# Multi-Objective Defocus Robust Source and Mask Optimization Using Sensitive Penalty
## 3 Simulation Results and Discussion
### Optimization Results and Analysis  
In order to illustrate the negative influence of defocus, Figure 3a-f shows the optimization proceeded by initial SMO which merely operated at the best focus plane [7], and the evaluation of the printed image was under different defocus planes. Figure 3a,b show the optimized source for initial SMO and optimized mask for initial SMO, respectively. Figure 3c shows the printed image at the best focus plane, and Figure 3d-f shows the printed image under 50 nm, 70 nm, and 100 nm defocus, respectively. It clearly shows that the PAE increased extremely with an increase of defocus, proving that the initial SMO could not compensate the defocus distortion because the cost function was not involved in the defocus term. However, the defocusing error inevitably existed in the actual lithography process, thereby it was necessary to gain a better and more robust defocusing via DRSMO. Figure 2: Two test patterns used in the simulation: (**a**) Test pattern 1; (**b**) Test pattern 2. The PW calculation positions are marked at the yellow lines. Similarly, Figure 3g-l illustrates [PERSON]'s [14] SMO method which merely operates at an assigned defocus plane (100 nm defocusing). In this method, the established cost function can be formulated as the weight sum of the nominal term and defocus term  
\[F=\|\overline{Z}-Z(0)\|_{2}^{2}+\omega\|\overline{Z}-Z(100)\|_{2}^{2}, \tag{12}\]  
Figure 3g,h show the optimized source and mask, respectively. Figure 3i-l shows the printed image under 0 nm, 50 nm, 70 nm, 100 nm defocus, respectively. Compared to the initial SMO, lower  
Figure 3: Target 2 simulation results of the initial SMO, SMO under assigned focus plane (100 nm defocusing), and DRSMO with \(\omega=0.2\), respectively. distortion and PAE were acquired in each defocus plane. However, since this method merely operated at an assigned defocusing plane, the global fidelity was not so good. In Figure 3j under 50 nm and Figure 3k under 70 nm defocusing, apparent hot spots existed, shown in the center of the red circles. Finally, the optimization results of the proposed DRSMO with \(\omega=0.2\) are show in Figure 3m-r. It clearly shows that the distortion and PAE further declined in each defocus plane, so that more robust source and mask were designed through this optimization. Compare to SMO under an assigned defocusing plane, the DRSMO guaranteed better global fidelity in a wide range of defocus. To further prove a robust improvement, Figure 4 depicts the defocus-PAE curves in the evaluation range of 0-100 nm for target 1 optimized systems. It should be noted that each weight factor \(\omega\) corresponds to a set of optimized source and mask. The slope of each curve reflected the process robustness, and a lower slope meant lower sensitivity for the optimized systems to focus on shifting. It should be noted that the slope of each curve gradual decreased in the order of initial SMO (blue curve), DRSMO with \(\omega=0\) (green curve), DRSMO with \(\omega=0.1\) (red curve), DRSMO with \(\omega=0.2\) (azury curve), and DRSMO with \(\omega=0.3\) (purple curve). We concluded that DRSMO is beneficial to reduce defocusing sensitivity and to gain a more uniform exposure pattern within a long range of defocusing, which means a better system robustness against uncertain and changeable focus shifts in a real lithography process. The core idea of the DRSMO is to introduce the defocusing sensitivity \(Y\) to constrain the uniformity of printed patterns in different defocus variations. Thus, simulations further compare the optimization performance of DRSMO, which merely operate at the fidelity part \(F\) (\(\omega=0\)) and DRSMO driven by the sensitivity penalty (\(\omega\
eq 0\)). For instance, comparing the DRSMO with \(\omega=0\) (green curve) and DRSMO with \(\omega=0.1\) (red curve) in Figure 4, the slope of the red curve is lower than that of the green curve, which infers that the introduction of the sensitivity penalty \(Y\) can further improve pattern uniformity with a wide range of defocus variations.