Gradient-based source mask and polarization optimization with the hybrid Hopkins–Abbe model

Source mask and polarization optimization (SMPO) is a promising extension of the widely used resolution enhancement technology, source mask optimization (SMO), to further enhance chip manufacturability beyond 28-nm node. Our work is aimed to develop an efficient gradient-based SMPO method by employing the hybrid Hopkins–Abbe imaging model to fulfill the goal. In addition to source and mask variables, the model is adapted to also include polarization variables to realize the optimization. Compact formulas for forward and backward model application are derived. The computation benefits from precomputed transmission cross coefficients and features high efficiency. Validity of the method is confirmed by case studies. For dense array pattern case, the optimal source and polarization can be found analytically. SMPO optimized results match well with the theoretical expectations. In addition, process window, mask error enhancement factor, and normalized image log-slope for the studied cases all get improved over the counterpart SMO results, which employ commonly used polarization. Runtime analysis shows the method is computationally efficient. Our work provides a valid way to optimize polarization together with source and mask.