# Metalens Antenna Based on an Ultrathin Dual-Polarized Double-Layer Huygens' Metasurface
## 2 Theoretical Analysis and Design of Huygens' Meta-Atoms  
The design of Huygens' metasurface is essentially an application of the Huygens' principle. Each meta-atom on the metasurface acts like a secondary wave source in Huygens' principle, alters the local wavefront, and thereby achieves precise control over the overall wavefront of the propagation waves. The schematic diagram of the Huygens' metasurface is illustrated in Figure 1. The domain is divided into two regions by the metasurface. The fields in region 1 are the incident waves \((\mathbf{E}_{\mathrm{r}}\mathbf{H}_{\mathrm{i}})\) and the reflected waves \((\mathbf{E}_{\mathrm{r}},\mathbf{H}_{\mathrm{r}})\), and the fields in region 2 are the transmission waves \((\mathbf{E}_{\mathrm{r}}\mathbf{H}_{\mathrm{i}})\). The fields in both regions induce the electric surface current densities \(\mathbf{J}_{\mathrm{s}}\) and the magnetic surface current densities \(\mathbf{M}_{\mathrm{s}}\) on the metasurface, which should satisfy the following expressions:  
\[\hat{z}\times(\mathbf{H}_{2}-\mathbf{H}_{1})=\mathbf{J}_{\mathrm{s}}=\tilde{ Y}_{\mathrm{es}}\cdot\mathbf{E}_{\mathrm{av}}^{\mathrm{tan}} \tag{1a}\] \[-\hat{z}\times(\mathbf{E}_{2}-\mathbf{E}_{1})=\mathbf{M}_{\mathrm{s}}=\tilde{ Z}_{\mathrm{ms}}\cdot\mathbf{H}_{\mathrm{av}}^{\mathrm{tan}} \tag{1b}\]  
where the subscripts 1 and 2 denote the field on the metasurface in region 1 and region 2, respectively. Here, \(\mathbf{\Phi}_{1}=\mathbf{\Phi}_{\mathrm{r}}+\mathbf{\Phi}_{\mathrm{i}}\) and \(\mathbf{\Phi}_{2}=\mathbf{\Phi}_{\mathrm{r}}\), \(\mathbf{\Phi}\) represents \(\mathbf{E}\) or \(\mathbf{H}\). \(\mathbf{E}_{\mathrm{av}}^{\mathrm{tan}}\) and \(\mathbf{H}_{\mathrm{av}}^{\mathrm{tan}}\) are the surface average tangential electric and magnetic fields. The electric surface admittance tensor \(\tilde{Y}_{\mathrm{es}}\) and the magnetic surface impedance tensor \(\tilde{Z}_{\mathrm{ms}}\) can fully characterize the material properties of the metasurface. Depending on the given incident polarization, the tensors \(\vec{Y}_{\text{es}}\) and \(\vec{Z}_{\text{ms}}\) are simplified to scalar quantities. Accordingly, they can be extracted from the transmission \((T)\) and reflection \((R)\) coefficients [13]:  
\[Y_{\text{e}}^{i}=\eta Y_{\text{es}}^{i}=\frac{2(1-T^{i}-R^{i})}{1+T^{i}+R^{i}}, \quad Z_{\text{m}}^{i}=\frac{Z_{\text{ms}}^{i}}{\eta}=\frac{2(1-T^{i}+R^{i})}{ 1+T^{i}-R^{i}} \tag{2}\]  
where \(Y_{\text{e}}\) and \(Z_{\text{m}}\) are the normalized admittance and impedance, respectively. The superscript \(i\) represents the polarization direction of the incident wave, which is denoted as \(x\) or \(y\). \(\eta\) represents the free-space impedance. When designing a transmissive metasurface, it is generally desirable to achieve zero reflection \((R=0)\) and near-unity transmission \((|T|=1)\). Research demonstrates that when the real parts of \(Y_{\text{e}}\) and \(Z_{\text{m}}\) are zero and their imaginary parts are equal, the Huygens' resonance is excited, which will achieve the near-unity [13]. Figure 2 shows the structure configuration of the designed Huygens' meta-atom. The meta-atom consists of double-layer metallic patches on each side of a dielectric substrate. The top and the bottom metallic patches have identical geometric structures, comprising a double-cross shape and a corner-ring shape. The double-cross shape consists of two parallel horizontal rectangles intersecting with two parallel vertical rectangles. The parallel rectangles have a length of \(l=3.05\) mm, with the spacing of \(m=0.6\) mm. On the other hand, the size of the corner-ring-shaped patches is \(s=4.6\) mm, with gaps between adjacent corner-ring patches. The gaps along different directions are denoted as \(g_{x}\) and \(g_{y}\), respectively. Both the double-cross-shaped patches and the corner-ring-shaped patches use the same width of \(w=0.2\) mm and thickness of \(0.018\) mm. The metallic patches are fabricated using copper (electrical conductivity \(\sigma=5.8\times 10^{7}\) S/m) with a thickness of \(0.018\) mm.