## 2 Analysis of the Relationship between Beam Walk and DWS Signal  
### DWS Interference Signal Model in Practical Situation  
As deduced from the prior discussions, the DWS signal undergoes interference from both the local Gaussian beam and the distant Gaussian flat-top beam [15,18]. Utilizing a rectangular spatial coordinate with the laser source at the origin, the primary mode of Gaussian beam propagating along the z-axis can be represented as follows:  
\[E_{G(x,y,z)}=\frac{A_{0}}{\omega\big{(}z_{g}\big{)}}\cdot e^{-\frac{x^{2}+y^{ 2}}{\omega^{2}(z_{g})}-i\mathbb{R}\Big{[}\frac{x^{2}+y^{2}}{2R(z_{g})}+i\phi \big{(}z_{g}\big{)}+z_{g}\Big{]}} \tag{1}\]  
where \(A_  
The laser emitted from the distal satellite platform will diffuse after far-field transmission, resulting in a spot size much larger than the aperture of the local satellite platform telescope. Therefore, the laser entering the telescope can be approximated as a parallel beam [18]. Within the same rectangular spatial coordinate system, anchored at the laser source, the complex amplitude of the flat-top beam propagating along the z-axis can be succinctly represented as follows:  
\[E_{ft(x,y,z)}=E_{0}e^{-ikz/\varpi(x^{2}+y^{2}\leq\omega_{f}{}^{2})} \tag{2}\]  
where \(E_{0}\) is the amplitude, \(\omega_{f}\) is the radius of the flat-top beam spot on the QPD surface, and \(z_{f}\) is the propagation distance of the flat-top beam. In the actual case, the distal incident beam's Gaussian flat-top profile gets truncated by the circular aperture of the receiving telescope, illustrated by the dotted blue circle in Figure 2a. Given the marginal intensity of the interferometric signal in the peripheral integration zone, this study simplifies the analysis by assuming that the Gaussian flat-top beam is truncated by a square aperture, represented by the blue box in Figure 2b. This approximation does not significantly alter the qualitative analysis. This study primarily examines the beam walk in the x-axis direction to understand its relationship with the DWS signal, though conclusions drawn are equally applicable to the pitch direction. The interferometric signal's complex amplitude within the left semi-quadrant can be expressed as follows:  
\[F_{left}=\int{}_{S_{left}}dS\cdot E_{G(x,y,z)}E_{ft(x,y,z)} \tag{3}\]  
The complex amplitude for the interferometric signal in the right semi-quadrant is given by the following:  
\[F_{right}=\int{}_{S_{left}}dS\cdot E_{G(x,y,z)}E_{ft(x,y,z)} \tag{4}\]  
Figure 2: Representation of interference dynamics amidst x-axis beam walk. (**a**) Depicts the QPD surface within the black circle. The centrally injected local Gaussian beam is denoted by the red circle, while the blue circle with the dotted line demonstrates the beam walk of the distant Gaussian flat-top spot, establishing an angle \(\varphi\) between the two beams. (**b**) Exhibits Gaussian flat-top beam post-square-aperture truncation, encapsulated in the blue box. Here, the y-axis integration ranges for both left and right quadrants extend from \(-\omega_{f}\) to \(\omega_{f}\), whereas the x-axis spans from \((x_{0}-\frac{\omega_{f}}{\cos(\varphi)},-h)\) to \((h,x_{0}+\frac{\omega_{f}}{\cos(\varphi)})\).