# Research on Nanometer Precision Measurement Method of High Order Even Aspheres
## 2 Methods
### Correction of Mapping Distortion for the Result of Surface Shape  
Mapping distortion often occurs in null tests on optical aspheres and freeforms. It is characterized by a nonlinear mapping of the transverse coordinates on the test surface to the transverse coordinates on the imaging surface of the interferometer. Mapping distortion is mainly caused by the incomplete imaging of light as it passes through the optics of the interferometer or null optics [29; 30]. The occurrence of mapping distortion means that the image of the interferometer does not truly reflect the height distribution of the surface shape of the test surface in the transverse direction. That is unacceptable for high-precision deterministic processing, as the measurement result does not guide the process to accurate material removal at the exact location on the test surface. Consequently, it is necessary to correct the mapping distortion of the measurement results when using CGH for null tests on optical aspheres.  
A pure ray tracing method is the mapping distortion correction method used in this paper. Dense rays are applied to trace the coordinate of the test surface to ensure that a sufficient number of feature points are used for correction. Since the transverse coordinates on the test surface and the imaging plane of the interferometer are not simply linearly proportional, we choose the TF as a transit surface in order to facilitate the calculation of the mapping relationship, as shown in Figure 3. Denote the coordinate of the point on the test surface as \((x_{t},y_{t})\), the point on the CGH as \((x_{c},y_{c})\), the point on the TF as \((x_{t},y_{t})\), and the point on the imaging plane as \((x_{p},y_{p})\).  
Find two feature points \((x_{p1},y_{p1})\) and \((x_{p2},y_{p2})\) on the imaging plane, corresponding to \((x_{t1},y_{t1})\) and \((x_{t2},y_{t2})\) on the test surface, respectively. Then, \((x_{f1},y_{f1})\) and \((x_{f2},y_{f2})\) are the corresponding points on the TF traced through rays. The transverse coordinates on the TF are linearly related to those on the imaging plane by the following relationship:  
\[a=\frac{\sqrt{\left(x_{f1}-x_{f2}\right)^{2}+\left(y_{f1}-y_{f2}\right)^{2}}}{ \sqrt{\left(x_{p1}-x_{p2}\right)^{2}+\left(y_{p1}-y_{p2}\right)^{2}}} \tag{1}\]  
Figure 2: The optical layout of the optical asphere null test. The blue line represents the test beam. The red line and the purple line represent the positioning beam. The green line represents the alignment beam.  
Based on the relationship \(a\), the coordinates of all the points on the TF corresponding to the imaging plane can be calculated as follows:  
\[x_{f} =a\big{(}x_{p}-x_{p1}\big{)}+x_{f1} \tag{2}\] \[y_{f} =a\big{(}y_{p}-y_{p1}\big{)}+y_{f1} \tag{3}\]  
The tracing rays can be used to establish a one-to-one correspondence between the points on the test surface (\(x_{f}\), \(y_{f}\)) and the points on the TF (\(x_{t}\), \(y_{t}\)). At this point, the correspondence between the transverse coordinates of the points on the test surface and those of the points on the imaging plane has been successfully established. The tracing rays can be set dense enough to avoid a lack of fiducial marks. Finally, the correction is completed by using linear interpolation for data completion to form a complete error map in the height direction of the test surface. Zemax can be used to track the rays.