## 2 Modeling of DUV Lithography
### Mathematical Modeling  
To mitigate the impacts of mask absorber errors and increase the PW, an alternate optimization framework of the source, NA, and process parameters are built to improve the optimization performance. Consequently, we develop different cost functions for the alternate optimization  
Figure 1: The 21 basis functions for representing the source pattern.  
procedure. We evaluate the lithographic imaging performance by measuring both the image fidelity and the process robustness.  
The cost function for source optimization consists of two parts: the CDE for the lithography system with random MAT error and SWA error to evaluate the image fidelity, and NILS to improve the PW. Thus, the weighted cost function is formulated as  
\[F_{S}=\varepsilon\{\alpha_{1}P_{CDE}(\mathbf{c};MAT,SWA)-\alpha_{2}P_{NILS}( \mathbf{c};MAT,SWA)\}, \tag{5}\]  
where \(\varepsilon\{\cdot\}\) is the expectation operator to the random absorber errors including the MAT and SWA error which follow the Gaussian distribution; \(P_{CDE}(\mathbf{c};MAT,SWA)\) and \(P_{NILS}(\mathbf{c};MAT,SWA)\) represent the absolute value of resist image CDE and aerial image NILS with random mask MAT and SWA error, respectively; \(\alpha_{1}\) and \(\alpha_{2}\) are the corresponding weighting coefficients; \(MAT\) and \(SWA\) represent the value of mask absorber thickness error and absorber sidewall angle error, respectively.  
To lighten the computation burden, we calculate \(F_{S}\) by discretizing \(MAT\) and \(SWA\) to a set of values \(MAT_{i}\) and \(SWA_{j}\) with corresponding probabilities \(\xi_{i}\) and \(\eta_{j}\), that is,  
\[F_{S}=\sum_{i}\sum_{j}\xi_{i}\eta_{j}\cdot\{\alpha_{1}P_{CDE}(\mathbf{c};MAT_{ i},SWA_{j})-\alpha_{2}P_{NILS}(\mathbf{c};MAT_{i},SWA_{j})\}, \tag{6}\]  
Then we denote the NA and process parameter to be optimized as vector \(\mathbf{x}\):  
\[\mathbf{x}=\left[x_{1},x_{2},\cdots,x_{n}\right], \tag{7}\]  
The cost function for NA and process parameters optimization consists of the DOF at a specific exposure latitude (EL) with random MAT and SWA errors to further increase the PW. Thus, the cost function is formulated as  
\[F_{NP}=\varepsilon\{-P_{DOF}(\mathbf{x};MAT,SWA)\}, \tag{8}\]  
where \(P_{DOF}(\mathbf{x};MAT,SWA)\) represents the absolute value of the DOF with random MAT and SWA error.  
Similarly, the \(F_{NP}\) is calculated by discretizing \(MAT\) and \(SWA\) to a set of values \(MAT_{i}\) and \(SWA_{j}\) with the corresponding probabilities \(\xi_{i}\) and \(\eta_{j}\), that is,  
\[F_{NP}=-\sum_{i}\sum_{j}\xi_{i}\eta_{j}\cdot P_{DOF}(\mathbf{x};MAT_{i},SWA_{j }), \tag{9}\]  
Therefore, the source optimization problem is formulated as searching for the coefficient vector \(\mathbf{c}\) to minimize the cost function \(F_{S}\). It can be expressed as  
\[\overset{\wedge}{\mathbf{c}}=\text{arg min }F_{S}(\mathbf{c}), \tag{10}\]  
where \(\overset{\wedge}{\mathbf{c}}\) is the optimum result of \(\mathbf{c}\).  
The NA and process parameters optimization problem is formulated as seeking the parameter vector \(\mathbf{x}\) to minimize the cost function \(F_{NP}\). This can be expressed as  
\[\overset{\wedge}{\mathbf{x}}=\text{arg min }F_{NP}(\mathbf{x}), \tag{11}\]  
where \(\overset{\wedge}{\mathbf{x}}\) is the optimum result of \(\mathbf{x}\).