Analysis of the positioning error on lateral shearing surface reconstruction with a Fizeau interferometer

Abstract

The performance of synchrotron beamline optics is often limited by the accuracy in the figuring and finishing of the optical surfaces. In consequence, a very sensitive and accurate characterization of the optics is required during manufacturing and testing. Such characterization can only be done with instruments like long trace profilometers or Fizeau interferometers. In the case of the Fizeau interferometer, the accuracy is mainly limited by the quality of the reference surface. In this work, we propose a new method for improving the accuracy of the surface reconstruction by using the lateral shearing technique. It consists on measuring the sample surface several times, applying different displacements. By subtracting these measurements each other, the error introduced by the reference surface can be removed and the profile of the sample mirror can be reconstructed. Then, the accuracy of the reconstruction is limited by the imprecisions of the linear stage used to shift the sample mirror. The positioning error is analyzed regarding the shearing transfer function and the Natural Extension. Small displacements are more sensitive to the positioning error, not only because the error is comparatively bigger, but also because the error using Natural Extension is bigger than using large displacements. Using the proposed technique, a statistical analysis regarding the positioning error has been performed. Its conclusion is that the accuracy in metrology of x-ray mirrors is improved by at least a factor of 18 compared to that achieved with the Fizeau interferometer and a standard λ/20 reference surface, giving a reconstruction error lower than 1.8 nm peak to valley.