# Integrated Deep Ultraviolet Doublet Metalens for Projection Imaging
## 2 Materials and Methods  
The principle of the metalens is shown in Figure 1. The propagation phase structure is used to design a deep ultraviolet all-dielectric metalens. The unit structure of the SiO\({}_{2}\) cylinder is placed on the SiO\({}_{2}\) substrate. SiO\({}_{2}\)[51] has higher transmission efficiency and lower absorption in the deep ultraviolet band, which can improve the performance of the metalens. It is common for a cylinder [52; 53; 54; 55] to be used as the unit structure for a metalens. By adjusting the geometric parameters, such as the radius and height of the cylinder, phase coverage of 2\(\pi\) can be obtained to achieve the arbitrary control of the wavefront. The finite difference time domain (FDTD) method is used to simulate the phase control element and the phase and intensity of the metalens. Theoretical simulations are conducted on the transmittance and phase of the phase-regulating unit, as well as on the transmittance and projection intensity of the metalens and doublet metalens. The simulation of the element structure uses periodic boundary conditions along the \(x\) and \(y\) axes and a perfect matching layer along the \(z\) axis. The refractive index of SiO\({}_{2}\) comes from the Palik manual, emitting plane waves from the bottom of the metasurface. The parameters of the unit structure are shown in Figure 2. By adjusting the geometric parameters such as the unit size and height, 2\(\pi\) phase coverage can be obtained, and arbitrary control of the light field can be achieved. To realize the phase control under the design wavelength \(\lambda\) = 193 nm, we choose the SiO\({}_{2}\) material with high transmittance as the substrate and unit structure pillar. We select the cylindrical structure to adjust and scan its radius and height. Based on observations, it can be seen that within a radius of 20-70 nm, the ability to adjust the phase becomes more robust with an increase in height. Therefore, selecting a height that offers higher transmittance overall is advisable. After careful consideration, choosing a height of 550 nm and a radius between 20 and 65 nm is recommended. The phase delay covers a 2\(\pi\) interval, the overall transmittance is above 84%, and the average transmittance reaches 92%. In the design of ultraviolet metalenses [56; 57; 58; 59], the element structure's amplitude and phase control results are quite acceptable. The full-wave simulations (E, H, and P) of meta-units with different radii (30, 40, 50, and 60 nm) are shown in Figure 2C. In this paper, the propagation phase is controlled by changing the radius of the SiO\({}_{2}\) cylinder. To achieve focus, the phase of the deep ultraviolet all-dielectric metalens needs to satisfy the following equations:  
\[\Phi=\frac{2\pi}{\lambda}\bigg{(}f-\sqrt{f^{2}+x^{2}+y^{2}}\bigg{)} \tag{1}\]  
Among them, \(\lambda\) is the wavelength of the incident light, \(f\) is the focal length, and \((x,y)\) is the position coordinate on the transparent substrate. When the exit light interface is located in a medium with a refractive index of n, the phase profile must satisfy the following:  
\[\Phi=n\times\frac{2\pi}{\lambda}\bigg{(}f-\sqrt{f^{2}+x^{2}+y^{2}}\bigg{)} \tag{2}\]  
The phase shift introduced by the unit structure can be expressed as  
\[\Phi=\frac{2\pi}{\lambda}nH \tag{3}\]  
Figure 1: (**A**) A schematic illustration of a deep ultraviolet integrated doublet metalens. (**B**) Side view of deep ultraviolet integrated doublet metalens. (**C**) Side view, top view, and three-dimensional view of the unit structure, respectively. The height of the cylinder is h, the radius is r, and the unit structure period is D. (**D**) The relationship between the phase control of the unit structure and the radius. where \(H\) is the unit structure height.