# Measurement of High Numerical Aperture Cylindrical Surface with Iterative Stitching Algorithm
## 1 Introduction  
The requirements for optical instruments are constantly increasing. Many optical instruments need one-dimensional shaping of the light source realized by cylindrical lenses. Cylindrical lenses have been widely used in high intensity laser systems and spectroscopic and interferometric devices. Their manufacturing processes, which include cutting, grinding, and polishing, are more complex and difficult than for spherical lenses [1; 2]. Therefore, quality control and characterization are required. Tactile measurements are usually recorded with a coordinate measuring machine or a cylindricity measuring instrument. This technology is mature as has been in use for a long time. These methods are highly accurate and provide various error compensation, but only have a small number of sampling points in a certain direction [3]. Besides, it is necessary to be extraordinarily careful to avoid damaging the precision instrument. Non-contact measurements are realized with optical interferometric techniques known for their high accuracy and dynamic measurement capability coupled with non-destructiveness. At present, plane and spherical interferometers, like that developed by Zygo Co. (Middlefield, CT, USA), are relatively accurate. However, there are some difficulties in using interferometry to measure large optical surfaces or high numerical apertures due to the problems in manufacturing referenceoptics for these purposes. The sub-aperture stitching technique [4; 5; 6; 7; 8; 9; 10] has to be applied in this case. In [PERSON] et al. [11; 12], stitching interferometry of high numerical aperture cylindrical optics without using a fringe-nulling routine was proposed. An outgoing plane wave from an interferometer is converted into a cylindrical wave by using a Computer Generated Hologram (CGH) [13; 14]. Although this work simplified the measurement procedure without meeting the null test condition [8], the described method cannot totally remove high-order errors. [PERSON] et al. [15] introduced the first order cylindrical coordinate transformation approximation method to measure the complete 360\({}^{\circ}\) cylinder surface. The wavefront of the light carrying information on the large aperture was divided into several sub-parts, and the whole surface profile of the metal cylindrical shaft was obtained by the stitching algorithm. However, the sub-aperture stitching technique, based on the null test method, produces errors when the axes of the lens, the wavefront, and the rotation stage do not coincide with each other. A multi-dimensional adjustment device was used to satisfy the null test condition by adjusting the position of the axis of rotation and to obtain the interferogram with a minimal number of fringes in the measured part (sub-aperture) of the lens. The actual errors of the turntable movements can cause miss-arrangement of the sub-apertures. In order to overcome this problem, we propose an iterative algorithm to correct the mismatch and to establish an interferometric stitching measurement system for cylindrical lens examination, where a CGH cylinder null is used as the cylindrical wave converter. We obtain a more accurate stitching result of the surface of the cylindrical lens. The remainder of the paper is organized as follows. In Section 2, we introduce the principles of the stitching algorithm that have been used in the past. The interferometric system for cylindrical measurement is described in this section. Considering the adjustment errors caused by the null test, we describe a cylinder stitching model based on the iterative algorithm, which can constantly adjust the position of the sub-aperture in the global coordination. In Section 3, we compare the results of the simulations obtained using the proposed iterative algorithm with the traditional algorithms. The results of the proposed method show good performance, with the errors we added to the sub-apertures being almost eliminated.