# Measurement Vertex Position of a Large Aspheric Mirror
## 3 Experiment and Results  
The preliminary experiment is carried out on a concave aspheric mirror 1 m in diameter and 2.1 m in RoC. The mirror is supported by a whiffletree structure on a steel plate and can be rotated manually. The form error of the mirror is 0.1\(\lambda\) (rms, with piston, tilt, power and coma terms removed, \(\lambda\) = 632.8 nm). The optical target for MAT is mounted on a hollow reference cylinder (see Figure 2). An OGP SmartScope ZIP 250 vision measuring system (maximum permissible error (MPE): 1.8 + 6 L/1000 \(\upmu\)m, where L is the measured length in mm) is used to measure the eccentricity between the target and reference cylinder. The vision measuring system measures the reference cylinder's outer edge (roundness is 3 \(\upmu\)m) and analyzes cylinder's center position, then it aims at the central circular dot of the target with a crossline to determine the eccentricity. The eccentricity is measured and adjusted to 3 \(\upmu\)m in x direction and \(-\)2 \(\upmu\)m in y direction. A rectangular light emitting diode (LED) is fixed in the reference cylinder and it provides uniform illumination light which is suitable for observation. An API Radian laser tracker (range: 40 m) [6] is used to align the target to mirror's mechanical center. The laser tracker is placed 2.5 m away from the mirror's mechanical center. The laser tracker measures the outer edge of the mirror and the reference cylinder and analyzes center positions  
Figure 1: Vertex position measurement process. Figure 2: Target for MAT mounted on a hollow reference cylinder. respectively. The eccentricity of two center positions is repeatedly measured and adjusted. The MPE of laser tracker is 10 + 5 L \(\upmu\)m (L is measured length in m) and the eccentricity of two center positions is finally controlled within 50 \(\upmu\)m.  
A Zygo GPI-XP/DPMI (test beam diameter: 4 inch) [7] is placed horizontally on the test tower with a 45\({}^{\circ}\) fold mirror which makes easy to align the optical path of heavy PMI system (see Figure 3). To align the optical path, a large 5-axis table which enables precision translation in the X, Y and Z axes, as well as tip-tilt adjustment for the PMI measurement is used. The refractive null corrector is also placed on the large 5-axis table and it is aligned to the PMI with a small 5-axis table by using interference fringe in the first place, then the large 5-axis table is precisely adjusted to align the optical path until the residual coma and defocus terms are within 0.2\(\uplambda\). A MAT with built in optical micrometers (graduations: 0.02 mm, error: \(\pm\)0.002 mm) from [PERSON] [8] is used to observe the lateral displacements of the target, as shown in Figure 3, the MAT is placed beside the PMI on a small 3-axis table, 3 m away from the mirror. The angle between the PMI axis and MAT axis is approximately 7 degrees in x direction, due to 'Cosine error', the actual displacement in x direction was 1.01 times the measured displacement. The measurement results are listed in Table 1.  
The measurement uncertainty analysis is as follows: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **0 Degrees (\(\upmu\)m)** & **180 Degrees (\(\upmu\)m)** \\ \hline \(\chi_{\text{tar-ref}}\) & 3 & – \\ \(\chi_{\text{tar-ref}}\) & \(-2\) & – \\ \(\chi_{\text{ref-mir}}\) & 21 & – \\ \(\chi_{\text{ref-mir}}\) & 42 & – \\ \(\chi_{\text{coma}}\) & 13 & –7 \\ \(\chi_{\text{coma}}\) & \(-6\) & 14 \\ \(\chi_{\text{mat}}\) & 40 & 120 \\ \(\chi_{\text{mat}}\) & 140 & 320 \\ \(\chi_{\text{vertex}}\) & & 74 \\ \(\chi_{\text{vertex}}\) & & 120 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of the vertex position measurement. \(\chi_{\text{tar-ref}}\), \(\chi_{\text{tar-ref}}\) are displacements between the target center and reference cylinder center in x, y directions; \(\chi_{\text{ref-mir}}\), \(\chi_{\text{ref-mir}}\) are displacements between the reference cylinder center and mirror center in x, y directions. Figure 3: Setup to measure vertex position of aspheric mirror.