Regularized Recursive Solutions for Prediction of Aberrations Associated With Projection Optics

For an improved reduction of thermally induced wavefront aberrations associated with the projection lens in deep ultraviolet lithography, this article proposes a regularization-based recursive linear estimation strategy for the model parameters of a predictive control scheme. The linearity of estimation is ensured by estimating only the gains of an approximate exponential regression model that effectively describes the spatial cooling behavior, where the time constants are preset to specific values such that the actual exponentials are collectively well described. Owing to its suitability for recursive formulations, the corresponding solution involves the celebrated Tikhonov regularization, with its regularization term additionally consisting of a scaling term that relates to the magnitudes of parameters so that the same regularization parameter can be used at different exposure settings and time moments. Moreover, for a proper tuning of this regularization parameter, the method of generalized cross-validation is adopted. By means of comparative analysis with respect to both synthetic and real data, the treated method is demonstrated to outperform the traditional Kalman filter, as well as nonlinear approaches such as the extended and unscented Kalman filters.