# Infrared All-Dielectric Metasurface Beam Splitter Based on Transflective Structures
## 2 Structural Design and Simulation  
The basic working principle of metasurfaces is to introduce phase discontinuity along the interface by designing the geometry shape of unit cells and arranging them according to certain rules, which can control the wavefronts of the reflected or transmitted beams. When the incident light field transmits through the metasurface, it will inspire the Mie resonance of the all-dielectric unit cell, which will greatly change the effective refractive index of the unit and lead to a certain phase delay. The process can be described by the truncated waveguide effect of the unit cell [38; 39]:  
\[\phi_{tp}=k_{0}n_{eff}H \tag{1}\]  
where \(\phi_{tp}\) is the phase shift introduced by the transmission phase; \(k_{0}\) is the wave vector; \(n_{eff}\) is the effective refractive index of the metasurface unit waveguide; and \(H\) is the metasurface unit height. According to generalized Snell's law, the departure angles of outgoing beams at two ends can be expressed by the following equation [40]:  
\[n_{t}\sin(\theta_{1})=n_{r}\sin(\theta_{2})=n_{t}\sin(\theta_{i})+\frac{\lambda _{0}}{2\pi}\frac{d\phi_{tp}}{dx} \tag{2}\]  
where \(\theta_{i}\) is the incident angle; \(n_{t}\), \(n_{r}\), and \(n_{i}\) are the refractive indices of the transmissive, reflective, and incident media, respectively; and \(d\phi_{tp}\) is the phase change. When the beam is normal incident on the metasurface in air, and due to the transflective structure of the metasurface, the \(\theta_{1}\) = 0, \(n_{t}=n_{r}\) and Equations (1) and (2) can be written as:  
\[\sin(\theta_{1})=\sin(\theta_{2})=\frac{\tau}{n_{t}P}\frac{\lambda_{0}}{2\pi} \tag{3}\]  
where \(\tau=P\times\frac{d\theta_{p}}{dx}\) is the phase interval and \(P\) is the spacing period. The splitting ratio of the metasurface BS can be defined as:  
\[S_{R}=\frac{T}{R} \tag{4}\]  
Here \(T\) and \(R\) represent the transmittance and reflectivity of the transflective metasurface BS. Based on the above theory, each unit cell on the metasurface is built as a circular Si nanocolumn on a SiO\({}_{2}\) (\(n\) = 1.44) substrate with a thickness of 6 \(\upmu\)m. Si was chosen as the nanocolumn material because of its high refractive index (\(\sim\)3.48) and low absorption coefficient (4.44 \(\times\) 10\({}^{-16}\)) at the 1.55 \(\upmu\)m wavelength, which give a substantial refractive index contrast with the background and hence improve the ability of phase modulation [41]. The period P of the unit cells is set to 1 \(\upmu\)m, the height H of the nanocolumn is set to 3.2 \(\upmu\)m, and the final metasurface beam splitter beam-splitting schematic is illustrated in Figure 1,b. We chose to vary the equivalent refractive index by changing the radius of the nanocolumn to accomplish the needed phase slope design for deflected beam splitting since the height of the unit cell is constant. Figure 2a depicts the variation of phase and transmittance with the radius of the nanocolumn. Due to the relatively stable variation of phase and transmittance, the range from 0.25 \(\upmu\)m to 0.4 \(\upmu\)m is selected as the radius variation range of the nanocolumn. Since the Fabry-Perot resonance generated by the unit cells at a specific radius will cause the interference extinction phenomenon of the outgoing light, it can cause significant transmittance troughs, and since the nanocolumns have the same period but different radii, the transmittance of two adjacent nanocolumns also changes with the distance as shown in Figure 2b. They both help to reduce the overall transmittance of the metasurface. The main approach to achieving a transflective metasurface is to reduce the overall transmittance of the metasurface without significantly impacting its performance, which is achieved by selecting a small number of specific low transmittance unit cell size structures and controlling the distance between unit cells.