Reduced-order modeling of thermally induced deformations on reticles for extreme ultraviolet lithography

Ranking already among the dominant causes of imaging errors in photolithography, deformation of reticles due to inevitable heating is becoming progressively more crucial in extreme ultraviolet (EUV) lithography as the source power continually increases, leading to higher levels of absorption of the EUV light by reticles. In order to mitigate its impact on exposed layers, accurate predictions to be the inputs of a control scheme are essential. To serve this purpose, a large-scale thermo-mechanical model in partially linear state-space form is derived by using the finite element method (FEM). The temperature-dependent coefficient of thermal expansion of materials produces the only nonlinearity in the model that is present in the static output equations. Since only low-order models are feasible for real-time use, this model is undergone several model reduction techniques to arrive at the best compact model with respect to its prediction performance, compaction rate, and easiness of computation. Treating the simulation outputs from a FEM software as the benchmark, the proper orthogonal decomposition approach combined with the discrete empirical interpolation method is selected as the most suitable route for the studied application.