# Multi-Objective Defocus Robust Source and Mask Optimization Using Sensitive Penalty
## 3 Simulation Results and Discussion  
### Simulation Conditions  
We illustrate the DRSMO method in two test patterns as shown in Figure 2. The critical dimension (CD) of each pattern was 45 nm. Resistance patterns and mask were represented by a \(201\times 201\) matrix with a resolution of 5.625 nm \(\times\) 5.625 nm and 22.500 nm \(\times\) 22.500 nm per pixel, respectively. The imaging system parameters were set to be \(\lambda=193\) nm and \(NA=1.2\). The freeform source was represented by a \(21\times 21\) matrix which uses TE-polarization illumination. In this paper, the whole training set \(\beta=\{\beta_{i}\}\) consisted of 900 random sampling points in the range of (\(-100\) nm, \(100\) nm), since this sample range was extremely larger than initial DOF, which was without optimization. Taking  
\begin{table}
\begin{tabular}{l} \hline \hline
**SGD procedure** \\ \hline
**1. Initialization:** Assign the starting source parameter \(\Omega_{J}\), mask parameter \(\Omega_{M}\), the source step size \(s_{J}\), the mask step size \(s_{M}\), the upper limit iteration number \(l_{smo}\) \\
**2. Optimization:** Simultaneously update the source and mask patterns: \\
**_While \(k\leq l_{smo}\)_ \\ \(k\gets k+1\) \\ Randomly generate the defocus value \(\beta_{i,k-1}\) \\ Calculate the generate \(\
abla_{J}G_{i}^{k-1}\), \(\
abla_{M}G_{i}^{k-1}\), respectively; \\ Update the source and mask parameters \\ \(\Omega_{J}^{k}=\Omega_{J}^{k-1}-s_{J}\frac{\
abla_{J}G_{i}^{k-1}}{|\
abla_{J}G_ {i}^{k-1}|_{2}}\) \\ \(\Omega_{M}^{k}=\Omega_{M}^{k-1}-s_{M}\frac{\
abla_{M}G_{i}^{k-1}}{|\
abla_{M}G_ {i}^{k-1}|_{2}}\) \\ \end{tabular} \\
**3. Output:** the optimized source and mask parameters. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Stochastic gradient descent (SGD) and mini-batch gradient descent (MBGD) optimization procedure.  
into account both optimization speed and accuracy, the batch number \(l_{batch}\) was set to be three per iteration and 300 iterations totally.  
To evaluate the imaging fidelity, pattern error (PAE) refers to the Euclidean distance between the target pattern and the actual pattern in the resist. Generally, the smaller PAE means the higher fidelity of the lithographic imaging. It can be formulated as  
\[PAE=\left\|\overline{Z}-Z(\beta)\right\|_{2}^{2}, \tag{10}\]  
where \(\overline{Z}\) is the binary target resist pattern and \(Z(\beta)\) is the actual resistance pattern under defocus \(\beta\). Meanwhile, to evaluate the defocusing sensitivity quantitatively, we defined the defocus sensitivity factor \(S_{\beta}\) as the change ratio of PAE to defocusing  
\[S_{\beta}=\frac{\partial PAE}{\partial\beta}, \tag{11}\]  
Moreover, to evaluate the process robustness in the actual exposure process, the PW was introduced to describe the restrictive relation between dose variation and focus variation. It was composed of two parameters, DOF and exposure latitude (EL). Exposure latitude is the allowable range of dose variation under a fixed defocus. Similarly, DOF is the largest acceptable defocus range under a fixed dose. Thus, PW consists of all pairs of DOF and EL which satisfy the exposure quality specification. Generally, taking the DOF when corresponding to ELs equal to 5% or 10% as process evaluation standard. Meanwhile, the PW representative calculation positions are marked at yellow lines in Figure 2.