Monte Carlo simulations applying rigorous coupled-wave analysis for tolerance analysis of diffractive optical elements

Rigorous coupled-wave analysis (RCWA) has been used for determining periodic grating structures of diffractive optical elements (DOE) such as rectangular surface-relief gratings. When we observed a cross-sectional image of a manufactured rectangular grating by the scanning transmission electron spectroscopy (STEM), we found that the rectangular structure was deformed to a trapezoidal shape with non-zero corner radii. Therefore, we assumed that we could derive more accurate diffraction efficiencies by RCWA using parameters of the trapezoid such as the height, base angles, and the corner radii as well as the grating pitch. However, each parameter varies in distribution, resulting in a variation in diffraction efficiencies. Tolerance analysis in the design stage is effective in quantitatively predicting that the diffraction efficiencies are in the allowable ranges. Therefore, we have developed a Monte Carlo simulation program by applying RCWA, which can handle distributions of these trapezoid parameters. The program can also handle the distributions of the thickness and refractive indices of the thin films deposited on the grating. The distributions of the trapezoid parameters were obtained through multiple measurements of the substrates without thin films using atomic force microscopy. The distribution of the thickness and refractive indices of the thin films were obtained from analyzed data measured in the vacuum deposition equipment. According to our measurements of the diffraction efficiencies of the manufactured gratings, the diffraction efficiencies have been in the allowable range, and have also been consistent with the Monte Carlo simulations for the tolerance analysis.