## 2 Principle
By combining the phase difference and DWS technology to calculate the angle deflection amount, the measurement and compensation of the incident angle deviation of the signal laser beam can be achieved. A detailed derivation of the DWS angular phase conversion process is given below, based on Figure 1c. This was first analyzed based on the plane wave model in the QD photosensitive surface. The local oscillation laser (green beam) is incident collimated, the signal laser (red beam) has an incident angular deviation \(\alpha\), and the spot radius is \(R\). The QD origin is \(O\), and the channel width is \(d\). For any point \(Q\) in the spot area, the distance from the origin is \(y\), and the angle between \(OQ\) and the detector's azimuthal axis is \(\theta\). For the pitch axis direction, the optical range difference between point \(Q\) and point \(O\) is \(y^{\star}\times sin\theta\times sin\alpha\), and the corresponding phase difference is \(y\times sin\theta\times sin\alpha\times 2\pi/\lambda\)[19]. Integral averaging of the phase difference between each point and point \(O\) in each quadrant of the spot area yields the output wavefront phase value \(\psi\). The expressions for the output wavefront phase values \(\psi_{A}\), \(\psi_{B}\), \(\psi_{C}\), and \(\psi_{D}\) in each quadrant are as follows. \[ \begin{split}\psi_{A}=\psi_{B}&=\frac{1}{1/4\pi R^{ 2}}\times\frac{2\pi}{\lambda}\times\int_{0}^{1/2\pi}\int_{d/2}^{R}y\times\sin \theta\times\sin\alpha\times rdrd\theta=\frac{8R^{3}-d^{3}}{3\lambda R^{3}} \sin\alpha\\ \psi_{C}=\psi_{D}&=\frac{1}{1/4\pi R^{2}}\times\frac {2\pi}{\lambda}\times\int_{\pi}^{3\pi/2}\int_{d/2}^{R}y\times\sin\theta\times \sin\alpha\times rdrd\theta=-\frac{8R^{3}-d^{3}}{3\lambda R^{3}}\sin\alpha \end{split}  \tag{6}\]  
The relationship between the pitch axis phase difference \(\Delta\psi_{PIT}\) and the angular deflection \(\alpha\) is established by the wavefront phase in each quadrant and can be expressed as Equation (7). A similar relationship exists between the azimuthal phase difference \(\Delta\psi_{AZI}\) and the angular deflection \(\alpha\). \[ \begin{array}{l}\Delta\psi_{PIT}=\frac{1}{2}(\psi_{A}+\psi_{B}-\psi_{C}-\psi_{ D})=\frac{16R^{3}-2d^{3}}{3\lambda R^{2}}\sin\alpha\\ \Delta\psi_{AZI}=\frac{1}{2}(\psi_{A}+\psi_{D}-\psi_{B}-\psi_{C})=\frac{16R^{3} -2d^{3}}{3\lambda R^{2}}\sin\alpha\end{array}  \tag{7}\]  
In the small angle deflection case, the approximation is: \(sin\alpha\approx\). The relationship between the QD wavefront phase difference \(\Delta\psi\) and the angular deflection \(\alpha\) can be expressed as Equation (8). This relation is mainly affected by the gap width \(d\) and spot radius \(R\). \[\Delta\psi\approx\frac{16R^{3}-2d^{3}}{3\lambda R^{2}}\alpha \tag{8}\]  
In the heterodyne interferometry system in the field of actual gravitational wave detection, after the signal laser beam is transmitted over an ultra-long distance in space, it will approximately follow a flat-top distribution when it reaches the detector at the receiving end. The local oscillator laser beam will follow a Gaussian distribution when reaching the detector at the receiving end. The DWS signal will be affected simultaneously by the far-end flat-top beam and the local Gaussian beam [20; 21]. The laser source is located at the origin in the spatial Cartesian coordinate system, and the z-axis represents the direction of transmission of the beam perpendicular to the photosensitive surface of the detector. The complex amplitude of the local Gaussian beam traveling along the Z-axis can be expressed as Equation (9) [22], where \(A_{0}\), \(z_{g}\), \(\omega(z_{g})\), and \(R(z_{g})\) represent the amplitude, propagation distance of the Gaussian beam, spot radius, and wavefront radius of curvature at position \(z_{g}\), respectively. The term \(\varphi(z_{g})\) denotes the additional optical phase.