# Tunable Beam Splitter Based on Acoustic Binary Metagrating
## 2 Materials and Methods  
Figure 1a shows the schematic illustration of the proposed ABM. It consists of only two different elements, \"0\" and \"1\", compactly arranged along the \(x\) direction, providing the binary phase shifts of 0 and \(\pi\), respectively. Here, the length of the element \"0\" or \"1\" is \(t\), and the period length in the ABM is \(a~{}=~{}2t\). When a given acoustic wave arrives at the ABM, it is split into two predesigned directions: beam 1 (\(-x\) direction) and beam 2 (\(+x\) direction). A general grating theory is provided here to demonstrate the physical mechanism underlying the beam splitting. When the acoustic waves impinge on the ABM with the incident angle of \(\theta_{i}\), the reflected waves should follow the grating Equation [1,3]:  
\[k_{0}(\sin\theta_{r}~{}-~{}\sin\theta_{i})~{}=~{}nG \tag{1}\]  
where \(k_{0}=~{}2\pi/\lambda_{0}\) is the wave number in the air, \(\lambda_{0}\) is the operating wavelength in the air, \(\theta_{r}\) is the reflected angle, \(n\) is the diffraction order (integer), and \(G~{}=~{}2\pi/a\) is the reciprocal lattice vector. As depicted in Figure 1b, for the normal incidence of \(\theta_{i}=~{}0^{o}\), by limiting \(-1~{}<~{}\sin\theta_{r}<~{}1\), it can be derived from Equation (1) that only the range of \(k_{0}/2~{}<~{}G~{}<~{}k_{0}\), corresponding  
Figure 1: **(a)** Schematic illustration of a beam splitter based on the ABM. **(b)** Diagram of realizing a beam splitter in the ABM. The black cross on the black arrow indicates that the specular reflection waves are totally suppressed. to \(\lambda_{0}<~{}a~{}<~{}2\lambda_{0}\), may give rise to the target pressure field. In the target pressure field, only the diffracted waves of the orders of \(n=0\), \(n=-1\), and \(n=1\) are the propagating waves. The diffraction order in the ABM taking a value of \(n=-1\) corresponds to the split beam 1 (\(-x\) direction), \(n=1\) corresponds to the split beam 2 (\(+x\) direction), and hence, the beam splitting occurs. In addition to the filter of the redundant diffracted waves by choosing the reciprocal lattice vector \(G\), i.e., period length \(a\) in the ABM, the specular reflection of \(n=0\) should also be totally suppressed here to realize the beam splitting. The structure we proposed to construct the ABM is schematically illustrated in Figure 2a, which is a planar hard surface having pitch \(a\) in the \(x\) direction. In each period, only one straight-walled groove is etched on the planar hard surface. The depth and width of the groove in each period are denoted as \(t\) and \(d\), respectively. The region of the hard surface provides no phase shift of the incident waves, corresponding to the element \"0\". Thus, to construct an ABM, the region of the groove should provide the phase shift of \(\pi\), corresponding to the element \"1\". Since the phase delay is \(2k_{0}d\) after the incident waves propagating along a round trip in the groove, the phase shift of \(\pi\) can be provided with the groove of the depth \(d~{}=~{}0.25\lambda_{0}\). Now we optimize the geometrical parameter of the groove width \(t\) in the ABM. A series of numerical simulations were performed to calculate the reflection of the diffracted waves of \(n=0\), \(n=-1\), and \(n=1\) as a function of the ratio \(t/a\). Here, the full wave simulations based on the finite-element analysis were performed using the COMSOL Multiphysics electromagnetic waves module. Equivalence between acoustics and electromagnetics (TM polarization) in the two-dimensional systems should be used. To clarify this equivalence, [PERSON]'s equations are presented:  
\[\
abla\cdot(\varepsilon E)~{}=~{}0. \tag{2}\]  
\[\
abla\cdot(\mu\mathbf{H})~{}=~{}0. \tag{3}\]  
\[\
abla\times\mathbf{E}~{}=~{}j\omega\mu\mathbf{H}. \tag{4}\]  
\[\
abla\times\mathbf{H}~{}=~{}-j\omega\varepsilon E. \tag{5}\]  
Figure 2: **(a)** Schematic illustration of the proposed ABM, which is constructed by etching one straight-walled groove per period on a planar hard surface.