# Measurement Vertex Position of a Large Aspheric Mirror
## 3 Experiment and Results
The measurement model for the vertex position is shown in Equation (1), while in the experiment,  
\[ \begin{split}\text{x}_{\text{target}}&=\text{x}_{\text{ tar-ref}}+\text{x}_{\text{ref-mir}}\\ \text{y}_{\text{target}}&=\text{y}_{\text{tar-ref}}+ \text{y}_{\text{ref-mir}}\end{split}  \tag{2}\]  
The uncertainty model of the vertex position measurement is shown in Equation (3) [9]:  
\[\text{u}_{\text{vertex}}^{2}=\text{u}_{\text{mat}}^{2}/2+\text{u}_{\text{ coma}}^{2}/2+\text{u}_{\text{tar-ref}}{}^{2}+\text{u}_{\text{ref-mir}}{}^{2} \tag{3}\]  
According to the MPE value of the SmartScope ZIP 250 (assigning rectangular distribution), the displacement between the target center and reference cylinder center measurement uncertainty is (Type B evaluation of uncertainty):  
\[\text{u}_{\text{tar-ref}}=2\ \text{\mu}\text{m} \tag{4}\]  
According to the MPE value of the API Radian laser tracker (assigning rectangular distribution), the displacement between the reference cylinder center and mirror center measurement uncertainty induced by the laser tracker is (Type B evaluation of uncertainty):  
\[\text{u}_{\text{ref-mir}\_\text{lt}}=14\ \text{\mu}\text{m} \tag{5}\]  
The displacement between the reference cylinder center and the mirror center measurement uncertainty evaluated by statistical analysis of 5 measurements is (Type A evaluation of uncertainty):  
\[\text{u}_{\text{ref-mir}\_\text{repeatability}}=24\ \text{\mu}\text{m} \tag{6}\]  
According to the measurement experience, the surface coma measurement uncertainty is 60 nm, to this mirror, the correlation coefficient of measured coma and lateral displacement is 227. The measurement uncertainty induced by coma measurement is (Type B evaluation of uncertainty):  
\[\text{u}_{\text{coma}}=14\ \text{\mu}\text{m} \tag{7}\]  
The uncertainty arises from the MAT reading is evaluated by measuring a target which mounted on a linear slider at a 3 m range. The statistical analysis of the repeatability and ranging test gives (Type A evaluation of uncertainty):  
\[\text{u}_{\text{mat}\_\text{reading}}=10\ \text{\mu}\text{m} \tag{8}\]  
The drift of relative orientation between the MAT axis and test optics axis is evaluated by the repeatability test. The statistical analysis of 6 times reading from the PMI and MAT (in an hour) gives (Type A evaluation of uncertainty):  
\[\text{u}_{\text{mat}\_\text{drift}}=45\ \text{\mu}\text{m} \tag{9}\]  
The combined uncertainty of the vertex position measurement is as follows:  
\[\text{u}_{\text{vertex}}=((\text{u}_{\text{mat}\_\text{reading}}{}^{2}+\text{ u}_{\text{mat}\_\text{drift}}{}^{2})/2+\text{u}_{\text{coma}}{}^{2}/2+\text{u}_{ \text{tar-ref}}{}^{2}+\text{u}_{\text{ref-mir}\_\text{l}}{}^{2}+\text{u}_{ \text{ref-mir}\_\text{repeatability}}{}^{2})^{0.5}=44\ \text{\mu}\text{m} \tag{10}\]  
The expanded uncertainty (k = 2) is as follows:  
\[\text{U}_{\text{vertex}}=\text{k}\ \times\ \text{u}_{\text{vertex}}=88\ \text{\mu}\text{m} \tag{11}\]  
The analysis indicates the vertex position expanded measurement uncertainty is 88 \(\text{\mu}\text{m}\). For comparison, the mirror is also measured with a CMM (MPE: 2.2 + L/350 \(\text{\mu}\text{m}\), where L is measured length in mm). The vertex position x displacement is \(-90\ \text{\mu}\text{m}\) and y displacement is 170 \(\text{\mu}\text{m}\). The vertex position measurement uncertainty of the CMM method is evaluated using statistical way (rotate the mirror 3 times by 90 degrees and use a CMM to measure the vertex position at 4 directions) and vertex position measurement uncertainty is 0.29 mm in x direction and 0.12 mm in y direction. Thus, the result of the proposed method is within the uncertainty of the CMM measurement.