Integration Method Using Three Nodes Spaced at Heterogeneous Intervals and Application to Highly Precise Measurement of Mirror Surface Aberrations

Abstract

The Hartmann test for measuring mirror surface aberrations accumulates integration errors when calculating the aberration on the basis of gradient data. For this reason, highly precise integration is desired. Here, we present a grating projection method modified from the Hartmann test. It involves using ray tracing and fringe scanning methods, in which sampling points can be set at arbitrary intervals. The authors propose a highly precise integration method appropriate for the grating projection method. In this integration method, three nodes are used, and Simpson's rule and the Gauss quadrature can be described in a unified manner. Furthermore, the relationship between the integration error and the coefficient that determines the interval between nodes is shown. We provide the coefficient conditions for rendering the integration error smaller than that of Simpson's 3/8 rule; this is a typical integration method. In addition, the advantage of this method is demonstrated by simulation experiments. The proposed integration method can be applied to any general precise numerical integration problem.