Lens heating induced aberration prediction via nonlinear Kalman filters

Lens heating induced aberrations rank among the dominant causes of image deteriorations in photolithography. In order to accurately counteract them via the available manipulators within the projection lens, it is crucial to employ a predictive model that is identified with relatively small errors. In this paper, parameters of a phenomenological model are recursively updated with respect to the measurements taken at the end of a wafer and are subsequently utilized in aberration predictions for the dies of the next wafer. To serve this purpose, two suboptimal Bayesian strategies, namely, the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), are applied to the nonlinear system at hand. In addition, the classical Kalman filter is tested on an approximate linear model. Filter performances are evaluated using both synthetic and experimental data and compared with respect to the posterior Crame´r-Rao lower bound. When synthetic measurements are in use, the UKF moderately outperforms the EKF. Moreover, they both perform significantly better than the classical Kalman filter. However, due to model imperfections, these gains decrease and may even vanish when real measurements are processed. If the computational costs are also considered, then the EKF becomes more preferable over the other options.