Phase-shifting masks: automated design and mask requirements

In this paper we present a computationally viable algorithm for the rapid design of phase- shifting masks for arbitrary two-dimensional patterns. Our approach is based on the construction of a class of optimal coherent approximations to partially coherent imaging systems described by the Hopkins model. We show that for partially coherent imaging systems with coherence factor (sigma) <EQ 0.5, the associated approximation error in the image is quite small (< 10%). A fast iterative algorithm is used to generate (suboptimal) phase- shifting masks using the approximate imaging system model. The computational effort required per iteration is O(N log N), where N is the number of discrete image points considered. Analytical results related to practical requirements for phase-shifting masks are also presented. These results address questions related to the number of discrete phase levels required for arbitrary patterns, and provide some insight into alternative phase-shifting strategies. A number of phase-shifting mask design examples are also discussed.

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