## 2 Materials and Methods
When the angle changed from 0 to \(\pi\), the conversion efficiency hardly changed for both GST states, which proves the high stability of the unit cell. In contrast, the co-polarization efficiency was considerably high (94%) when GST was in a crystalline state, while the co-polarization efficiency was only 14% for the amorphous state, as shown in Figure 3e. Figure 3f proves that the co-polarization light does have any phase modulation, as the theory predicted. This means that the incident light can pass through the unit cell just like passing through a transparent dielectric when GST is in a crystalline state. All the simulation results showed that the designed unit cell can very well meet the requirements of tunable function. Then, the designed unit cell needs to be arranged appropriately to satisfy the hyperbolodial phase profile Equation (1) with \(\theta(x,y,f)=~{}\varphi(x,y,f)/2\). In our design, the radius of the single metalens was 30 \(\upmu\)m, and the focal length was 60 \(\upmu\)m with the numerical aperture (NA) of 0.45. Next, a single metalens was considered as the secondary unit cell to compose the metalens array. For simplicity, a \(3\times 3\) tunable metalens array was designed as shown in Figure 4. The green blocks represent the amorphous state GST, which can convert the incident LCP light into RCP light, then focus it at the focal plane. The red blocks represent the crystalline state GST, which could hardly convert the polarization of the incident light, thus no light was focused at this area. The proposed structure can be fabricated using electron beam lithography (EBL) and atomic layer deposition (ALD) methods. The combination of two methods can ensure that the structure has better conformality. The ITO layer can be sputtered on the substrate. As a thin film of transparent electrode, it has sufficient conductivity to allow the electrical Joule heating of the GST, so the state of the GST can be changed reversibly by applying an appropriate electrical current pulse (equivalent to \(\sim\)0.3 V) [29] through the conductive layer; at the same time, it has little influence on the phase modulation. On the other hand, a high-repetition-rate femtosecond laser is also a good choice to switch the proposed device [23]. Figure 4: The diagram of the tunable metalens array. The green blocks represent amorphous GST, which can focus the incident light, and the red blocks represent crystalline GST, which cannot focus the incident light. By actively tuning the phase state of GST, the performance of the metalens array can be controlled. Figure 3: (**a**) Conversion efficiency versus l and w when \(p\) = 3 \(\upmu\)m and h = 2.5 \(\upmu\)m. (**b**) The relationship between conversion efficiency and wavelength when GST stays in an amorphous state (green line) and crystalline state (red line). (**c**) The relationship between phase modulation and rotating angle \(\theta\). The phase modulation linearly covers the whole 2\(\uppi\) range when the angle rotates from 0 to 180 degrees. (**d**) The relationship between conversion efficiency and rotating angle \(\theta\). (**e**) The relationship between co-polarization efficiency and rotating angle \(\theta\). (**f**) The relationship between co-polarization phase and rotating angle \(\theta\). Figure 2: (**a**) The real part of permittivity for amorphous GST and crystalline GST. (**b**) The imaginary part of permittivity for amorphous GST and crystalline GST.