Error estimation for lattice methods of stage self-calibration

摘要

For more than a dozen years now, stage self-calibration methods have been proposed for ameliorating overlay alignment problems. The idea is that a stage, calibrated to an accurate Cartesian standard, can reduce butting errors and, in mix-and-match, eliminate the need to force all line-of-duty lithography tools to conform to an identically distorted standard, with all of the re-adjustments and bookkeeping that entails. For all of their promise, there are important issues that must be resolved before self-calibration strategies can be trusted. The major problem is the determination of rigorously established error bounds - self-calibration procedures can be useful only if they assuredly improve the accuracy of the machines they are used to calibrate. However, in a previous paper the author has shown that a basic operation in so-called lattice methods of self-calibration procedures, namely rotation of a symmetric artifact, leads to an unstable set of equations, raising the question whether such procedures can possibly yield improved accuracy. This paper addresses that problem explicitly and thereby demonstrates a method for deriving the needed error bounds. The example used is a rotationally symmetric (regular) polygon. For the purpose of exposition, qualitative algebraic, not numeric, results are emphasized, but the method can be applied to produce the necessary numbers for general calibration lattices. The method can be applied to error estimation for practical stage self-calibration.