Approximation of Hopkins equation in Hilbert space and its application in polarized illumination modelling

Abstract

In this article, we derive an analytical approximation to compute the aerial images on general geometric domains. Under the framework in Hilbert space, we develop a general approximation model for aerial image intensity, which is particularly valid for the computation of the aerial image with polarized illumination source. Our mathematical result is a generalization of classical Mercer's theorem provided that the integral kernel is Hilbert-Schmidt. By applying Synopsys's simulation tool Progen, we demonstrate the effectiveness of our proposed approximation with polarized illumination sources (65 nm and below). Since our approach setting is quite general, the proposed approximation in this article is suitable for modification in various applications.

Model-based OPC using the MEEF matrix III

“…Thus, an approximation 𝑋 𝑖𝑗 ≈ 𝑎 𝑖𝑗 /𝑠𝑙𝑜𝑝𝑒(𝑃 𝑖 ) follows. Since the transformation 𝑚 → 𝐼 is in essence a quadratic operator [7], fast algorithms are possible for computing the Jacobian 𝐴, and hence for the cross MEEF matrix 𝑋.…”