# Measurement Vertex Position of a Large Aspheric Mirror
## 1 Introduction  
Accurate measurement of the vertex position of a large aspheric mirror is often critical to the performance of the optical system in which the mirror is used. Take the monolithic primary tertiary mirror as an example, the Large Synoptic Survey Telescope (LSST) uses a unique opto-mechanical design that places the primary and tertiary mirrors on a single glass substrate, the alignment of primary and tertiary mirrors is controlled at the time of manufacture; the tolerance in lateral displacement of the two vertices is 1 mm [1; 2; 3]. Another example is the space telescope, which has very limited alignment space, and the vertex eccentricity of the primary mirror is often required to within less than a millimeter from nominal.  
The displacement of the vertex causes shear of the spherical aberration terms, resulting in coma in surface form [4]. However, it is difficult to determine vertex position from measurement of surface form because of low sensitivity. Take a parabolic mirror with 1.3 m in diameter and 3.5 m in radius of curvature (RoC) as an example, an error of 1 mm in the measurement of the vertex position only induces 2 \(\upmu\)m (Peak-to-Valley, PV) surface form error.  
A variety of methods exists to measure the vertex position of an aspheric mirror. The mechanical method uses a coordinate measuring system, such as a coordinate measuring machine (CMM) or laser tracker to measure the surface coordinates and calculate the best-fit vertex position; the mechanical method is less efficient and often not applicable for large polished mirrors. The second method uses a phase measuring interferometer (PMI) and a null corrector. When the test optics are aligned to the mirror surface, the null corrector accurately defines the optical axis. The position and orientation of the null corrector and the position of the mirror's mechanical axis can be measured by a portable coordinate measuring system (CMS, such as a laser tracker), and the vertex position can be subsequently calculated. A shortcoming of this method is that a small error in the measurement of the null corrector's orientation results in a large error in the calculation of the vertex position due to Abbe error [5]. The third method consists of rotating the mirror by 180 degrees about its mechanical axis and measuring the change in coma. Two drawbacks associated with this method are as follows. First, the method depends on a highaccuracy rotating table and precise alignment of the test piece so that the PMI can measure the mirror surface both at 0 degrees and 180 degrees without adjustment. Second, the method relies on stable positioning of the test optics, which is difficult in large-scale dimensional measurement.  
The need to carefully match the vertex position of a large aspheric mirror presents an interesting challenge for the optics shop. Accurately measuring the vertex position using conventional methods is extremely difficult. In the following sections, we present a technique to use a PMI, a laser tracker and a micro alignment telescope (MAT) to perform vertex position measurement on an aspheric mirror. The method eliminates the aforementioned drawbacks, and accurate measurement of the vertex position is achieved.