## 1 Introduction  
For the convenience of reading, most acronyms and symbols used here are listed in Table 1.  
Satellite laser communication received global attention for its high-speed communication rate, good confidentiality, small size, and lightweight terminal. Its communication rate is two orders of magnitude higher than conventional satellite microwave communication, which can reach the Gbps level [1; 2; 3]. In the early stage of the proposition of this technique, the communication terminals were mainly used for principle verification, and scientists have conducted many experimental verifications between different links, such as LEO-LEO (Low Earth Orbit), LEO-MEO (Medium Earth Orbit), LEO-GEO (Geosynchronous Earth Orbit) and GEO-GEO links. However, after decades of development, networking communication engineering applications, such as the Star Link plan of Space X Ltd., Cardiff, UK, the Europe data relay system (EDRS) [4], and China's sky and earth integrated information network gradually stepped onto the stage [5]. Therefore, the development of technology has put forward a higher standard for the performance of the terminal [6]. In a typical satellite laser communication system, a communication modulation subsystem, a beam-pointing subsystem (also known as APT (Acquisition, Pointing, and Tracking)), and a telescope subsystem are required. The beam-pointing subsystem is mainly responsible for controlling the incoming and outgoing beam direction for the acquisition, tracking, and pointing of the target in advance, but the pointing beam is very narrow, and the beam width used in communication links is usually tens of microradians, hence a little pointing error angle will lead to an increase in the communication error rate, or even cause the interruption of the link. Therefore, the beam-pointing accuracy is very important [7]. Thus, precise pointing mechanisms (such as a fast-steering mirror) are usually used to improve the beam-pointing angle when the terminal is launched into orbit [8]. Before the system begins to communicate, the target must be stably tracked (i.e., the incoming beam angle must be accurately obtained) by a fine tracking path. The point-ahead mirror controls the communication beam, causing it to rotate by a small angle relative to the direction of the incoming beam and point towards the target satellite. Therefore, if there is an inherent error in the angle between the transmitting and receiving paths, the point-ahead beam direction will deviate from the ideal angle during communication, which is called a coaxiality error (CE). A CE will significantly affect the performance of the APT system, such as acquisition probability and link stability. Before a satellite is launched into orbit and prepared to capture the target, the CE requires rigorous calibration such that the transmitting and receiving paths are parallel to each other. The beam can then point precisely toward the target bearing. However, various factors influence coaxiality errors, such as assembly, vibration and detection system errors [9], making it difficult to reduce the CE. Hence, rigorous corrective measures are required to reduce the CE based on the measurement results. Generally, the cube corner retro-reflector (CCR) is used in coaxiality calibration because of its good beam back-reflection characteristic. [PERSON] et al. [10] used the CCR to measure the accuracy of the CE and point-ahead angle. The laser source emits communication light, and the light is transferred throughout the APT system. The CCR is set at the surface of the telescope, then the communication light is reflected back into the APT system by CCR and transferred to the surface of the acquisition detector. In this way, the CE can be determined by measuring the displacement between the centroid of the incoming and outgoing beam faculae. Thus, the errors leading to CE can be introduced in every unit when the beam passes through.