Theory of transformation-invariant compact process modeling

Modern one-digit technological nodes demand strict reproduction of the optical proximity corrections (OPC) for repeatable congruent patterns. To ensure this property the optical and process simulations must be invariant to the geometrical transformations of the translation, rotation, and reflection. Simulators must support invariance both in theory, mathematically, and in practice, numerically. In the first part of this study we aim to examine manner and conditions under which optical approximations, such as Sum of Coherent Systems (SOCS) and Abbe decomposition, preserve or violate intrinsic invariances of exact imaging. In this age of asymmetrical pixelated sources, complex Jones pupils, and tilted chief rays, the full rotational invariance is observed only in some cases of annular illuminations; otherwise, it is rare in contemporary optics. In the second part we consider more important topic: invariances of compact process modeling (CPM) operators. Translational, rotational, and reflectional invariances are obligatory in CPM, because photoresist processing effects do not have preferential lateral direction, origin, or reflectional axis. The invariance of CPM operators has never been scrutinized before. We fill this gap by expanding generic translationally-invariant CPM operator into Volterra series, and then examine linear, quadratic, and high-order terms. Contrary to the straightforward invariance conditions for linear operators, the morphology of invariance for the second and high order operators is non-trivial. We found necessary and sufficient conditions for the invariance to take place, and provide examples of non-linear Volterra operators that can be used as atomic construction blocks in neural networks for CPM.