# Tunable Beam Splitter Based on Acoustic Binary Metagrating
## 3 Results  
Here, how the incident angle affects the beam splitting is investigated. It can be derived from the Equation (1) that \(\sin\theta_{r}=~{}\sin\theta_{i}+nG/k_{0}\). As the geometrical structure of the proposed ABM is symmetric in the \(x\) direction, the range of the incident angle of \(\theta_{i}~{}\geq~{}0\) is investigated. It should be emphasized that the key to realize the beam splitting is that the dominant propagating modes are the diffracted waves of \(n=-1\) and \(n=1\). Since the reflection of the diffracted waves of \(n=0\) (specular reflection) is extremely small, by limiting \(-1~{}<~{}\sin\theta_{i}+nG/k_{0}<~{}1\) for \(n=-1\) and \(n=1\), the possible range of the incident angle for the beam splitting can be derived that \(0~{}<~{}\sin\theta_{i}<~{}1~{}-~{}\lambda_{0}/a\). The numerical simulations were performed to verify the mechanism on the beam splitting in the ABM. Figure 4 shows the case of \(a~{}=~{}1.4\lambda_{0}\). The possible range of the incident angle for the beam splitting can be derived that \(0~{}<~{}\theta_{i}<~{}16.6^{\circ}\). The depth and width of the groove in each period are \(t=0.5a\) and \(d~{}=~{}0.25\lambda_{0}\), respectively. The operating wavelength is set as \(\lambda_{0}~{}=~{}100\) mm. Here, the full wave simulations based on finite-element analysis were performed using the COMSOL Multiphysics pressure acoustics module. The plane-wave radiation boundary condition is used on the top boundary. The sound hard boundary condition is used on the bottom boundary. The background pressure field is used on the whole region with the incident angle of \(\theta_{i}=~{}0^{\circ}\), \(\theta_{i}~{}=~{}5^{\circ}\), and \(\theta_{i}~{}=~{}10^{\circ}\). The scattering pressure field distributions with the incident angle of \(\theta_{i}=~{}0^{\circ}\), \(\theta_{i}~{}=~{}5^{\circ}\), and \(\theta_{i}~{}=~{}10^{\circ}\) are shown in Figure 4a-c, respectively. The white arrows show the directions of the incident and the reflected waves. The polar plots of the normalized intensity of the scattering waves in the exterior field with the incident angle of \(\theta_{i}=~{}0^{\circ}\), \(\theta_{i}~{}=~{}5^{\circ}\), and \(\theta_{i}~{}=~{}10^{\circ}\) are shown in Figure 4d-f, respectively. Here, \(D(\theta_{r})\) is the normalized intensity of the scattering waves as a function of the reflected angle. All the intensities are normalized to the maximal intensity of the scattering waves of the normal incidence. It is noted that the beam splitting effects are obvious. For all these incident angles, when a given acoustic wave arrives at the ABM, it is split into two predesigned directions: split beam 1 (\(-x\) direction) and split beam 2 (\(+x\) direction). With the increase in the oblique incident angle, the reflected angle of the split beam 1 (\(-x\) direction) decreases gradually, while the reflected angle of the split beam 2 (\(+x\) direction) increases gradually. Figure 5 shows the case of \(a~{}=~{}1.8\lambda_{0}\). The possible range of the incident angle for the beam splitting can be derived that \(0~{}<~{}\theta_{i}<~{}26.4^{\circ}\). The depth and width of the groove in each period are \(t=0.5a\) and \(d~{}=~{}0.25\lambda_{0}\), respectively. The operating wavelength is set as \(\lambda_{0}=~{}100\) mm. Here, the full wave simulations based on finite-element analysis were performed using the COMSOL Multiphysics pressure acoustics module. The plane-wave radiation boundary condition is used on the top boundary. The sound hard boundary condition is used on the bottom boundary. The background pressure field is used on the whole region with the incident angle of \(\theta_{i}=0^{\circ}\), \(\theta_{i}=5^{\circ}\), \(\theta_{i}=10^{\circ}\), \(\theta_{i}=15^{\circ}\), and \(\theta_{i}=~{}20^{\circ}\). The scattering pressure field distributions with the incident angle of \(\theta_{i}=~{}0^{\circ}\), \(\theta_{i}=~{}5^{\circ}\), \(\theta_{i}=~{}10^{\circ}\), \(\theta_{i}=~{}15^{\circ}\), and \(\theta_{i}=~{}20^{\circ}\) are shown in Figure 5a-e, respectively.