Mathematical modelling of indirect measurements in periodic diffractive optics and scatterometry

Abstract

In this work, we illustrate the benefits and problems of mathematical modelling and effective numerical algorithms to determine the diffraction of light by periodic grating structures. Such models are required for reconstruction of the grating structure from the light diffraction patterns. With decreasing structure dimensions on lithography masks, increasing demands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical method offer access to the geometrical parameters of periodic structures including pitch, side-wall angles, line heights, top and bottom widths. The mathematical model for scatterometry is based on the Helmholtz equation derived as a time-harmonic solution of Maxwell's equations. It determines the incident and scattered electric and magnetic fields, which fully specify the light propagation in a periodic two-dimensional grating structure. For numerical simulations of the diffraction patterns, a standard finite element method (FEM) or a generalized finite element method (GFEM) is used for solving the elliptic Helmholtz equation. In a first step, we performed systematic forward calculations for different varying structure parameters to evaluate the applicability and sensitivity of different scatterometric measurement methods ...