# Integrated Deep Ultraviolet Doublet Metalens for Projection Imaging
## 2 Materials and Methods
### Analysis of Flatness of Outgoing Light  
In recent years, the design of doublet metalenses [6; 61; 62] has also been proposed. To reduce the simulation data, in this paper, we choose the diameters of the two metalenses as \(R_{1}\) = 3 \(\upmu\)m, \(R_{2}\) = 0.75 \(\upmu\)m, and the focal lengths as \(f_{1}\) = 6 \(\upmu\)m and \(f_{2}\) = 1.5 \(\upmu\)m. Among them, Figure 5A is the intensity map of the \(x\)-\(z\) plane. The shaded part of the box is the position of the two metalenses. The dashed line indicates the focal plane position of metalens L\({}_{1}\), located at a distance of 6 \(\upmu\)m from metalens L\({}_{1}\). It is precisely focused on the focal position of metalens L\({}_{2}\) to obtain the outgoing parallel light. Figure 5B shows the phase distribution on the \(x\)-\(z\) plane. A simulation material with total reflection properties is added to the periphery of the two metalenses as a diaphragm to limit stray light. The reflected light is messy and affects the substrate's phase distribution. However, according to Figure 5B, we can observe that after passing through metalens L\({}_{2}\), the phase distribution of the outgoing light is flat. Amplifying and analyzing the phase distribution in the purple box in Figure 5B, the phase of the outgoing light is very flat and close to the phase of parallel light. Based on the above analysis, we quantitatively analyze the flatness of the emitted light. We select the phase of the outgoing light propagation distance dz of 1, 1.5, 2, 2.5, 3, 3.5, 4 \(\upmu\)m, and dz is the distance after metalens L\({}_{2}\). We define the difference between the actual and ideal phases at different propagation distances after the integrated doublet metalens as the wave phase difference (WPD). As shown in Figure 6A, the phase fluctuates slightly when the propagation distance is relatively short. Nevertheless, after the propagation distance reaches 2 \(\upmu\)m, the propagation phase tends to be flat, and the maximum fluctuation is controlled within 0.03 times 2\(\upmu\)r (the phase of one wavelength). The average value (AV), variance, and standard deviation of the phase difference curve at different propagation distances are analyzed in Figure 6B. The standard deviation and variance can reflect the degree of dispersion of each phase difference curve dataset. When the propagation distance dz is greater than 2 \(\upmu\)m, the flatness of the emitted light is better, and the average value is controlled within 0.022 times 2\(\upmu\)r.  
Table 1 lists the AV of the wave phase difference and the peak-valley (PV) value under different propagation distances. The PV value is the difference between the maximum and minimum values in the phase difference curve data. When the propagation distance reaches 2 \(\upmu\)m, we obtain the wave phase difference with the minimum average value, and its variance and standard deviation are also considered the smallest. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**dx (\(\upmu\)m)** & **1** & **1.5** & **2** & **2.5** & **3** & **3.5** & **4** \\ \hline AV (2\(\upmu\)r) & 0.08588 & 0.0872 & 0.00995 & 0.01426 & 0.01818 & 0.01991 & 0.02225 \\ PV (2\(\upmu\)r) & 0.15311 & 0.16616 & 0.02268 & 0.02145 & 0.02411 & 0.02465 & 0.03073 \\ \hline \end{tabular}
\end{table} 
Table 1: Average value and peak-valley value of wave difference after different propagation distances. Figure 5: The simulation results of the integrated doublet metalens. (**A**) The energy distribution when the incident light is parallel (\(x\)-\(z\) plane). (**B**) The phase distribution when the incident light is parallel (\(x\)-\(z\) plane). The right figure shows the outgoing light’s phase distribution after the integrated doublet metalens (\(x\)-\(z\) plane). At the same time, we analyze the efficiency of the integrated doublet metalens. The transmittance of the integrated deep ultraviolet doublet metalens is 46.21%.