## 2 Analysis of the Relationship between Beam Walk and DWS Signal
### Numerical Simulation Analysis of Beam Walk  
The phenomenon of beam walk, associated with the Gaussian flat-top beam, alters the region of spot overlap on the QPD detection surface, consequently impacting the integration area for the DWS signal. In the numerical simulation, the radius of the QPD detector's image plane \(L\) is set to 1 mm with the gap \(h\) of 25 \(\upmu\)m. The laser wavelength is 1064 nm. The radius of the Gaussian spot and the Gaussian flat-top spot are 0.5 mm. The propagation distance of the Gaussian beam is 0.5 m with a power of 20 mW. The power of the Gaussian flat-top beam is 100 pW. With a beam walk of \(x_{0}\) in the yaw direction from the distant Gaussian flat-top beam, as depicted in Figure 2b, the integration areas along the x-axis are delineated as \((x_{0}-\frac{\omega_{f}}{\cos(\varphi)},-h)\) and \((h,x_{0}+\frac{\omega_{f}}{\cos(\varphi)})\). GAO [18] indicated, in the context of the Taiji program, that the angle deviation for the incident beam, relative to QPD, ranges between 300 \(\upmu\)rad and \(-\)300 \(\upmu\)rad. The angle measurement discrepancy attributed to the DWS methodology must not exceed 300 nrad. Hence, any area where the error remains below 300 nrad is classified as the linear region. \[|\Delta\varphi(\varphi)|=\left|\frac{DWS(\varphi)}{k}-\varphi\right|\leq 300 \text{ nrad}\cdots\forall\varphi\in[-300\ \upmu\text{rad},300\ \upmu\text{rad}] \tag{12}\]  
where \(\varphi\) is the angle between the two beams, \(\Delta\varphi(\varphi)\) is the angular measurement error caused by the nonlinearity of the DWS technique, DWS(\(\varphi\)) is the measured DWS signal value, and \(k\) is the slope of the linear curve fitting. The DWS signal and angle measurement error \(\Delta\varphi(\varphi)\) are illustrated in the Figure 3 without the beam walk. When the measured light beam undergoes pointing jitter, it is considered to be the motion around the exit pupil of the telescope, namely the entrance pupil motion of the optical platform. Due to the extremely small angle of deflection and the fact that the optical platform only refracts the incident light beam, the beam walk caused by the image plane is approximately equal to the arc length. It can be obtained by the following formula:  
\[\Delta l\approx\Delta\varphi\cdot R \tag{13}\]  
\(\Delta l\) represents the offset of the light beam relative to the center point of the QPD, \(\Delta\varphi\) denotes the angular offset, and \(R\) stands for the beam propagation distance. Given a fixed beam propagation distance, as the angular offset changes, the beam walk increases with the increasing angular offset. Assuming an axial propagation distance of 2 m, the relationship between the beam walk at different degrees and the numerical results of the DWS technique, as well as phase angle conversion results, can be established, as shown in the Figure 4.  
The angular measurement error under different beam offsets can be obtained by subtracting the numerical results of the DWS technique from the phase angle conversion results, as shown in Figure 5.  
Figure 4: DWS signals for various degrees of beam walk during light propagation over a 2 m axial distance. Figure 3: Variation of angle measurement error \(\Delta\varphi(\varphi)\) in the absence of beam walk (0 \(\upmu\)m Scenario). It can be observed that when the beam walk is approximately within \(\pm\)116 \(\upmu\)m, it meets the requirement of angular measurement error within \(\pm\)300 \(\upmu\)rad. For the maximum beam offset, since the axial propagation distance of the beam is 2 m, the resulting beam angular deviation can be calculated to be approximately \(\pm\)58 \(\upmu\)rad using Equation (13), which is significantly below the required \(\pm\)300 \(\upmu\)rad range for DWS nonlinearity specified by the Taiji program. As the propagation distance decreases, theoretically, the beam walk caused by the same angle will also decrease. Thus, reducing the distance may be effective in reducing measurement errors.