# Integrated Deep Ultraviolet Doublet Metalens for Projection Imaging
## 3 Results  
To further verify the possibility of applying the design to deep ultraviolet lithography, we set up a \"mask\" in the simulation. We added light-shielding materials in front of metalens L\({}_{1}\) (in front of the overall structure) to observe the imaging conditions. For better verification, we chose the \"F\" pattern, as shown in Figure 7. The \"F\" pattern has asymmetry, which can be used to analyze the imaging capability of the doublet metalens system more clearly. Due to the limitations in computer simulation capabilities, the overall size of the doublet metalens that we designed was small, so the diffraction effect was unavoidable. To reduce the influence of diffraction as much as possible, we set its position close to the front side of metalens L\({}_{1}\). As shown in Figure 7A, we set a = b = 1 \(\upmu\)m of the \"F\" pattern for simulation. Figure 8 displays the various intensity patterns at different distances behind metalens L\({}_{2}\): 1.5, 2, 2.5, and 3 \(\upmu\)m. According to the simulation results, we can observe a more explicit \"F\" pattern in the inverted upside-down images. Figure 6: Analysis of the parallelism of the emitted light. (**A**) The WPD at different propagation distances after the integrated doublet metalens. (**B**) The average, standard deviation, and variance of the WPD at different propagation distances after the integrated doublet metalens. However, simultaneously, the imaging also tends to increase as the distance increases. This phenomenon does not conform to the imaging law of the afocal system that we designed. We believe that it is due to the small overall size of the doublet metalens, which has the diffraction effect. Analyzing the propagation distance, we suggest that the diffraction type is Fresnel diffraction. To verify our judgment, we perform an inverse diffraction calculation. Using the intensity simulation results at 3 \(\upmu\)m as the original image, we calculate the intensity pattern before 3 \(\upmu\)m at different locations only affected by diffraction. The transfer function of Fresnel diffraction is  
\[H=\exp(ikz)\exp\Bigl{[}-i\pi\lambda z\Bigl{(}f_{x}{}^{2}+f_{y}{}^{2}\Bigr{)} \Bigr{]} \tag{8}\]  
Among them, \(k=2\pi/\lambda\) is the wavenumber, \(z\) represents the distance between the observation screen and the diffraction screen, and \(f_{x}\) and \(f_{y}\) are the spatial frequencies of the complex amplitude distribution in the x and y directions. Then, the transfer function of Fresnel inverse diffraction is  
\[H_{1}=\exp(-ikz)\exp\Bigl{[}-i\pi\lambda z\Bigl{(}f_{x}{}^{2}+f_{y}{}^{2} \Bigr{)}\Bigr{]} \tag{9}\]  
By applying the abovementioned formula, we can calculate the complex amplitude using the intensity and phase data at a distance of 3 \(\upmu\)m from L\({}_{2}\). This can be used to perform inverse diffraction, which allows us to obtain the intensity distribution at different distances from the source plane. The resulting distribution is illustrated in Figure 9. We compare the actual simulation results at different positions in Figure 8 with the results obtained by the inverse diffraction calculation in Figure 9. It can be seen that the outer frame size of the pattern is the same for the same propagation distance. This indicates that the gradual increase in the pattern with increasing propagation distance is due to the diffraction effect. The diffraction effect will be significantly reduced when the size of the two metalenses is large enough compared with the working wavelength. However, the consideration of how to further reduce the influence of diffraction by optimizing the design is essential for our next step. Meanwhile, based on the pattern calculated by the inverse diffraction method at 0 \(\upmu\)m, we can find that the overall pattern size has reached the expected ratio of 4:1. However, according to the analysis of Figures 5 and 6, it can be seen that the phase of the near field is uneven to an extent. As a result, the pattern after the inverse diffraction calculation is also somewhat unclear.