Flatness error control technology based on random forests

In a dual-beam super-resolution laser direct-writing lithography system, the flatness error of the motion stage during XY-plane scanning (small fluctuations along the Z direction) is a critical factor that degrades writing accuracy. This work presents an integrated method combining dual-cylindrical-lens astigmatic measurement and machine-learning-based feedforward compensation. By employing the dual-cylindrical-lens astigmatic approach together with a four-quadrant detector (FQD) to capture variations of the spot profile, nanometer-scale Z-axis errors of the stage are converted into changes in the focus error signal (FES), enabling precise detection of flatness errors. To exploit the predictability of the error, this study further introduces random forest regression from ensemble learning. A prediction model is constructed through analysis of historical data to accurately estimate the spatial distribution of the stage flatness error, and the predicted error is then used to drive a piezoelectric actuator for feedforward compensation. Experimental results demonstrate that the proposed random-forest-based feedforward strategy markedly reduces flatness errors generated during planar motion, thereby improving both the writing accuracy and operational stability of the dual-beam super-resolution laser direct-writing lithography system.