# Tunable Beam Splitter Based on Acoustic Binary Metagrating
## 2 Materials and Methods
The periodicity is along the \(x\) direction, and the period length is \(a\). The depth and width of the groove in each period are denoted as \(t\) and \(d\), respectively. Simulated reflection responses as a function of the ratio \(t/a\) (**b**) in the case of \(a~{}=~{}1.4\lambda_{0}\) and (**c**) in the case of \(a~{}=~{}1.8\lambda_{0}\). Here, \(d~{}=~{}0.25\lambda_{0}\), \(\lambda_{0}=~{}100\) mm. where \(\mathbf{\varepsilon}=(\varepsilon_{\,x},\,\varepsilon_{y},\,\varepsilon_{z})\) and \(\mathbf{\mu}=(\mu_{\,x},\,\mu_{y},\,\mu_{z})\) are the parameters of permittivity and permeability, respectively. \(\omega\) is the angular frequency. \(\mathbf{E}=(E_{\,x},\,E_{y},\,E_{z})\) is the electric field. \(\mathbf{H}=(H_{\,x},\,H_{y},\,H_{z})\) is the magnetic field. For the two-dimensional electromagnetic wave (TM polarization) in the \(x\)-\(z\) plane of \((E_{\,x},\,E_{z},\,H_{y})\), the corresponding parameters are \((\varepsilon_{\,x},\,\varepsilon_{z},\,\mu_{y})\). Considering the harmonic factor \(e^{-\,j\omega t}\), there are:  
\[\frac{\partial(\varepsilon_{\,x}E_{x})}{\partial x}+\frac{\partial(\varepsilon _{\,z}E_{z})}{\partial z}=\,0. \tag{6}\]  
\[\frac{\partial E_{x}}{\partial z}-\frac{\partial E_{z}}{\partial x}=\,j\omega \mu H_{y}. \tag{7}\]  
\[\frac{\partial H_{y}}{\partial z}=\,j\omega\varepsilon_{x}E_{x}. \tag{8}\]  
Figure 1:0  
Acoustic wave equations are expressed as:  
Figure 1:1  
Figure 1:2  
where \(P\) is the pressure. \(\mathbf{v}\,\,=\,\,(v_{\,x},\,v_{y},\,v_{z})\) is the velocity. \(\omega\) is the angular frequency. \(\rho\) is the mass density. \(B\) is the bulk modulus. For the two-dimensional acoustic wave in the \(x\)-\(z\) plane of \((v_{\,x},\,v_{z},\,P)\), the corresponding parameters are \((\rho_{\,x},\,\rho_{z},\,B)\). Considering the harmonic factor \(e^{-\,j\omega t}\), there are:  
Figure 1:3  
Figure 1:4  
Figure 1:5  
By comparizing the Equations (6)-(9) of the electromagnetic wave in the \(x\)-\(z\) plane and the Equations (12)-(14) of the acoustic wave in the \(x\)-\(z\) plane, the equivalence can be derived as shown in Table 1[20; 21; 22]. For the isotropic medium of air, there are \(\rho=\rho_{\,x}=\rho_{\,z}\) and \(\varepsilon=\varepsilon_{x}=\varepsilon_{z}\). The parameters of density \(\rho\), bulk modulus \(B\), and pressure \(P\) in acoustics are equivalent to the parameters of permittivity \(\varepsilon\), reciprocal permeability \(\mu^{-\,1}\), and magnetic field component \(H_{y}\) in electromagnetics (TM polarization), respectively. Thus, the material parameters in the COMSOL Multiphysics electromagnetic waves module are set as \(\varepsilon\prime\,=\,\rho_{air}/\varepsilon_{0}\) and \(\mu\prime\,=\,1/(\mu_{\,0}B_{air})\). Here, \(\varepsilon\prime\) is the relative permittivity, \(\mu\prime\) is the relative permeability, \(\varepsilon_{0}\) is the permittivity of vacuum, \(\mu_{0}\) is the permeability of vacuum. \(\rho_{air}=1.21\,\mathrm{kg}/\mathrm{m}^{3}\) is the \begin{table}
\begin{tabular}{c c} \hline \hline
**Acoustic Wave** & **Electromagnetic Wave (TM Polarization)** \\ \hline Mass density \(\rho_{\,x}\) & Permittivity \(\varepsilon_{\,z}\) \\ Mass density \(\rho_{\,z}\) & Permittivity \(\varepsilon_{\,x}\) \\ bulk modulus \(B\) & Reciprocal permeability \(\mu^{-\,1}\) \\ Pressure \(P\) & magnetic field component \(H_{y}\) \\ Velocity field component \(v_{x}\) & electric field component \(-E_{z}\) \\ Velocity field component \(v_{z}\) & electric field component \(E_{x}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The equivalence between the acoustic wave and the electromagnetic wave (TM polarization). mass density in the air. \(B_{air}=~{}\rho_{air}c_{air}^{2}\) is the bulk modulus in the air. \(c_{air}=~{}343~{}\mathrm{m/s}\) is the sound velocity in the air. In the COMSOL Multiphysics electromagnetic waves module, the numerical model as shown in Figure 3 is used. The top boundary is set as the period port with the wave excitation on. The input quantity is the magnetic field (0, 0, 1).