# Multi-Objective Defocus Robust Source and Mask Optimization Using Sensitive Penalty
## 2 DRSMO Modeling  
Figure 1 illustrates the DRSMO optimization framework which is composed of the forward calculation to evaluate pattern fidelity and the inverse optimization to update the source and mask parameters. In current lithography processes, source and mask are freeform pixel-based configurations. Therefore, both the source and mask can be represented by matrix form \(J\) and \(m\), respectively. Each source element \(J^{x_{s},y_{s}}\in(0,1)\) represent normalized light intensity and each mask element \(m_{rs}\) is subject to 0-1 binary distribution. To overcome the complexity of the constrained optimization problem, a parameters transfer was made to convert \(J^{k}\) and \(m^{k}\) to unconstrained sources mask parameters \(\Omega_{J}^{k}\) and \(\Omega_{M}^{k}\) (see Appendix B and Equation (A9)) in the \(k\)th iteration, respectively [8].  
As for the forward calculation process, the printed resist pattern was calculated by the given light source and mask parameters through the corresponding physical process. Then, the printed pattern was compared with the target resist pattern to evaluate the pattern fidelity and CD error. Under the Abbe imaging principle, the aerial image that takes into account defocusing can be represented as \(t^{defocus}(\beta)\). For the model-based SMO method, the scalar imaging model is inaccurate in hyper-NA (NA \(>\) 1) immersion lithography systems [22]. Thus, we previously studied the vector imaging model for aerial image calculation [16], and it can be formulated as  
\[l^{defocus}(\beta)=\frac{1}{J_{sum}}\sum_{x_{s}}\sum_{y_{s}}\left(P^{x_{s},y_{ s}}\times\sum_{p=x,y,z}\left\|E_{p}^{nofer}\right\|^{2}\right), \tag{1}\]  
where \(J_{sum}\) is the summation of all the source intensity and is used as a normalization factor. \(E_{p}^{nofer}\) is the electric fields (x-, y-, and z-directions) in the exposure plane which can be expressed as  
\[\begin{array}{l}E_{p}^{nofer}=\mathcal{F}^{-1}[2\pi/n_{w}\times R\times C \times V(x_{s},y_{s})\odot U_{ideal}\odot Def\\ \odot\mathcal{F}\left[M_{near}(x_{s},y_{s})\right]\odot E_{i}(x_{s},y_{s})], \end{array} \tag{2}\]  
where \(n_{w}\) is the index of refraction, the magnification of \(R=4\) normally, \(V(x_{s},y_{s})\) is the vector matrix for hyper-NA systems, \(C\) is the irradiance correction factor, \(U_{ideal}\) is the ideal pupil filter, \(\mathcal{F}[M_{near}(x_{s},y_{s})]\) is the mask diffraction near field, \(E_{i}(x_{s},y_{s})\) is the electric field of the source that is represented by a \(2\times 1\) vector. Operators \(\odot\), \(\mathcal{F}[\ ]\), and \(\mathcal{F}^{-1}[\ ]\) are represented by matrix entry-by-entry multiplication, forward Fourier transform, and inverse Fourier transform, respectively.  
It should be noted that the impact of defocusing on resulting aerial image _Def_ can be described as an aberration of a sort [23]. This causes a distribution of the phase change on the ideal aperture which can be described as  
\[Def=\exp\left[j\frac{2\pi n_{w}\beta(1-y_{i})}{\lambda}\right], \tag{3}\]  
where \(y_{i}\) is the direction cosine in the propagation direction and \(\beta\) is the defocus value.  
Next, the resist image was adopted to the continuous derivable sig model [24], which can be expressed as  
\[Z(\beta)=sig\Big{[}l^{defocus}(\beta)\Big{]}=\frac{1}{1+\exp\{-a\Big{[}l^{ defocus}(\beta)-t_{r}\Big{]}\}}. \tag{4}\]  
Figure 1: Forward calculation and inverse optimization process for defocus robust SMO (DRSMO).  
where \(a\) indicates the steepness of the sigmoid function, and \(t_{r}\) is the threshold.