## 2 Principle
\[E_{G(z,y,z)}=\frac{A_{0}}{\omega\big{(}z_{g}\big{)}}e^{-\frac{x^{2}+y^{2}}{ \omega^{2}(z_{g})}-k_{1}^{2}\frac{x^{2}+y^{2}}{2R(z_{g})}+i\varphi(z_{g})+z_{g }]} \tag{9}\]  
The complex amplitude of the distal flat-top beam traveling along the z-axis can be expressed as Equation (10), where \(E_{0}\) denotes the amplitude, \(\omega_{f}\) denotes the spot radius of the flat-top beam in the detector's photosensitive plane, and \(z_{f}\) denotes the distance traveled by the flat-top beam. \[E_{ft(x,y,z)}=E_{0}e^{-kiz_{f}}\quad\left(x^{2}+y^{2}\leq\omega_{f}{}^{2}\right) \tag{10}\]  
In a practical system, the signal laser beam from the far end will be truncated by the circular aperture of the optical antenna at the receiving end, thus presenting a circular spot pattern in the QD photosensitive surface. To simplify the analysis, it was assumed that the signal laser beam was truncated by a square aperture tangent to the actual circular aperture, causing it to present a square spot pattern in the QD. This approximation did not change the results of the qualitative analysis. To illustrate the derivation and analysis of the resulting mathematical analytical equation, let us consider the angular deflection of the signal laser beam along the azimuthal axis. It can be shown that this equation will also be applicable to the pitch axis direction. The complex amplitudes of the heterodyne interferometric signals in the left and right half quadrants of the QD can be expressed as Equation (11), respectively, where \(\alpha\) denotes the deflection angle of the signal laser beam along the azimuth axis. \[ \begin{array}{l}E_{left}=\int_{S_{left}}E_{G(x,y,z)}\times E_{ft(x,y,z)}dS=E _{0}\times C_{1}\times C_{2}\times C_{3}\\ E_{right}=\int_{S_{right}}E_{G(x,y,z)}\times E_{ft(x,y,z)}dS=E_{0}\times C_{1 }\times C_{2}\times C_{4}\end{array}  \tag{11}\]where \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\) have the following expressions, respectively. \[\left\{ \begin{aligned} C_{1}&=\frac{A_{0}}{ \omega\big{(}z_{g}\big{)}}e^{[-i\varphi\big{(}z_{g}\big{)}-ikz_{g}-ik\sin{( \phi)}x_{0}+ikz_{f}]}\\ C_{2}&=\int_{-\omega_{f}}^{\omega_{f}}e^{-\frac{y^{2 }}{\omega^{2}(z_{g})}-\frac{iky^{2}}{2R(z_{g})}}dy\\ C_{3}&=\int_{x_{0}-\frac{R}{\cos{(\alpha)}}}^{-d} \frac{R}{\cos{(\alpha)}}e^{-\frac{x^{2}}{\omega^{2}(z_{g})}-\frac{ikx^{2}}{2R( z_{g})}+ik\sin{(\alpha)}x}dx\\ C_{4}&=\int_{d}^{x_{0}+\frac{R}{\cos{(\alpha)}}}e^{- \frac{x^{2}}{\omega^{2}(z_{g})}-\frac{ikx^{2}}{2R(z_{g})}+ik\sin{(\alpha)}x}dx \end{aligned} \right. \tag{12}\]  
According to the above equation, the DWS signal can be expressed as follows. \[\Delta\psi_{AZI}=\arg\Big{(}E_{right}\Big{)}-\arg\Big{(}E_{left}\Big{)} \tag{13}\]  
As demonstrated by the derivation results outlined above, it can be concluded that the DWS signal is predominantly influenced by the signal laser beam deflection angle \(\alpha\), the QD channel width \(d\), the spot radius \(R\), and the offset of the signal spot center of mass in the QD photosensitive surface \(x_{0}\). Subsequent to the conversion of the angular deflection into a phase difference in the QD photosensitive surface, the phase tracking and detection module utilizes a Costas phase-locked loop to measure the received signal phase, denoted as \(\psi_{S}\), with a high degree of accuracy, as shown in Figure 1d. Within a specific quadrant of the QD, the input signal \(V_{S}(t)\) of the phase-locked loop can be expressed as follows:  
\[V_{S}(t)=A_{S}\sin(\omega_{het}t+\psi_{i})\ \ \ \ (i=1,2,3,4) \tag{14}\]  
The NCO in the phase-locked loop generates in-phase \(V_{NCOI}(t)\) and quadrature branch signals \(V_{NCOQ}(t)\) (\(\omega_{NCO}\approx\omega_{het}\)) of a frequency that is similar to the input signal based on the initial frequency word. \(V_{NCOI}(t)\) and \(V_{NCOQ}(t)\) can be expressed as Equation (15). The phase tracking is achieved through constant feedback adjustment \(\omega_{NCO}\) during the loop locking process. \[