# Measurement of High Numerical Aperture Cylindrical Surface with Iterative Stitching Algorithm
## 2 Principles  
### Experimental Setup and Error Analysis  
In the null test, when the tested and reference surfaces have the same profile, the interferogram is a uniform zero fringe pattern. A non-zero fringe pattern indicates differences between the two surfaces. In order to minimize the misalignment aberrations during the null test, the sub-apertures must be adjusted so that the interferograms contain minimal fringes. In this paper, a CGH was chosen as a null corrector to measure the cylindrical lens. Figure 1 presents the cylindrical interferometric stitching system. Firstly, the plane wave from the interferometer is converted into a cylindrical wave by the CGH and projected on the cylindrical lens to be tested. Then, after interaction with the cylindrical lens surface, the reflected light returns to the CGH, where the cylindrical wavefront is converted into the plane wavefront and directed back into the interferometer. After, the discrepancy between the measured and reference cylindrical surfaces are obtained by analyzing the recorded interferograms. In order to measure cylindrical lenses with large apertures, it was necessary to have a CGH with a small f/number (the ratio of the system's focal length to the diameter of the entrance pupil, usually greater than 1). The CCH used in our experiment had a f/number of 3, which enables observation and detection within 20\({}^{\circ}\) and, as a result, prevents obtaining the whole surface profile of a cylindrical lens by scanning once. The problem mentioned above is usually solved with the sub-aperture stitching technique [4; 5; 6; 7; 8; 9; 10]. Such an approach is also helpful for characterization of optical components with high numerical apertures. Multiple scanning of a sample requires preliminary planning of dividing the whole aperture into sub-apertures at different angles of observation. To ensure the whole tested surface is covered during the measurements and to control the stitching process, the adjacent segments from the sub-apertures must have overlapping regions. The reconstructed profiles of these overlapping regions have to be identical to smoothly and continuously supplement the information obtained from the non-overlapping regions. In the measuring approach, we set the global Cartesian coordinate system so that the X-axis coincides with the focus of the CGH, the optical axis of the tested sample, and the axis of rotation of the rotatable stage (Figure 2a). \(Z\) represents the direction of defocus and \(Y\) is the horizontal direction. Since the actual positions of the sub-apertures are unknown in the traditional algorithm [5], we assumed that the areas covered by the sub-apertures were uniformly allocated on the tested surface. To obtain uniform allocations of the overlapped and non-overlapped regions, the rotation axis of the multi-dimensional adjustment device platform is adjusted with the optical axis of the cylinder and the focus of the CGH (Figure 2b). Figure 1: Schematic view of the cylindrical interference system. Figure 2: (**a**) Schematic diagram of sub-aperture layout; (**b**) Schematic view of a multi-dimensional adjustment stage. However, in real situations, a deviation may exist in the adjustment. As a result, the mentioned three axes are usually mismatched. Sub-aperture data acquisition in the rotating process may introduce relative position errors, including: (1) turntable radial error caused by inaccuracies in \(\Delta y\) and \(\Delta z\), (2) the turntable axial runout, caused by \(\Delta x\), (3) the rotating error of the turntable shaft caused by the change in \(\Delta\beta\) and \(\Delta\gamma\) (Figure 2b), and (4) the turntable angle error caused by the change in \(\Delta\alpha\) (Figure 2b). The real coordinates on the surface of the cylindrical lens, \(\rho_{2}\), \(\Theta_{2}\), and \(X_{2}\), can be expressed through the real coordinates, \(\rho_{1}\), \(\Theta_{1}\), and \(X_{1}\), respectively, according to the following formula [16]:  
\[\left[ \begin{array}{c}\rho_{2}\\ \Theta_{2}\\ X_{2}\end{array} \right]=\left[