# Parameter Design for Inter-Satellite Laser Link Acquisition Under Weak Light Conditions
## 2 Acquisition System Influencing Factors
### Laser Spot Location Error  
As shown in Figure 3, at a certain moment during the scanning process, the scanning beam passes through the receiving SC's telescope, crosses the aperture and imaging system, and is imaged onto the acquisition camera. The position of the spot formed on the camera can be calculated using the centroid algorithm: \(x_{a}=\sum_{i=1}^{n}x_{i}N_{i}/\sum_{i=1}^{n}N_{i}\), \(y_{a}=\sum_{i=1}^{n}y_{i}N_{i}/\sum_{i=1}^{n}N_{i}\), where \(n\) is the total number of pixels calculated, \((x_{i},y_{i})\) is the position of the \(i\)th pixel, and \(N_{i}\) is the number of electrons detected at the \(i\)th pixel. Thus, we can calculate the beam offset angle based on the reference position and the measured actual position [20]:  
\[\theta_{y}=\frac{\theta_{y}^{\prime}}{M}\approx\frac{y_{offset}}{f_{img}M}= \frac{y_{a}-y_{r}}{f_{img}M}, \tag{6}\]  
where \(M\) is the magnification of the telescope. The FOV of the QPD is smaller than the divergence angle of the laser beam. Therefore, accurately locating the spot's center is essential for ensuring the incident beam enters the QPD's FOV, which is the primary objective of camera acquisition. The Taiji program has selected a QPD FOV of \(\pm 1\)\(\upmu\)rad, and the acquisition precision before the telescope must be better than \(1\)\(\upmu\)rad. The corresponding angular measurement precision must be an order of magnitude better, reaching \(0.1\)\(\upmu\)rad. Considering \(M=100\), the angular measurement precision before the acquisition camera is \(\varphi=10\)\(\upmu\)rad. Figure 3: Diagram of the acquisition system. For clarity, only the pitch direction is shown here, where \(\theta_{y}\) is the beam deviation angle in the pitch direction, \(\theta  
Based on the maximum permissible deviation of the incident beam, we can calculate the allowable laser spot centroid error as follows:  
\[\Delta x=\varphi\cdot\frac{f_{img}}{L_{pixel}}, \tag{7}\]  
where \(L_{pixel}\) is the pixel size of the acquisition camera, and \(\Delta x\) is the corresponding number of pixels. In the Taiji program, the preliminary design parameter is set to \(f_{img}=250\) mm and \(L_{pixel}=15\) um. The calculation shows that the accuracy of the laser spot's center positioning algorithm is 0.17 pixels. Taking mounting errors and redundancy into account, the algorithm requires an accuracy better than 0.1 pixels. The traditional center-of-mass algorithm is sensitive to noise and struggles to achieve high-precision positioning. [PERSON] et al. (2014) proposed an improved centroid method for weak light conditions by reducing the number of pixels introduced into the center-of-mass calculation. To minimize the impact of the dark current noise introduced during computation, it is essential to select a suitably sized window around the actual centroid position. This ensures the entire spot remains within the window while reducing the number of involved pixels. To determine the appropriate window size, we first need to estimate the size of the spot. The diameter of the telescope and the focal length of the imaging system dictate the size of the Airy spot formed on the camera, which can be expressed as follows:  
\[R_{spot}=\frac{1.22\lambda f_{img}}{2R_{clip}}, \tag{8}\]  
where \(R_{clip}\) represents the spot size clipped by the telescope and can be concluded by \(R_{clip}=R_{tel}/M\). Consequentially, the number of pixels corresponding to the spot diameter length can be determined by \(N_{spot}=2*R_{spot}/L_{pixel}\). Spot centroid position errors can be categorized into two components. The first is a fixed measurement offset, \(\Delta=x_{0}-E(x)\), which represents the difference between the actual center position \(x_{0}\) (without noise) and the expectation of measurement position \(E(x)\). The second component is the position fluctuation around \(E(x)\) because of the randomness of the noise. The fixed measurement offset is the primary influencing factor.