2026-05-29 Latest Published
By Qi Sun
Ying Wang
Ziyin Ma
Shujie Liu
Degui Li
Zhonglei Mei
David H. Wei
Progress In Electromagnetics Research, Vol. 185, 110-117, 2026
Abstract
Accurate simulation of partially coherent imaging is crucial for computational lithography, with Abbe and Hopkins as the two main formulations being used. Although the two methods are equivalent in theory, practical simulators making independent choices between Abbe and Hopkins could hardly produce consistent results that match the desired accuracy owing to the inherently different ways of numerically representing, discretizing, and truncating the illumination source and lens pupil function, etc. Moreover, classical Hopkins models require prior construction and/or eigen decomposition of the high-dimensional transmission cross coefficient (TCC), the prohibitive costs of which hinder timely model verification. To address these challenges, we developed a unified Abbe-Hopkins formulation in conjunction with a TCC-free Hopkins pointwise sampler for efficient cross-model validation. Our formulation supports both Abbe and Hopkins modeling in a single unified framework, with the two simulation modes using exactly the same numerical representations of the illumination source and projection lens. Cross-model verification for randomly sampled points is performed efficiently by evaluating the Hopkins quadratic form through a fast Fourier transform of an image and a few pointwise multiplications between images, without ever explicitly constructing a TCC and eigen-analyzing it. Numerical tests show that the Abbe and Hopkins results agree up to the machine precision level.
