## 2 Materials and Methods  
In the design process, the P-B phase shift was adopted to generate the abrupt phase. The amplitude and phase were related to the materials to construct the unit cell. As introduced above, the permittivity of GST will be changed greatly when its state is tuned between the amorphous state and crystalline state. Therefore, the patterned GST can be used directly to construct as the tunable meta-atoms instead of using it as a dielectric layer. This design method has decreased the complexity of the device further, making it more suitable for the integration of nano-devices. The structure arrangement diagram of the unit cell is shown in Figure 1a. The green blocks represent the patterned GST. Figure 1b shows one of the unit cells. The patterned GST is settled on the substrate SiO\({}_{2}\), and a thin indium tin oxide (ITO) layer, sandwiched between the substrate and GST, is used as an electrical heating layer to tune the phase state of GST. Figure 1c is the top view of the unit cell. The patterned GST forms an angle \(\theta\) with the \(x\) axis, which can be used to generate an abrupt phase. When a left/right circularly polarization (LCP/RCP) incident light propagates along the \(+z\) direction, the output light field can be simply expressed as:  
\[\begin{array}{ll}\begin{array}{ll}\begin{array}{ll}\begin{array}{ll} \begin{array}{ll  
However, the parameters of the unit cell need to be designed carefully to satisfy the tuning function. When the patterned GST is in an amorphous state, the conversion efficiency from LCP to RCP should be high enough and the phase modulation should strictly satisfy the linear relationship of \(\varphi=2\theta\). When the patterned GST is in a crystalline state, the conversion efficiency from LCP to RCP should be low enough. Due to the low conversion efficiency, the corresponding phase modulation relations can be ignored. The simulation of the unit cell was performed using the finite-element method in the frequency domain with the unit cell boundary in the \(xy\) direction and the open boundary in the \(z\) direction. In the simulation, the incident LCP was settled propagating along the \(+z\) direction through the substrate. The GST data were obtained from the experimental data in [37], where the dielectric function \(\varepsilon(\omega)\) was investigated by infrared spectroscopy and spectroscopic ellipsometry. The big differences of \(\varepsilon(\omega)\) between amorphous GST and crystalline GST at the wavelength of 4-5.4 \(\upmu\)m are shown in Figure 2. Considering that the parameters p, l, w, and h are designed to determine the conversion efficiency, we needed to sweep the parameters, picking up the parameters where the conversion efficiency was high enough. Due to the four freedom, the data are complicated. Here, only the conversion efficiency versus l and w is depicted in Figure 3a, when p was fixed as 3 \(\upmu\)m and h is fixed as 2.5 \(\upmu\)m. Finally, the parameters of the unit cell were determined as \(p=3\)\(\upmu\)m, \(l=1\)\(\upmu\)m, \(w=0.7\)\(\upmu\)m, and \(h=2.5\)\(\upmu\)m. Figure 3b shows the relationship between the conversion efficiency from LCP to RCP and wavelength when GST is in amorphous state and crystalline state. The conversion efficiency exceeded 75% in the whole range from 4.5 \(\upmu\)m to 5.2 \(\upmu\)m when GST was in an amorphous state, while it was less than 10% for crystalline GST. At the designed wavelength of 4.6 \(\upmu\)m, when the angle \(\theta\) changed from 0 to \(\pi\), the phase modulation of the unit cell for both GST states are shown in Figure 3c. While the GST was in an amorphous state, the abrupt phase covered the whole 2\(\pi\) range linearly. Figure 3d shows the relationship between the conversion efficiency and rotating angle \(\theta\). When \(\theta=0\), the conversion efficiency for amorphous GST was as high as 89.1%, while the conversion efficiency for crystalline GST was only 9.2%.