Computational Lithography

Chapter 6

Regularization Framework

Inverse lithography is an ill-posed problem where numerous input patterns can lead to the same binary output pattern. Regularization in ILT seeks to bias the solution space to sample solutions that have some favorable properties [40, 41, 68]. According to Chapter 5, the described OPC and PSM optimization algorithms result in the optimized mask with continuous amplitude and phase, referred to as the continuous mask, which is not physically realizable. To overcome this limitation, the post-processing steps are used to quantize the amplitude and phase of the optimized mask to several discrete levels, resulting in the pole-level mask. However, these post-processing steps are suboptimal with no guarantee that the pattern error is under the goal [42, 43, 68]. To reduce the error increase contributed by the post-processing step, it is desired to obtain an optimized continuous mask, which is close to the optimized pole-level mask. Furthermore, the OPC and PSM optimization algorithms in Chapter 5 lead to optimized mask patterns containing numerous details, which may bring difficulty to mask fabrication. Most of the details consist of singular transmission and opaque pixels. To control the manufacturing cost, we would like to reduce the complexity of the optimized masks.

One approach to obtain the optimized solution with prescribed properties is through regularization during the optimization process [83]. Regularization framework is formulated as follows: ...