# Optical Pupil Shift Correction Method for Large Ground-Based Optical Telescopes
## 2 Structural Optics Hybrid Modeling of the Telescope  
The diagram of the 4 m class ground-based telescope system described in this article is illustrated in Figure 1, the optical system of which is in the form of a Cassegrain. The dimensions of the system are 12.5 m \(\times\) 5.8 m (height \(\times\) diameter) and a total mass of approximately 90 tons. The optical structure of the telescope mainly includes a tracking frame, primary mirror, secondary mirror, Couder system, adaptive optical imaging system, and multiple imaging terminals. To simplify the analysis process, we only considered the first five parts. The tracking frame is the carrier of the telescope's optical system, which precisely tracks targets in select areas in the sky through the high-precision rotation of the elevation and azimuth axis systems. The interconnection between the tracking frame and each optical element is depicted in Figure 1.  
The complexity of the 4 m class telescope system makes it difficult to model every component in detail during optical analysis, despite eliminating other imaging terminals. Therefore, the model is simplified as follows. 1. The overall gravity- and temperature-induced deformations of the telescope's tracking frame are uniformly affected by the deviation in the position of the secondary mirror from the primary mirror. This assumption is used as the basis for discussion on the optical-axis deviation of the primary system. 2. The telescope azimuth axis and elevation-bearing-load margin are extremely high, and it is regarded as an ideal body in the process of tracking, i.e., the influences of bearing friction and clearance on the optical system are dismissed. 3. The positional accuracy of the main mirror of the whiffletree hydraulic support system has an extremely high resolution. The main mirror's optical axis does not change. The hydraulic correction process has a short response time. In the process of following the elevation movement, the hydraulic correction process is regarded as the ideal movement, dismissing the main mirror's position offset error and correction delay. With these assumptions, we applied the ray-tracing method to establish the overall optical model of the telescope (Figure 2). Figure 1: The simplified telescope structure. According to the working condition of a telescope, the positional accuracy of the secondary mirror plays a decisive role in changing its optical axis. The position of the secondary mirror relative to the primary mirror is expressed by the finite element analysis of the position of the nodes on the surface of the secondary mirror (Figure 3). Therefore, in the process of the ray-tracing of a telescope, we consider the change in the secondary mirror's nodal position relative to the primary mirror. The primary and secondary mirror nodal displacements can be expressed as:  
\[\left\{ \begin{matrix}q_{f}\\ q_{s}\end{matrix} \right\}=\left[ \begin{matrix}\Phi_{f}\\ \Phi_{s}\end{matrix} \right]\left\{x\right\}, \tag{1}\]  
Figure 3: The finite element analysis model of the telescope. The model includes the telescope mechanics, primary mirror, primary mirror cell, secondary mirror, and hexapod system. Figure 2: The overall optical model of the telescope. The primary mirror is the pupil of the system, conjugated to the deformation mirror and the wavefront detection system. Beam expansion system is used for different size transfer between different optical pupils. Wavefront detection system refers to Hartmann detector. where \(\left\{q_{f}\right\}\) and \(\left\{q_{s}\right\}\) are the nodal displacement vectors of the primary and secondary mirrors, and \(\left[\Phi_{f}\right]\) and \(\left[\Phi_{s}\right]\) are the selection matrices associated with the primary and secondary mirrors. The element at the mirror nodal position is 1, and the rest of the elements are 0.