Mathematical Foundations of a Geometric Theory of Diffraction for Light Scattering from a 3D Topographic Photomask

Abstract

Applications in the field of computational lithography require an accurate and extremely fast method to compute light scatterings from a photomask, which is laterally sized on the order of 10 cm in each of two dimensions and consists of dense sub-100 nm geometric patterns. All but every practical solution approaches this challenging problem by integrating scattering effects of knife-edges in association with the mask pattern boundaries. Photomasks in the near future will require integrating effects of curved edges. This paper presents a rigorous mathematical formulation of a geometric theory of diffraction for light scattering from a photomask, as an attempt to not only unify previous edge-based Mask3D models and methods, but also lay the theoretical foundations for continuing technology developments, especially Mask3D modeling of the upcoming curvilinear patterned masks. As a by-product, this paper should be useful for computational electromagnetics practitioners, and computational lithography engineers in particular, to get familiar with some of the powerful mathematical tools in differential forms, tensor analysis, and Riemann surfaces.