# Parameter Design for Inter-Satellite Laser Link Acquisition Under Weak Light Conditions
## 1 Introduction  
To detect GW signals in a frequency band ranging from 0.1 mHz to 1 Hz, several space-based missions are currently being actively pursued, including China's Taiji program [1, 2] and TianQin mission [3], as well as the ESA/NASA Laser Interferometric Space Antenna (LISA) mission [4]. These missions typically deploy three satellites to form an equilateral triangular constellation, with each arm containing a bi-directional laser link. By employing laser heterodyne interferometry technology, these missions measure the variations in the distance between free-floating test masses within the spacecraft (SC) along each arm, allowing them to detect GW signals with high precision. Establishing these laser links is a prerequisite for the measurement of distance fluctuations, which poses a significant challenge. Take the Taiji program as an example: the distance between the two satellitesis \(3\times 10^{6}\) km, and the laser pointing error must be suppressed to a level of \(1\)\(\mathrm{nrad}/\sqrt{\mathrm{Hz}}\). After long-distance transmission, the laser's power weakens to the order of just 100 pW. To establish laser links, the Taiji program/LISA uses a three-stage detector scheme for a single link: first, star trackers (STRs) are used to perform initial attitude determination, aligning the two satellites along an arm. Due to navigation errors and other factors, the remote satellite is positioned within an uncertainty cone (UC), which is larger than the divergence angle of the laser, making it challenging to directly establish a laser link between the two satellites. Next, acquisition cameras are used to detect the laser spot as the laser beam emitted by a remote satellite scans the entire UC. The required acquisition accuracy before the telescope is less than 1 \(\mathrm{\SIUnitSymbolMicro rad}\), which is smaller than the FOV of the quadrant photodiode (QPD), ensuring that the QPDs can receive continuous interference signals. Finally, QPDs are used to perform differential wavefront sensing (DWS), achieving high-precision mutual pointing at a level of \(1\)\(\mathrm{nrad}/\sqrt{\mathrm{Hz}}\). The second stage, known as acquisition, is crucial for establishing laser links and is the focus of this article. Several factors influence this stage, including scan speed, scan trajectory, the power of the laser, the threshold for acquisition judgement, and jitter caused by SC attitude positioning errors and noise from the micro-propulsion system. For spaced-based GW detection, laser beam orientation is controlled through a combination of SC attitude adjustments and in-plane angle modifications, known as the telescope-pointing scheme. In-plane angle modifications are adjusted by the pivot of the movable optical subassembly. Therefore, the scan speed is limited by the angular rotation speed, which is inherently slow. The optical power available for beam acquisition after ultra-long distance transmission is extremely low, and satellite jitter causes the scanning beam to deviate from its intended trajectory, making reliable and efficient acquisition a critical challenge under low scan speed conditions. The goal of this paper is to optimize parameter design to achieve the fastest acquisition time while accounting for the factors mentioned above. The process of establishing inter-satellite laser links has been extensively studied. [PERSON] et al. (2016, 2017) analyzed the scanning trajectory of the acquisition process, and [PERSON] et al. (2017) obtained the mean acquisition time and the optimum ratio of the UC to the standard deviation of receiving SC position distribution. A significant factor influencing the acquisition process is the jitter of the satellite platform or other moving bodies coupled to the outgoing beam, which has also been the focus of considerable research. [PERSON] et al. (2017) derived an analytical equation of the \"hit\" probability based on the assumption that the jitter follows a Gaussian distribution.