# Parameter Design for Inter-Satellite Laser Link Acquisition Under Weak Light Conditions
## 5 Monte Carlo Simulation  
The theoretical formula needs to be verified through simulation; we will introduce how to perform the simulation in this section. Since simulating a continuous process is not feasible, the scanning process needs to be split into smaller time intervals for simulation that are shorter than both the jitter data interval \(t_{j}\) and the acquisition camera integration time \(t_{int}\). To streamline the simulation process, the time interval \(t_{s}\) between two scan points can be chosen as the greatest common divisor of \(t_{j}\) and \(t_{int}\), or smaller if necessary.  
To generate a random time series of jitter, we introduce a random phase on the amplitude spectral density (ASD) and generate a random amplitude for each frequency bin. According to [25], we then perform the inverse Fourier transform (IFFT) to produce jitter data. The jitter frequency \(f_{s}\) is 10 Hz, and the jitter data are loaded into the scan points at regular integrals.  
Since the scanning process is divided into many small time segments during the simulation, the calculation of the energy accumulation can naturally be split into corresponding small time periods. According to Equation (5), the trajectory \(s\) is hard to determine theoretically. However, as shown in Figure 2, when the time integrals between scan points are small, we can approximate the integration along the trajectory \(s\) between two scanning points as trajectory \(s^{\prime}\). Based on the properties of Gaussian beams, we can treat the distance from the spot center to the trajectory \(s^{\prime}\) as the value \(x\) of the X-axis. Therefore, once the value \(x\) is determined, we only need to integrate along the Y-axis.  
The integration along the Y-axis can be divided into two scenarios: when the endpoints are on the same side of the X-axis or when they are on opposite sides of the X-axis:  
1. When the endpoints on the same side of the X-axis,  
\[y^{\prime}_{1}=\max(|y_{1}|,|y_{2}|),\ y^{\prime}_{2}=\min(|y_{1}|,|y_{2}|), \tag{25}\]  
\[S_{sig}=\frac{P_{0}Q_{e}}{h\upsilon}\int_{t^{\prime}}^{t^{\prime}+t_{s}}e^{- \frac{2^{\prime}\left(t^{\prime}\right)^{2}}{\upsilon_{0}^{2}}}dt=\frac{P_{0} Q_{e}}{\upsilon_{1}h\upsilon}e^{-\frac{2\upsilon^{2}}{\upsilon_{0}^{2}}}\int_{y^{ \prime}_{2}}^{y^{\prime}_{1}}e^{-\frac{2\upsilon^{2}}{\upsilon_{0}^{2}}}dy \tag{26}\]  
where \(t^{\prime}\) is the start time of this small time interval.  
2. When the endpoints are on opposite sides of the X-axis,  
\[S_{sig}=\frac{P_{0}Q_{e}}{h\upsilon}\int_{t^{\prime}}^{t^{\prime}+t_{s}}e^{- \frac{2^{\prime}\left(t^{\prime}\right)^{2}}{\upsilon_{0}^{2}}}dt=\frac{P_{0 }Q_{e}}{\upsilon_{1}h\upsilon}e^{-\frac{2\upsilon^{2}}{\upsilon_{0}^{2}}}\left( \int_{0}^{|y_{1}|}e^{-\frac{2\upsilon^{2}}{\upsilon_{0}^{2}}}dy+\int_{0}^{|y_{ 2}|}e^{-\frac{2\upsilon^{2}}{\upsilon_{0}^{2}}}dy\right). \tag{27}\]  
Figure 8: Effect of noise on the spot centroid error as the integration time varies.  
According to the above formula, we can sum the number of photoelectrons accumulated between scanning points during the integration time to calculate the number of photoelectrons gathered during the spatial scanning process as the beam sweeps through the receiving SC. This allows us to determine the photoelectrons. By comparing the photoelectrons to their threshold, we can assess whether the acquisition is complete.  
Since the integration time is relatively fixed, we explore the optimal acquisition system parameters by varying the scan speed and track width. In summary, we can perform a Monte Carlo simulation to mimic the real acquisition process, as shown in Figure 9.