## 2 Optical Model  
In the gravitational wave detection interferometer, a far-field beam (according to the properties of Gaussian beams, the far-field beam can be approximated as a flat-top beam) is expanded by the telescope system and emitted by the laser in the adjacent satellite. After the telescope intercepts and converges, it interferes with the local Gaussian laser on the QPD. When properly adjusted (according to the DWS signal of the QPD), the center deviation of the double beams can reach below the micrometer level [14; 15]. For the convenience  
Figure 1: A schematic diagram of the far-field beam received by the gravitational wave telescope. The far-field beam is received by the entrance pupil of the telescope, reflected by the telescope, and then passed through the imaging system to reach the detector. of research, we assume that the center positions of the two beams are consistent with the incident angle. The local Gaussian beam on the detector can be written as (in Cartesian coordinates):  
\[E(x,y,z)_{gauss}=A_{1}e^{-\frac{x^{2}+y^{2}}{w(z)^{2}}}e^{-i[k(\frac{x^{2}+y^{2} }{2R(z)}+\varphi_{1}]+i2\pi f_{1}t} \tag{1}\]  
where \(f_{1}\) is the frequency of the Gaussian beam, \(A_{1}\) is the amplitude. \(\varphi_{1}\) is the initial phase of the Gaussian beam, \(w(z)=w_{0}\sqrt{1+\left(z/z_{r}\right)^{2}}\) is the beam width of the Gaussian beam, \(R(z)=z(1+(\frac{z_{r}}{z})^{2})\) is the radius of curvature of the equiphase surface of the Gaussian beam, and \(z_{r}=(\pi w_{0}^{2})/\lambda\) is called the Rayleigh criterion or confocal parameter. In the formula, in addition to the time phase factor \(i2\pi f_{1}t\) and the initial phase \(\varphi_{1}\) due to optical element defects, the phase term \(\frac{x^{2}+y^{2}}{2R(z)}\) is similar to the defocus. Its magnitude depends on the distance z between the interference position and the beam waist. It can be seen from Formula (1) that when the propagation distance \(z\sim\infty\), we can get \(z>>z_{r}\) and \(R(z)\approx z\). This means that the Gaussian beam can be approximated as a plane wave emanating from the point \(z=0\) at the far field. Therefore, the far-field beams, separated by millions of kilometers in space gravitational wave detection, can be regarded as flat-top beams, which can be written on the detector as:  
\[E(x,y,z)_{flat}=\left\{ \begin{array}{ll}A_{2}e^{-ik(x\sin\alpha+z\cos\alpha) +i2\pi f_{2}t+\varphi_{2}}&x^{2}+y^{2}\leq R\\ 0&else\end{array} \right. \tag{2}\]  
where \(\varphi_{2}\) is the phase of the flat-top beam due to defects in the optics. For the convenience of derivation, in the subsequent theoretical derivation, the point to the angle \(\alpha\to 0\), \(A_{2}\), is the amplitude. Since the phase information is independent of the amplitude, we normalize the amplitudes of the two beams in the subsequent derivation. The phase demodulation after two-beam interference can be obtained by simply multiplying and integrating complex amplitudes. In the theoretical analysis, the time factor is irrelevant to the phase and can be ignored, so we assume that in a circular detection area of radius \(\rho\), the phase information can be written as:  
\[\text{int}=\int_{s}E_{flat}E^{*}_{gauss}dr^{2}=\int_{s}e^{-\frac{x^{2}+y^{2}} {w(z)^{2}}}e^{ik\Delta\varphi(\rho,\theta)}dr^{2} \tag{3}\]  
where \(\Delta\varphi(\rho,\theta)=\varphi_{2}-\varphi_{1}-\frac{x^{2}+y^{2}}{2R(z)}\) is the static wavefront aberration of the interference of the two beams.