## 3 Results and Discussion
(**a**) A supercell consists of six cells, (**b**) the designed metasurface constituted of \(3\times 2\) supercells. Based on the results above, we additionally design a polarization-dependent focusing metasurface to focus on the \(x\)- and \(y\)-polarized waves at different positions. By employing the proposed structure, metasurfaces with a hyperboloidal phase distribution can tailor the reflective wavefront and convert the spherical wave to a plane wave, or focus the plane wave at a point. It is widely known that the hyperboloidal phase distribution based on [PERSON]'s principle can be described as follows [31]:  
\[\varphi(x,y)=\frac{2\pi}{\lambda_{0}}(\sqrt{x^{2}+y^{2}+L^{2}}-L)+\varphi_{0} \tag{2}\]  
where \(\lambda_{0}\) and \(L\) are the incident wavelength and focal length, respectively; \(\varphi(x,y)\) and \(\varphi_{0}\) are the phase in the arbitrary position (\(x\), \(y\)) and the original phase on the metasurface, respectively. To achieve polarization-dependent focusing, different hyperboloidal phase distributions need to be arranged for \(x\)- and \(y\)-polarized waves, as shown below [31]. \[\varphi_{\mathrm{x}}(x,y)=\frac{2\pi}{\lambda}(\sqrt{x^{2}+y^{2}+L^{2}}-L)+ \xi_{1}\cdot x+\varphi_{0} \tag{3}\]  
\[\varphi_{\mathrm{y}}(x,y)=\frac{2\pi}{\lambda}(\sqrt{x^{2}+y^{2}+L^{2}}-L)+ \xi_{2}\cdot y+\varphi_{0} \tag{4}\]  
where \(\xi_{1}\) and \(\xi_{2}\) are extra linear-phase gradient responses along the \(x\)- and \(y\)-directions, to enable anomalous focusing for incident \(x\)- and \(y\)-polarized waves, respectively. \(\varphi_{\mathrm{x}}(x,y)\) and \(\varphi_{\mathrm{y}}(x,y)\) are the rearranged phase distributions at the metasurface for the incident \(x\)- and \(y\)-polarized waves, respectively. From Equations (5) and (6), if the incidence is a plane wave with both the \(x\)/\(y\)-components, it can be converted to spherical waves and focused at the target spot for \(x\)/\(y\)-polarization. Based on the designing principle, we artificially engineer different hyperbolical phase profiles for \(x\)/\(y\)-polarized waves on the metasurface by 19 \(\times\) 19 cells, as schematically presented in Figure 6a. In addition, the phase distributions for the \(x\)- and \(y\)-polarized waves is plotted in Figure 6b,c, respectively. In these designs, we set the focal length \(L=40\) mm, \(\varphi_{0}=-345.6^{\circ}\), and the incident wavelength \(\lambda=20\) mm. The phase gradient \(\xi_{1}\) and \(\xi_{2}\) are set as \(-\pi/(3p)\) and \(\pi/(3p)\), respectively. Thus, the position of the designed focusing metasurface is \((22,0)\) and \((0,-22)\) mm for the \(x\)- and \(y\)-polarized waves, respectively. To investigate the polarization-dependent focusing ability, we perform numerical simulations by using CST Microwave Studio, where the absorbing boundary condition was applied at all boundaries. Figure 7a,b shows the near-field distributions of electric intensity in the \(x\)-\(y\) plane for the \(x\)- and \(y\)-polarized waves, respectively, demonstrating a perfect polarization-dependent focusing effect. In addition, Figure 7c,d are the near-field distributions of electric intensity in the \(x\)-\(z\) and \(y\)-\(z\) planes for the \(x\)- and \(y\)-polarized waves, respectively. The position of the deflective focusing spot in the \(x\)-\(y\) plane (\(z=40\) mm) is approximately \((29,0)\) and \((0,-29)\) mm for the \(x\)- and \(y\)-polarized waves, respectively. The difference in the focusing position between the theory and simulation is attributed to a phase convergence of merely \(320^{\circ}\) and the coupling effect between neighboring unit cells. To some  
Figure 5: The near-field distribution of electric field for (**a**) \(x\)- and (**b**) \(y\)-polarized waves. (**c**) Return loss of the \(x\)- and \(y\)-polarized waves. extent, the simulation results agree well with the theoretical result.