Theoretical analysis of errors in correction algorithms for periodic nonlinearity in displacement measuring interferometers

Two different measure-and-correct algorithms, the Chu–Ray algorithm and the Eom algorithm, were simulated to estimate their effectiveness for correcting periodic nonlinearity applied a simulated wafer positioning system. Both algorithms, which demonstrated no periodic nonlinearity in some example constant velocity and constant acceleration datasets, and their specific implementations are described in detail. The algorithms were tested for both slit and scan directions using estimated wafer stage trajectories for ramping up to a constant scanning velocity with no velocity in the slit direction. Three different heterodyne frequencies and three different sampling frequencies were simulated to estimate the optical system and data acquisition parameters. The initial phase offset was also considered in the simulations and had a significant effect on both the scan and slit offset errors. The offset error appears to increase for certain initial phase values which manifests from compensating the periodic errors combined with the high stage dynamics. High noise levels and displacement jumps were observed when high velocity changes during a correction period suggesting increasing the minimum velocity threshold and limiting the maximum acceleration can improve the effectiveness of these algorithms.

► Periodic error correction is dependent on absolute phase (absolute position).
► More than 1.25 fringes of displacement is needed for effective correction.
► Algorithms are not as effective under high dynamic stage motions.