Absolute Measurement of Rectangular Flat Mirrors Using a Unidirectional Multi-Shifted Method

With the development of advanced light sources, surface smoothness is becoming crucial for large- sized and high-precision mirrors. In this study, a two-dimensional (2D) absolute metrology method is developed that is based on the laser interferometer translational shear method for high aspect ratio rectangular flat mirrors used in X-rays. In this method, translational measurements in orthogonal directions are replaced with multiple shifted measurements in a single direction. In addition, a multi-matrix augmented zonal method reconstruction algorithm is derived to obtain the absolute surface shape of high-precision rectangular flat mirrors. Through simulations, the root mean square ( RMS) of the residuals between the reconstructed absolute surface shape and initial surface shape is 0. 03 nm (lambda/20000) without considering noise. Surface shape recovery under different numbers of translations and translation distances when considering Gaussian noise is then simulated and analyzed, and an experimental verification is performed for a rectangular plane mirror of 120 mmx40 mm. The measured absolute surface shape recovery is determined as 1. 07 nm (lambda/591) using the RMS of the absolute surface shape residual obtained by the three-plane method. Both simulations and experiments show that the proposed method can effectively obtain the 2D absolute surface shape of a high-precision rectangular planar mirror. Unidirectional translation thus lays the foundation for establishing multi- aperture stitching measurements based on absolute measurements.