# Multi-Objective Defocus Robust Source and Mask Optimization Using Sensitive Penalty
## 3 Simulation Results and Discussion
### Optimization Results and Analysis
It was proved that the validity of introduction sensitivity penalty \(Y\) further improves the optimization performance. In brief, to maximize the DRSMO optimization performance, both the fidelity part \(F\) and the penalty term \(Y\) must be introduced into the optimization framework, and the weight factor \(\omega\) must be chosen appropriately. In actual lithography processes, PW is one of the critical evaluation criteria which refers to the exposure error tolerance. Figure 5 shows the PWs for the conventional SMO (blue curve), the DRSMO with the weight factor \(\omega=0\) (green curve), \(\omega=0.1\) (red curve), \(\omega=0.2\) (azury curve), and \(\omega=0.3\) (purple curve). It is illustrated that the PW of the proposed DRSMO was evidently larger than that of  
Figure 4: The defocus–pattern error (PAE) curves of target 1 for initial SMO (blue curve), the DRSMO with the weight factor \(\omega=0\) (green curve), \(\omega=0.1\) (red curve), \(\omega=0.2\) (azury curve), and \(\omega=0.3\) (purple curve). Optimizations were proceeded by the MBGD algorithm. the initial SMO. For the initial SMO, the maximal EL was less than 5%, which was far below the actual exposure requirements. By using DRSMO, the EL maximal increased from 4.5% to 10.5%. Similar results were found when comparing the difference of DRSMO without the sensitive penalty and DRSMO with the sensitive penalty. For example, when comparing PW with \(\omega\) = 0 (green curve) and \(\omega\) = 0.1 (red curve), a wider PW was found for the red curve than the green curve. It is inferred that the sensitive penalty is helpful for improving system robustness so that it indirectly boosts PW. However, because the cost function does not involve terms which directly relate to EL and DOF, the relationship between weight factor \(\omega\) and PW is uncertain and unclear. In this way, although DRSMO with \(\omega\) = 0.3 had the best defocus robustness, the PW was shrunk because the weight factor was too large that it led to overfitting during the training process. This illustrates that a well-chosen weight factor \(\omega\) is important to simultaneously improve robustness and PW. Table 2 summarizes the target 1 comparison of optimization results for conventional SMO, DRSMO with \(\omega\) = 0, \(\omega\) = 0.1, \(\omega\) = 0.2, and \(\omega\) = 0.3, respectively. It should be noted that PAE sensitivity factor \(S_{\beta}\) declined significantly with the \(\omega\) increase. Compare with initial SMO, the largest decrease of \(S_{\beta}\) in DRSMO with \(\omega\) = 0.3 is 63.5%. Integrated consider the improvement of both \(S_{\beta}\) and PW, \(\omega\) = 0.1 is a relative reasonable weight factor for maximize optimization performance. Target 2 consisted of a series of vertical and horizontal mixed lines. Figure 6 shows the defocus-PAE curves in the evaluation range of (0 nm, 100 nm) for target 2; optimizations were proceeded by the MBGD algorithm. It should be noted that the slope of the DRSMO with a \(\omega\) = 0.2 (azury curve) lower \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(S_{\beta}\) & **DOF (EL = 5\%)** & **DOF (EL = 8\%)** \\ \hline
**Initial SMO** & 56.1 & 0 & 0 \\
**DRSMO (\(\omega\) = 0)** & 36.6 & 102 & 87 \\
**DRSMO (\(\omega\) = 0.1)** & 25.5 & 122 & 107 \\
**DRSMO (\(\omega\) = 0.2)** & 23.8 & 138 & 105 \\
**DRSMO (\(\omega\) = 0.3)** & 20.5 & 102 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The target 1 optimized values of \(S_{\beta}\), depth of focus (DOF) (nm) corresponding to exposure latitude (El) equal to 5% and 8% for conventional SMO, and DRSMO with \(\omega\) = 0, \(\omega\) = 0.1, \(\omega\) = 0.2, and \(\omega\) = 0.3, respectively. Figure 5: Process window (PW) of target 1 for the initial SMO (blue curve), the DRSMO with the weight factor \(\omega\) = 0 (green curve), \(\omega\) = 0.1 (red curve), \(\omega\) = 0.2 (azury curve), and \(\omega\) = 0.3 (purple curve). Optimizations were proceeded by the MBGD algorithm.