# The Inverse Optimization of an Optical Lithographic Source with a Hybrid Genetic Algorithm
## 3 Simulations  
In this section, several simulation results involving aerial images with optimized sources are provided to verify the superior performance in optical lithography after utilizing the pixelated SO method based on the proposed hybrid GA approach. An imaging model for optical lithography is established, and this was based on a 193 nm immersion lithography system in the 45 nm node. In this model, the numerical aperture of the projection objective was 1.25. The annular illumination mode was employed for the imaging source in the initial stage to provide the initial aerial image, where \(\sigma_{out}=0.98\) and \(\sigma_{in}=0.65\). Furthermore, this pixelated annular source was set as a \(41\times 41\) matrix. To generate the resistance pattern, a sigmoid function was utilized in the optimization model, where \(\alpha=85\) and \(t_{r}=0.19\). Figure 4 illustrates the initial source and mask employed to accomplish the imaging process for optical lithography. For Figure 4a-c, the value of the pixels in the white region was set to 1; otherwise, their value was 0. To ensure the continuous variation of the valid pixels and to reduce the dispersion degree of the annular source in the optimization process, the blurred source shown in Figure 4b could be generated by performing a fuzzy operation on the binarized source in Figure 4a. To verify the optimization performance of the pixelated SO based on the proposed method, two patterns were employed as input feature patterns to form an aerial image for optical lithography; these were the horizontal block pattern (#M1) and the periodic optical grating pattern (#M2). They could be represented as real matrices with the size \(N\times N\), where \(N=521\). In these simulations, the proposed pixelated-SO-based method was carried out in three runs to evaluate its optimization performance. In each simulation, there were 150 iterations to optimize the intensity distribution of the lithographic source for the horizontal block pattern (#M1) and the periodic optical grating pattern (#M2). The final lithographic source, which was defined as the optimal source, was used as the input condition to complete the lithographic imaging process and to attain the resistance pattern. These simulation results are shown in Figure 5. Rows 1 and 2 present the source optimization results and the resistance patterns from three runs. The intensity distributions of the optimal sources for (#M1) and (#M2) are, respectively, displayed in the first and third columns of Figure 5. Their resistance patterns are, respectively, shown in the second and fourth columns, and they were obtained by utilizing the optimal source. In these optimized sources, the intensity distributions for #M1 were closer to a vertical quadrupole illumination mode. This is because, in the vertical direction, the pattern had a similar periodic grating structure to that of the vertical bipolar illumination mode. Horizontally, the middle area lacked pattern information for the horizontal illumination mode. For #M2, a 45-degree quadrupole illumination mode was more necessary for ensuring better image quality in the imaging process for optical lithography due to the periodic horizontal grating structure of the array. Figure 6 shows the simulation results when using two other methods with 150 iterations; these were a hybrid GA based on TS (TS-GA) and a method that combined the GA with the optimization strategy of PSO (PSO-GA). Figure 6a,e show the final source shapes for #M1 when utilizing TS-GA and PSO-GA, respectively. Similarly, Figure 6c,g were, respectively, achieved by using these two methods for #M2. Compared with the simulation results for the proposed method, the results shown in Figure 6 are not satisfactory. Although the tendencies of the source shapes were similar, there was a significant decrease in the intensity value for TS-GA and PSO-GA. Furthermore, it was obvious that the quality of the resistance patterns was not high enough when these optimal sources were employed as the lithographic imaging conditions.