Automated Layout and Phase Assignment Techniques for Dark Field Alternating PSM

We describe new, e cient algorithms for layout modi cation and phase assignment for dark eld alternating-type phase-shifting masks in the single-exposure regime. We make the following contributions. First, we give optimal and fast algorithms to minimize the number of phase con icts that must be removed to ensure 2-colorability of the con ict graph. These methods can potentially reduce runtime and/or improve solution quality, compared to previous approaches of Moniwa et al. and Ooi et al. Second, we suggest a new iterative 2-coloring and compaction approach that simultaneously optimizes layout and phase assignment. The approach iteratively performs the following steps: (i) compact the layout and nd the con ict graph; (ii) nd the minimum set of edges whose deletion makes the con ict graph bipartite; and (iii) add a new compaction constraint for each edge in this minimum set, such that the corresponding pair of features will no longer con ict. Third, we describe additional approaches to co-optimization of layout and phase assignment for alternating PSM. Preliminary computational experience appears promising.