# Optical Pupil Shift Correction Method for Large Ground-Based Optical Telescopes
## 1 Introduction  
The increasing knowledge of the observable universe is driving the demand for telescopes with large numerical apertures to observe farther objects in space. The Cassegrain-type telescope has a large aperture and is used in radio astronomy, Earth stations, and satellite communication. A recent example of such an instrument is the James Webb Telescope. The increasing aperture of Cassegrain telescopes, and consequently their truss weight and back intercept, has amplified the degradation of imaging quality caused by the misalignment error due to structural and temperature changes, especially for the adaptive optics high-resolution imaging system. The compensation method of the error has received special focus and caused difficulties in the research of ground-based large-aperture telescopes [1]. The gravitational bending and sinking of the secondary mirror, as well as changes in the relative position of the primary and secondary mirrors due to temperature changes, are the main causes of the misalignment error of an optical system. After the optical system is folded into the Couder imaging system, it causes the system optical axis, and consequently the system optical pupil position, to shift [2; 3; 4]. For the adaptive optics system, the Hartmann measurement instrument and deformation mirror are placed at the optical pupil position of the primary mirror. The optical pupil shift causes the pupil exit surface of the primary mirror to draw a circle on the surface of the deformation mirror. As the aperture of the primary mirror increases, and the back intercept distance of the system becomes longer and the circular motion begins to severely influence the corrective effect of the imaging system [5; 6]. In terms of offset error correction, the European Southern Observatory first used a piece of correction technology based on active optics on the New Technology Telescope (NTT). However, the active support technology for the primary mirror does not solve in principle the optical pupil shift problem. [PERSON] et al. analyzed the sensitivity of the third-order coma of the two-reflection system to off-centering [7; 8; 9]. The alignment of the optical pupil and field of view in an off-axis three-mirror system by computer-assisted misalignment was demonstrated by [PERSON] et al. [10; 11], while [PERSON] et al. undertook the active alignment of the Advanced Technology Solar Telescope by means of optical analysis [12]. For the alignment of off-axis optical systems, [PERSON] and [PERSON] et al. realized the alignment of off-axis two reflectors using aberration fields and nodal aberration theory [13; 14; 15]. All of the above methods are used during the period of preliminary alignment of the telescope; however, the optical pupil shift correction proposed in this paper is generated by the bending and sinking of the ground-based large-aperture telescope and the temperature change, so the method proposed here is more advanced and novel in terms of the compensation in the observation process. In addition, [PERSON] used the inverse optimization method to adjust a Cassegrain telescope, adjusting the wavefront aberration from 0.283\(\lambda\) to 0.194\(\lambda\). [PERSON] used the nodal aberration theory to study the aberration type corresponding to the freeform surface located at any position in the system. However, these methods did not consider the optical pupil shift, and consequently could not correct the misalignment error [16; 17]. The adjustment mechanism of the secondary mirror can compensate the system aberration produced by the deviation in the position of the mirror. However, from the system's perspective, this compensation-correction method cannot correct the change of the optical axis of the main system [18]. Therefore, to compensate the main system's misalignment error and the optical axis change, we proposed a method based on the coma-free theory of the Cassegrain main system and the position-aberration compensation of the secondary mirror.