Theoretical research on parameter calculation and error analysis of Hindle auxiliary sphere method for testing convex hyperboloid mirror

In this article, the measurement method of convex hyperboloid using Hindle sphere is further studied. Firstly, the precise analytical formulas for solving radius of curvature, the diameters of outer edge and inner hole of Hindle sphere mirror are developed. The formula can be degenerated to the common approximate formula under certain conditions. In addition, the semi exact analytical formulas for solving the parameters of auxiliary Hindle sphere mirror are derived under some conditions, and the approximate formula is further extended to other occasions of aspheric mirror test, and a feasible stitching detection scheme for the secondary mirror of the CFGT (China Future Giant Telescope) is given. Then, based on the aberration theory, using Hindle auxiliary sphere, the relationships and formulas of the measurement error of vertex radius of curvature of convex hyperboloid mirror caused by the adjustment of interferometer position, the mirrors spacing, and machining error of radius of curvature of the Hindle sphere mirror are obtained, and the Zemax optical software is used for simulation verification.