and experimentally, that this problem can be overcome by using a bilayer ventilated labyrinthine metasurface. By altering the spacing between two constituent single-layer metasurfaces and adopting asymmetric losses in them, near-perfect (98.6%) absorption is achieved at resonant frequency for sound waves incident from the front. The relative bandwidth of absorption peak can be tuned in a wide range (from 12% to 80%) by adjusting the open area ratio of the structure. For sound waves from the back, the bilayer metasurface still serves as a sound barrier with low transmission. Our results present a strategy to realize high sound absorption and free air flow simultaneously, and could find applications in building acoustics and noise remediation.


Results
Structures. The designed structure consists of two ventilated labyrinthine metasurfaces with a spacing d, as shown in Figs. 1a,b. Both metasurfaces are made of rigid body and immersed in air. The building block of each metasurface has a size a in both the x and y directions, and a thickness b in the z direction. Inside the building block, there exist a ventilation duct along the z direction and a one-end-closed channel curled in the x-z plane. The ventilation duct has a rectangular cross section with area of S o = (a − t)(a − h) with t being the wall thickness. The curled channel has an effective length L ≈ Nb , a width w = (h − Nt − t)/N , a folding number N = 2 , a rectangular cross section with area of S c = (a − t)w , and an aperture near the surfaces of the bilayer metasurface. Such a curled channel constitutes a labyrinthine resonator with a fundamental resonant frequency where c is the sound speed in air and R ≈ 4L is the fundamental resonant wavelength.
Theory. Consider the bilayer metasurface impinged by a plane sound wave with frequency f and at incident angle θ . For wavelengths longer than 2a ( = c/f > 2a ), only fundamental modes exist in the ventilation ducts and curled channels, and no diffracted propagating waves are generated by the structure (Fig. 1b). At the left side of the bilayer metasurface, the left-ward (right-ward) propagating wave has an complex amplitude C 1 ( D 1 ) in sound pressure. At the right side of the bilayer structure, the complex amplitude of sound pressure is C 2 ( D 2 ) for the left-ward (right-ward) propagating wave. Since the sound pressure and particle flow need to be continuous at each structural interface, the field ( C 2 , D 2 ) at the back (i.e. right) can be related to the field ( C 1 , D 1 ) at the front (i.e. left) by a 2 × 2 transfer matrix M is the transfer matrix for the region between the two metasurfaces, k = 2π/ is the wavenumber in air, and P 1 U 1 and U 2 P 1 are the transfer matrices for the front and back metasurfaces. The matrix P 1 is given by Here, a unit cell of the bilayer metasurface, consisting of two curled channels with sound loss β 1 and β 2 inside, is placed in an acoustic impedance tube. (d) Simplified models for cases with perfect absorption (I) and incomplete absorption (II).  where n is a positive integer 18 . We note that when the back metasurface has no absorption loss ( β 2 = 0 ), it can present complete reflection at resonant frequency f R . When the openings of the two labyrinthine resonators in a unit cell have a distance of (2n − 1) R /4 , the back resonator can cause zero particle velocity at the opening of the front resonator at f R . To further obtain complete absorption at f R , a critical sound loss ( β 1 = β 1m ) is required in the front curled channels.
Simulations. To verify the above theory, we perform simulations for bilayer ventilated labyrinthine metasurfaces, which have h = 0.5a , t ≪ a , and thus an open area ratio p o = 0.5 . Sound waves are incident normally from the front ( θ = 0 ). We first consider the case with a zero spacing between the two metasurfaces ( d = 0 ). In  Fig. 2b. We see that the absorption A R approach a maximum A Rm (99.5%) at β 1 = β 1m = 0.161 and β 2 = β 2m = 0 , agreeing well with analytic results from Eqs. (6) and (7) ( β 1m = 0.159 and β 2m = 0 ). We note that A R remains high around the optimal losses ( β 1m , β 2m ). If the front and back metasurfaces possess the same optimal losses ( β 1 = β 2 = 0.148 ) in the curled channels, the maximal absorption can still be as high as 95% (see Fig. 3). But if only the front metasurface exists, the maximal absorption will decrease to 71% even with using an optimal loss ( β 1 = 0.108 ) (see Fig. 3).
More results are shown in Fig. 2c for bilayer metasurfaces with different spacing d. It is found that the maximal absorption A Rm varies with increasing the spacing d. Near perfect absorption ( A Rm > 99% ) can be achieved in some ranges including 0 < d < 0.24L and 1.94L < d < 2.24L . But if the spacing is not in such optimized regions, low absorption will occur. For instance, at d = 0.66L and even with using optimal losses ( β 1m = 0.321 , β 2m = 0.115 ), the absorption A Rm can be only 59.3%, indicating the importance of the spacing d for achieving perfect absorption. The corresponding absorption spectrum is plotted as curve II in Fig. 2a, where two wide resonant peaks are visible at frequencies of 0.208c/L and 0.281c/L. It should also be mentioned that a very sharp absorption peak can occur at a critical spacing ( d = 0.7L ) with tiny optimal losses of β 1m = 0.0073 and β 2m = 0 (see curve III in Fig. 2c). We note that such a high-Q resonant mode can be viewed as an acoustic quasi bound state in the continuum (BIC), which can exist in various acoustic systems 43 .
To clarify the above results, the distribution of particle velocity in a unit cell of the bilayer metasurface is simulated at resonant frequency using a finite-element method (COMSOL Multiphysics), as shown in Fig. 2d. When d = 0 , the openings of the two labyrinthine resonators in the unit cell have a distance of about a quarter of resonant wavelength ( ∼ R /4 ). Since a large particle velocity occurs at the opening of the back resonator, zero velocity can be obtained at the opening of the front resonator (see case I in Figs. 1d and 2d). Therefore, a single absorption peak is visible (curve I in Fig. 2a) and its strength can be 100% with using an appropriate loss in the front resonator. In contrast, if the spacing between the two metasurfaces is not appropriately chosen, the ventilation duct can also serve as a resonator but with a zero loss ( β = 0 ) (see case II in Figs. 1d and 2d). Hence, two resonant peaks occur with imperfect strengths (see curve II in Fig. 2a).
Experiments. Based on the above theoretical results, we fabricated bilayer ventilated labyrinthine metasurfaces with polylactic acid (PLA) by means of 3D-printing technology (see Fig. 4a and b and Methods). The structural parameters are a = 43 mm, b = 107 mm, t = 1 mm, d = 0 , and p o = 0.5 . By using an impedance tube with a square cross section (see Figs. 1c, 4c, and Methods), the reflection, transmission, and absorption spectra www.nature.com/scientificreports/ were measured for a unit cell, as shown in Fig. 5a-c. For the unit cell, the back resonator is hollow whereas the front one contains an appropriate amount of porous media (with a mass of 0.1498 g; see Methods). It is found that for sound waves incident from the front, the unit cell exhibits a low reflection (17%) in a wide frequency range (100-800 Hz). Very low reflection ( R = 0.2% ), low transmission ( T = 1.2% ), and near-perfect absorption ( A = 98.6% ) can be seen at resonant frequency ( f R = 368 Hz). For sounds incident from the back, a lower absorption (67%) is found at resonant frequency due to the asymmetric loss of the unit cell ( β 1 > β 2 ). It should also be mentioned that almost the same transmission is observed for sounds from the front and back, agreeing with the theoretical expectation. Numerical calculations were conducted for the above sample based on the transfer-matrix method (Fig. 5d-f). In the calculations, a thin layer of air (thickness = 0.6 mm) is considered to be static at the surface of each wall. We calculated absorption spectra for different losses ( β 1 , β 2 ). It is found that when β 1 = 0.11 and β 2 = 0.03 are used in the front and back curled channels, the strength and width of the calculated absorption peak can match the experimental values. Since slight sound loss ( β 2 = 0.03 ) exists in the back channel, the back resonator cannot provide complete reflection. Hence, for sound incident from the back, a considerable absorption ( A back = 67% ) is observed in experiments. If an ideal asymmetric absorption (i.e. A back = 0 and A front = 100% ) is desired, the back channel should exhibit zero sound loss ( β 2 = 0). (b) A R as a function of β 1 and β 2 for bilayer metasurfaces with d = 0 . A R reaches a maximum A Rm at optimal losses ( β 1 = β 1m , β 2 = β 2m ). (c) β 1m , β 2m , and A Rm as a function of layer spacing d. (d) Simulated distributions of |v|/|v 0 | in the x-z plane for structures I and II at f R in (a). v represents particle velocity and v 0 is the particle velocity of the incident sound wave. www.nature.com/scientificreports/ Besides the absorption strength, the absorption bandwidth is also important for a sound absorber. For the above sample, the measured absorption is higher than 50% for frequencies ranging from 305 Hz to 435 Hz, corresponding to a relative band width � f /f R = 36% with the full width at half maximum f = 129.4 Hz. We note that the thickness of the bilayer metasurface is in a subwavelength scale (λ/5h) compared to the lower limit (305 Hz) of the absorption band. In addition, the relative bandwidth of absorption here is much larger than (7 times of) recent results with using a pair of Helmholtz resonators in the unit cell 37 . If the same wide absorption  The ventilation performance is further characterized for the bilayer labyrinthine metasurfaces. Here, a unit cell of the metasurface is placed in an aluminum tube with a square cross section of a size 44 mm. One end of the tube is seamlessly connected with the outlet pipe of an electric blower (9028E, Anjieshun, China). An anemometer (DP-1000-IIIB, Yiou, China) is used to monitor the air flow velocity at a fixed position in the tube. When the tube is hollow, the wind velocity is v wo (8 and 15 m/s were tested in our experiments). When the unit cell is placed in the tube, the wind velocity becomes v w . Thus, a ventilation rate can be defined as the ratio of wind velocity ( v wo /v w ) 27,37 . Figure 7b shows the measured wind velocity ratio for different unit cells. We can see that the wind velocity ratio increases with increasing the open area ratio of the structure. For the above five samples, the ventilation rates are larger than 0.6.

Discussion
In summary, we design, fabricate, and characterize a bilayer ventilated labyrinthine metasurface for perfect sound absorption and free air flow. Both a ventilation duct and a curled single-port channel exist in the constituent single-layer metasurface. By using asymmetric losses in the two single-layer metasurfaces and adjusting their www.nature.com/scientificreports/ spacing, perfect sound absorption is achieved at resonant frequency for sound waves incident from the front. The measured peak absorption is as high as 98.6% even at a relatively large open area ratio (50%). By tuning the open area ratio, the relative absorption bandwidth can be adjusted in a large range (from 12% to 98%). For sounds incident from the back, the bilayer metasurface still serves as a sound barrier with low transmission and partial reflection/absorption. Our work provides a strategy for achieving broadband perfect sound absorption in ventilated structures and could be extended to other fields such as electromagnetic waves and water waves.

Methods
Sample preparations. Each experimental sample is composed of two boxes, and each box contains a labyrinthine structure (see. Fig. 1a) and a side plate. The two parts are first fabricated with PLA by 3D-printing technology, then agglutinated together. The porous media placed in the box is polyester pillow stuffing (white 15d× 51mm hcs 100% polyester stuffing pp staple fiber filling pillow).

Sound absorption measurements.
A commercial impedance tube (Hangzhou Aihua, AWA6290T) was applied to measure the absorption of acoustic metasurfaces (see Figs. 1c and 4c). Here, a unit cell of the bilayer metasurface was placed in an aluminum impedance tube, which has a total length of 2.9 m, a wall thickness of 3 mm, and a square cross section with an inner size of 44 mm. A loudspeaker is placed at the left end of the impedance tube. The right part of the impedance tube is filled with sound-absorbing materials of a length of 1.4 m, so that the resulted reflection can be less than 1.5% in the frequency band of interest ( f > 250 Hz). Four 1/4-in. condensed microphones are situated at designated positions to sense local pressure. The loudspeaker was fed with a sinusoidal signal of which the frequency increases with increasing time. By analyzing the signals from microphones, the reflection, transmission, and absorption spectra can be obtained for the unit cell.