## 1 Introduction  
The LIGO scientific collaboration and the VIRGO team detected gravitational waves from the merger of two black holes on 14 September 2015 [1; 2]. This marked a significant leap forward for gravitational astronomy. Ground-based gravitational wave detectors are affected by seismic noise, correlated gravity gradient noise, and thermal noise from the final stages of suspension and test masses, generally limiting their detection capabilities to high-frequency gravitational waves above 10 Hz. Space-based gravitational wave detectors are situated in an excellent vacuum environment far from Earth-based noise sources. Space-based gravitational wave detection involves constructing constellations of interferometers with extremely long arm lengths while employing drag-free spacecraft technology. By using payloads such as inertial sensors and micro-thrusters to counteract non-conservative forces, these detectors can effectively detect low-frequency gravitational waves [3]. Notable examples include the LISA (Laser Interferometer Space Antenna) mission and China's Taiji program [3; 4]. Gravitational wave detection in space involves the deployment of Michelson interferometers, having arm lengths ranging from tens to millions of kilometers [5; 6; 7]. To discern gravitational wave signals, these systems measure relative distance variations at the picometer scale through heterodyne interferometry [3; 8]. Differential wavefront sensing is used to establish laser interferometric links. By measuring the phase difference between the local and distal incident beams and converting it into an angle, this information is used for satellite drive control. This forms the basis for establishing and maintaining stable laser links [4; 9; 10]. After the satellite is launched into space, it experiences jitter due to the influence of complex space environments such as solar wind and cosmic rays, which are non-conservative forces. The drag-free control system will ensure the satellite's attitude stability as much as possible [11]. However, residual errors will still couple into the outgoing laser. The telescope on the receiving satellite platform will amplify this jitter. Ultimately, this jitter will couple into the image plane of the Quadrant Photodiode (QPD) detector, causing beam walk of the distal incident measurement beam on the image plane. The overlap between the distal instant measurement beam and the local beam will be reduced, affecting the measurement results of the DWS technique. As illustrated in Figure 1a, pointing jitter leads to variations in the angle between the incident and local beam, manifesting as differential beam walk lengths on the QPD image plane.The measurement performance of the DWS technique in this situation is shown in Figure 1b. Measurement errors caused by beam walk may affect the results of measuring the angle between the distant beam and the local beam. The angle between the two beams serves as a direct feedback parameter for satellite link establishment. If there is a significant deviation between the DWS measured angle and the actual angle, it could impact both the establishment and stability of the link. This could potentially lead to the failure of gravitational wave detection [12; 13; 14]. Consequently, it is crucial to explore the correlation between beam walk and the DWS signal. Studies have been conducted on the technical characteristics of DWS and beam walk. Previous research by [PERSON] et al. [15] introduced an analytical model for the DWS technique centered on Gaussian beam propagation, emphasizing the significant nonlinear effects in DWS signals, which necessitate consideration during the system's design and calibration. In the context of the LISA program, [PERSON] et al. [12] in 2012 proposed employing a pupil relay imaging system to curtail beam walk arising from measurement beam pointing jitter. In 2018, [PERSON] et al.