# An Optically Transparent Metasurface for Microwave Amplitude-Phase Manipulation
## 2 Results
### Multiplane Complex-Amplitude Hologram  
In this section, as schematically shown in Figure 4a, the designed meta-atom featuring continuous amplitude control and eight-level phase control at 16 GHz is utilized to generate a spatially multiplexed complex-amplitude hologram that can reconstruct different holographic images of \"WH\" and \"UT\" with \(z_{1}\) = 130 mm and \(z_{2}\) = 270 mm away from the metasurface, respectively. In the microwave region, the Rayleigh-Sommerfeld diffraction formula is adopted to calculate the near-field distribution diffracted by the metasurface, which can be expressed as [39; 40]  
\[U(x,y,z)=\frac{1}{i\lambda}\iint_{S_{0}}U(x_{0},y_{0},0)\cos(\delta)\frac{\exp (ik|\mathbf{r}|)}{|\mathbf{r}|}dS_{0} \tag{4}\]  
where \(\lambda\) and \(k\) are the wavelength and wavevector, respectively, \(U(x,y,z)\) and \(U(x_{0},y_{0},0)\) represent the complex-amplitude distributions on the imaging plane \(S\) and metasurface plane \(S_{0}\), respectively, \(|\mathbf{r}|=\sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+z^{2}}\) represents the distance between points (\(x\), \(y\), \(z\)) and (\(x_{0}\), \(y_{0}\), \(0\)), and \(\cos(\delta)=z/\left|\mathbf{r}\right|\) is the inclination factor. By considering the inverse propagation operator, the field distribution on the metasurface plane can be calculated as  
\[U(x_{0},y_{0},0)=\frac{1}{i\lambda}\iint_{S}U(x,y,z)\cos(\delta)\frac{\exp(-ik |\mathbf{r}|)}{|\mathbf{r}|}dS \tag{5}\]  
Using Equations (4) and (5), the complex-amplitude distributions at a single imaging distance can be obtained. Furthermore, the spatially multiplexed technology is utilized  
Figure 3: (**a,b**) Simulated amplitude and phase of the cross-polarized reflection coefficient \(r_{yx}\) with different rotation angles \(\beta\) ranging from 0–180\({}^{\circ}\). Simulated amplitude and phase of \(r_{yx}\) with different arm lengths \(l\) ranging from 2–6.4 mm when (**c**) \(\beta\) = 45\({}^{\circ}\) and (**d**) \(\beta\) = 135\({}^{\circ}\). (**e,f**) Simulated amplitude and phase of \(r_{yx}\) as a function of \(l\) and \(\beta\). All results are obtained at 16 GHz. to enhance the information capacity, which can be obtained by using the superposition theorem denoted by [41; 42]  
\[U_{\text{tot}}(x_{0},y_{0},0)=\sum_{i=1}^{N}U_{i}(x_{0},y_{0},0) \tag{6}\]  
where \(U_{\text{tot}}(x_{0},y_{0},0)\) represents the required total field distribution on the metasurface. Here we choose the letters \"WH\" and \"UT\" as the target images to be separately generated in two imaging planes of 130 mm and 270 mm apart from the metasurface, and the target images and metasurface contain 45 \(\times\) 45 number of pixels covering an area of \(360\times 360\) mm\({}^{2}\). Calculated from Equations (4)-(6), the normalized continuous amplitude distribution and the discretized eight-level phase distribution of the metasurface are obtained, as shown in Figure 4b,c, respectively. By mapping the meta-atoms with different arm lengths \(l\) and rotation angles \(\beta\) to the calculated amplitude-phase profile accurately, the final arrangement of the entire metasurface can be attained. Following the above design process, the multiplane complex-amplitude metasurface hologram was fabricated and experimentally verified. The top layer of the metal-frame-based H-shaped resonator and the bottom metallic mesh layer were composed of 18 \(\upmu\)m thick copper grids with an electrical conductivity of 5.8 \(\times\) 10\({}^{7}\) S/m, which were deposited on 50 \(\upmu\)m thick PET films and fabricated by the mature and low-cost FPC technology. The microscopic images of the top metal frame layer and the bottom metallic mesh layer are shown in Figure 5b,c, demonstrating a mesh linewidth of approximately 45 \(\upmu\)m, which is slightly narrower than the designed value (50 \(\upmu\)m). From the previous analyses in Figure 2, such a slight discrepancy hardly affects the polarization conversion efficiency.