# Active Optics and Aberration Correction Technology for Sparse Aperture Segmented Mirrors
## 1 Introduction  
The completion of Keck I and II telescopes has shown the direction for the further development of large-aperture telescopes. Summarize the current shape of the segment in large segmented telescopes. Furthermore, the hexagonal and circular segments splicing schemes seem more suitable for future large-aperture segmented telescopes [1, 2]. Compared with hexagonal segments, circular segments use fewer types of segments for splicing. The processing and preparation process is more mature. The design of the support structure is easier to implement, transplant, and expand. And the R&D is more cost-effective. Meanwhile, [PERSON] [3] analyzed that the MTF of different aperture circular and hexagonal segments splicing mirrors are basically the same. This proves the feasibility of using circular segment splicing. In addition, the typical representative of a circular segmented telescope is Giant Magellan Telescope (GMT) [4, 5]. The co-phase maintenance technology for the active optics of the segmented telescope is the key to ensuring the imaging quality of the optical system of the segmented telescope. In the traditional model, this technique mainly monitors the relative position of the two segments in real time by the edge sensors, which are located at the edge of the two adjacent spliced segments. Then the correction amount of the displacement actuator is calculated by the active optical control system. Finally, the spliced sub-mirrors are readjusted to theco-phase by adjusting the displacement output of the displacement actuator state [6; 7]. Due to the participation of the edge sensor in this mode, the measurement error of the edge sensor will affect the co-phase maintenance accuracy of the mode. Also, the sensor needs to be recalibrated when the temperature and zero drift of the sensor occur. For example, the edge sensor temperature drift of 6 nm/week, 10 nm/week, and 5 nm/month was used by Keck, E-ELT, and TMT telescopes, respectively [8; 9; 10]. The WH technique can correct the wavefront error of the segmented mirrors with spatial frequencies close to zero. The technique was first applied in the Keck telescope. WH technique is also applied in the two segmented ground-based telescopes, E-ELT, and TMT, which are currently under construction. In the above-mentioned telescope, the correction process of the segmented mirror aberration by this technique is as follows. Firstly, estimate each segment's surface using the Shack-Hartmann wavefront sensor (SH-WFS) and WH influence function. Secondly, correct the aberration of each segment surface. Then achieve the segmented mirror co-phase by using the traditional co-phase maintenance technique. Finally, reduce the segmented mirror shape introduced due to the sub-mirror surface error [11; 12; 13; 14]. For large-scale segments, stitching of the primary mirror, the process of correcting each segment aberration individually, is very complicated, and the correction time will be very long. While the WH technique only has a certain correction effect on the low-order aberration of the segment surface, which corresponds to a relatively small number of aberrations in the stitching mirror. This paper presents a combination of theoretical derivation and simulation analysis. Firstly, the polynomial of the aberration of the sparse aperture splicing mirror, the correction principle of the WH technique, and the solution method of the correction amount are derived. Then the co-phase maintenance control model based on the change of the edge sensors indication value in the traditional mode of the sparse aperture splicing mirror is derived. Meanwhile, to address the problem of traditional co-phase maintenance techniques relying on edge sensor measurement accuracy. A co-phase maintenance model based on the segmented mirror aberration evaluation model is proposed, which does not involve edge sensors. The calibration accuracy of two segmented mirror co-phase maintenance models is compared.