Thermal deformation prediction in reticles for extreme ultraviolet lithography based on a measurement-dependent low-order model

In extreme ultraviolet lithography, imaging errors due to thermal deformation of reticles are becoming progressively intolerable as the source power increases. Despite this trend, such errors can be mitigated by adjusting the wafer and reticle stages based on a set of predicted deformation-induced displacements. Since this control scheme operates online, an accurate low-order model is necessary. However, finite element modeling of the reticle and its adjacent components leads to a large-scale thermo-mechanical model that should be simplified. First, parameters of the model's initial thermal condition are reduced to only a few from which numerous initial conditions can be accurately reconstructed. This entails placement of temperature sensors at the corresponding locations, and for this purpose, the discrete empirical interpolation method (DEIM) is utilized. Then, linear and nonlinear model reductions are performed via the proper orthogonal decomposition method and DEIM, respectively. The resultant model is employed in the Kalman filter to estimate the parameters of the reticle's temperature-dependent coefficient of thermal expansion from several displacement measurements and to subsequently predict the displacements that are used for control. By processing the outputs from the simulated large-scale model, this filter is shown to perform successfully, even in the presence of an unexpected initial condition.