Acoustic metasurfaces (Cummer et al., 2016; Assouar et al., 2018) are artificially designed structures composed of periodic subwavelength elements including groove structures (Shen et al., 2018), Helmholtz resonators (Li et al., 2018), labyrinthine structures (Xie et al., 2014), space coiling-up structures (Li et al., 2013; Chen et al., 2021), membranes (Ma et al., 2014), etc. In recent years, acoustic metasurfaces have attracted significant attention for their great potential applications in many fields attributed by their interesting and extraordinary acoustic properties (Zhao et al., 2022), such as anomalous reflection and refraction (Memoli et al., 2017; Zhu and Lau, 2019), acoustic focusing (Qi et al., 2017; Lombard et al., 2022), acoustic cloaking (Faure et al., 2016; Jin et al., 2019), sound absorption (Aurégan, 2018; Song et al., 2019), one-way acoustic propagation (Liang et al., 2010; Li et al., 2017; Song et al., 2019), beam splitting (Ding et al., 2021), etc. Particularly, acoustic beam splitters have attracted growing interest recently due to their applications in acoustic communication (Prada et al., 2007) and acoustic sensing (Dowling and Sabra, 2015) fields.

Acoustic beam splitters (Ni et al., 2019) are devices that can effectively split the incident beam into two or more beams. Acoustic metasurfaces capable of manipulating the amplitude and phase of acoustic waves provide a good method for realizing beam splitting. (Díaz-Rubio et al. 2019) proposed a power flow-conformal acoustic beam splitter by manipulating the surface impedance distribution of metasurface to split the incident wave into two reflected beams propagating along two different directions. Beam splitting with arbitrary energy ratios can be realized by non-local grooved metasurfaces by using deep learning algorithms (Ding et al., 2021). Bianisotropic metasurface was developed for near-perfect arbitrary beam splitting by introducing self-induced surface waves into scattered field (Li et al., 2020). In addition, coding acoustic metasurfaces are widely used for designing acoustic beam splitters. Reflection-type coding acoustic metasurfaces were designed for realizing broadband acoustic splitter by encoding the sequence of logical units (Zhang et al., 2022). Coding acoustic metasurfaces composed of hornlike helical unit were theoretically and experimentally demonstrated for realizing acoustic splitting (Fang et al., 2019). However, we can find that the beam splitters based on acoustic metasurfaces are inevitably limited in element resolution, manipulation efficiency, and operating angle. Furthermore, the most commonly used microstructures including resonator structures (Shen et al., 2018) and space coiling-up structures (Jia et al., 2018) have complex configuration, which may result in large intrinsic loss and low efficiency. In order to solve the above-mentioned problems existing in beam splitters based on acoustic metasurfaces, simple and high-efficiency acoustic metagratings (Torrent, 2018; Fan and Mei, 2021) based on diffraction theory provide a reliable method for achieving good acoustic performances. High-efficiency anomalous acoustic splitters based on genetic optimization algorithm and acoustic metagratings are designed to split the incident wave into different directions (Ni et al., 2019). Perfect beam splitting with unitary efficiency (Fan and Mei, 2020) was demonstrated by using acoustic metagrating composed of periodic iron cylinders and a sound soft plane placed in water. Beam splitter based on acoustic binary metagrating was designed to split an acoustic wave into two directions (Liu et al., 2022). A transmitting beam splitter with grooves was designed to split the incident wave into two waves with arbitrary amplitude ratio and phase differences (Cao and Hou, 2019).

From above discussion, we find that the beam splitters based on acoustic metasurfaces suffer from complicated configurations and inevitable intrinsic loss, which greatly limit their wide practical applications of beam splitters. The literature on metagrating-based acoustic beam splitter with changeable beam number also remain scarce. Therefore, it is necessary to explore new realization methods to overcome these limitations and design beam splitters with simplified structures, high efficiency, and good reconfigurability. In this paper, simple acoustic metagrating with periodic grooves is designed for realizing perfect two- and three-beam splitting with high efficiency, broad bandwidth, and easy fabrication, and demonstrating good beam manipulation performance by changing the groove height. The proposed acoustic metagrating has potential applications in the fields of acoustic communication (Prada et al., 2007) and acoustic sensing (Dowling and Sabra, 2015).

In this paper, we propose a simple acoustic metagratings for realizing perfect two-beam and three-beam splitting with specified reflected amplitudes and power flows, as shown in Figure 1. As the proposed acoustic metagrating is a periodic structure, the groove number will not greatly affect the beam splitting performances. Without loss of generality, we take the proposed acoustic metagrating containing 22 periodic grooves as an example, and incident wave is perpendicularly incident on the acoustic metagrating. Actually, the groove number can also be set as other values. Compared with the previously reported beam splitters, our proposed metagrating could manipulate the number of reflected beams, the reflected amplitudes, and the power flows by controlling the groove height based on local power conservation. For perfect two-beam splitting case, the acoustic metagrating can split a normally incident beam into two symmetric reflected beams with reflected angles of θ _-1 and θ ₊₁, the corresponding amplitudes are p _-1 and p ₊₁, respectively. For perfect three-beam splitting case, the acoustic metagrating can split a normally incident beam into three reflected beams with reflected angles of θ _-1, θ ₊₁, and θ ₀, the corresponding amplitudes are p _-1, p ₊₁, and p ₀, respectively. The inset in Figure 1 shows the two-dimensional configuration of the proposed metagrating, which contains periodic grooves. Because the reflected beams of −1 and +1 diffraction orders are completely symmetric beams, so it is enough for one unit cell only contains one groove in our design as demonstrated by Torrent (Torrent, 2018). To generate reflected beams with different amplitudes and realize more beam number, one unit cell should contain two or more grooves and the groove parameters should be changed. The groove width is w, the groove height is h, the thickness of acoustic metagrating is d, the period of acoustic metagrating is a. Careful choice of acoustic metagrating geometrical parameters can in principle scattered all acoustic energy into different diffraction orders according to the diffraction theory, so the amplitudes and power flows of reflected beams can be manipulated. As the metagrating with periodic grooves does not contain resonant (Shen et al., 2018) or narrow (Jia et al., 2018) structures, we can expect high efficiency and low intrinsic loss.

When a plane wave impinges on the periodic acoustic metagrating with the incident angle of θ _i , the incident wave will be diffracted into multiple diffraction orders and the scattered filed can be expressed as (Torrent, 2018): p s = ∑ n p n e − j k n ⋅ r (1) where p _n is the amplitude of nth-order diffracted wave, r is the position vector, k _n is the wave vector of nth-order diffracted wave and can be expressed as k n = k n x x + k n y y = k ⁡ sin ⁡ θ n x + k ⁡ cos ⁡ θ n y , k is the wavenumber in air, θ _n is the reflected angle of nth-order diffracted wave. According to the periodicity of acoustic metagrating and the grating theory, the x-direction wave vector components of incident and reflected waves should satisfy: k n x = k i x + G n (2) where k _nx = ksinθ _n is x-direction wave vector component of nth-order reflected wave, k _ix = ksinθ _i is x-direction wave vector component of incident wave, G _n = 2πn/a is the nth-order reciprocal vector. Therefore, the y-direction wave vector component of nth-order reflected wave is k n y = k 2 − k ⁡ sin ⁡ θ i + 2 π n / a 2 , it is noted that k _ny will be a real component only when k 2 > k n x 2 and the nth diffraction order corresponds to a propagating wave.

In our design, the incident wave is perpendicularly incident on the acoustic metagrating, so the incident angle is θ _i = 0°. To realize a beam splitting acoustic metagrating, we first calculate the period of the acoustic metagrating based on the Eq. 2 to guarantee the nth diffraction orders are propagating waves. Therefore, the period of acoustic metagrating is a = nλ/|sinθ _±1|, λ is the wavelength in air. By setting a = λ/|sinθ _±1|, the acoustic metagrating will only have −1, 0, and +1 diffraction orders as propagating waves for the case of θ _n > 30°. The n = −1 diffraction order corresponds to the reflected beam with the reflected angle of θ _-1, n = +1 diffraction order corresponds to the reflected beam with the reflected angle of θ ₊₁, n = 0 diffraction order corresponds to the reflected beam with the reflected angle of θ ₀. As an example, we choose the working frequency is 4,000 Hz and the corresponding wavelength is λ = 85.75 mm, the reflected angles or splitting angles are θ _-1 = −60° and θ ₊₁ = 60°, so the period of acoustic metagrating is a = 99 mm. By simultaneously changing the groove width w and groove height h of acoustic metagrating, the amplitudes of different diffracted waves can be effectively manipulated. For simplicity, the groove width is fixed to w = 0.6a = 59.4 mm, the groove height is a variable value. The thickness of acoustic metagrating is d = 50 mm.

As the groove height of acoustic metagrating could influence the amplitudes of different diffracted waves, so it is necessary to determine the groove heights for different beam splitting cases. The amplitude of incident wave is set to be 1. We separately calculate the amplitudes of different reflected beams normalized by the amplitude of normally incident wave, Figure 2A shows the amplitudes of −60°, 60°, and 0° reflected beams when the proposed acoustic metagratings are illuminated by a plane wave of 4,000 Hz, and the groove height gradually increases from 0 to 0.5λ. We can see the amplitudes of −60° (red line) and 60° (cyan circle) reflected beams remain same and gradually increase from 0 to 0.999 at the groove height of h = 0.22λ (point M), then reduce to the minimum amplitude at h = 0.5λ. However, the amplitudes of 0° (blue dotted line) reflected beam presents opposite variation tendency and reaches the minimum amplitude at the groove height of h = 0.22λ (point M′), which means 0° reflected beam is greatly suppressed and two-beam splitting with the same amplitude is achieved at h = 0.22λ. It is worth noting that −60°, 60°, and 0° reflected beams have the same amplitude of 0.707 when the groove height is h = 0.10λ, as denoted by point N in Figure 2A, which means three reflected beams are all effectively excited and perfect three-beam splitting with same amplitude is achieved.

On the other hand, the y-direction power flows of −60°, 60°, and 0° reflected beams are calculated and shown in Figure 2B with the groove height changing from 0 to 0.5λ. The y-direction power flow I _y is normalized by the power flow of normally incident wave. It is observed that the power flows of −60° (red line) and 60° (cyan circle) reflected beams still remain same and reach the maximum value of 0.5 at h = 0.22λ, while the amplitude of 0° (blue dotted line) reflected beam reaches the minimum value, which means two-beam splitting with same power flow is achieved. We can see that −60°, 60°, and 0° reflected beams have same power flow when the groove height is h = 0.12λ, as denoted by point P in Figure 2B, which means all three reflected beams are excited with same power flow and three-beam splitting is achieved. If the local power flow perpendicular to the acoustic metagrating is conserved, the incident amplitude and reflected amplitudes should satisfy p i 2 = p − 1 2 ⁡ cos θ − 1 + p + 1 2 ⁡ cos ⁡ θ + 1 + p 0 2 ⁡ cos ⁡ θ 0 , which means the power flow ratio of reflected beams is I − 1 : I + 1 : I 0 = p − 1 2 ⁡ cos θ − 1 : p + 1 2 ⁡ cos ⁡ θ + 1 : p 0 2 ⁡ cos ⁡ θ 0 . From above analysis, we predict that equal two-beam splitting ( p − 1 = p + 1 = 1 or I − 1 : I + 1 = 0.5 : 0.5 ) can be achieved at h = 0.22λ, three-beam splitting with same amplitude ( p − 1 = p + 1 = p 0 = 2 / 2 ) can be achieved at h = 0.10λ, and three-beam splitting with same power flow ( I − 1 : I + 1 : I 0 = 1 3 : 1 3 : 1 3 ) can be achieved at h = 0.12λ. Local power flow is conserved in both two- and three-beam splitting cases for realizing perfect beam splitting.

To intuitively show the two-beam and three-beam splitting performances, numerical simulations of the proposed acoustic metagratings are performed in COMSOL Multiphysics software. The proposed acoustic metagratings are placed in air background and the intrinsic loss is not considered, all boundaries are set to be sound hard boundaries, the air density is 1.21 kg/m³, the sound speed in air is 343 m/s.

In conclusion, we demonstrate an acoustic metagrating with periodic grooves can realize perfect two-beam splitting and three-beam splitting with conserved local power. The proposed acoustic metagrating will only allow −1, 0, and +1 diffraction orders as propagating waves by setting the period of acoustic metagrating. By changing the groove height, the amplitudes and power flows of different reflected beams can be effectively manipulated. Numerical results clearly show that the proposed metagrating can split the normally incident beam into two or three beams by suppressing or exciting the mirror reflected wave. Equal two-beam splitting and three-beam splitting with same amplitude or same power flow are achieved, numerical simulations agree well with the theoretical analysis. The reflected angle gradually decreases with the increase of working frequency. This design method can also be applied to design beam splitters with other splitting angles and working frequencies.