## 3 Aberration Theory Analysis  
From the complete Formula (6) of the phase signal, it is not difficult to see that, in theory, the wavefront aberration between the far-field beam and the Gaussian beam will be considered the initial phase difference in gravitational wave detection. However, due to factors such as the space environment and satellite jitter, the final performance of the telescope's mirror surface deformation and the increase in the distance between the primary and secondary mirrors can be equivalent to the changes in the telescope's aberrations [16]. The telescope's aberrations will always have subtle changes. Therefore, unstable aberration changes will be an inevitable source of errors in gravitational wave detection. In 2020, [PERSON] et al. used the first 25 items of the Zernike polynomial to fit the random aberration size of \(\lambda/20\)-\(\lambda/30\) using the Monte Carlo algorithm on phase demodulation [12; 13; 14]. However, it only analyzed static aberrations and did not have a detailed decomposition of specific aberrations. This paper will start from the basic Zernike polynomial and derive the effect of its various parts on phase demodulation. In actual situations, the situation of wavefront aberrations is extremely complicated, and the Zernike polynomial is a very good basis function for fitting the wavefront; for example, the spherical aberration and coma mentioned above can be easily fitted with it [17]. Its expression can be written as:  
\[Z_{i}(\rho,\theta)= \begin{cases}\sqrt{2(n+1)}R_{n}^{m}(\rho)\cos(m\theta)&\text {$i$ is even and $m\
eq 0$}\\ \sqrt{2(n+1)}R_{n}^{m}(\rho)\sin(m\theta)&\text{$i$ is odd and $m\
eq 0$}\\ \sqrt{n+1}R_{n}^{0}(\rho)&\text{$m=0$}\end{cases}  \tag{8}\]  
where \(R_{n}^{m}(\rho)\) is a polynomial concerning the wavefront curvature, \(m\) and \(n\) are positive integers, and \(n\geq m\). \(\rho=r/r_{0}\) is the normalized radial coordinate. The index \(i\) is a polynomial-ordering number and a function of \(n\) and \(m\). The specific expression \(R_{n}^{m}(\rho)\) is:  
\[R_{n}^{m}(\rho)=\sum_{s=0}^{(n-m)/2}\frac{\left(-1\right)^{s}(n-s)!}{s!(\frac {n+m}{2}-s)!(\frac{n+m}{2}-s)!}\rho^{n-2s} \tag{9}\]  
According to the above phase demodulation formula, the Zernike polynomial is divided into three parts for calculation. After substituting the first part \(\sqrt{2(n+1)}R_{n}^{m}(\rho)\cos(m\theta)\) into (6), (7), we obtain (10):  
In the actual phase demodulation, the detector can only detect the light intensity information, and the phase information cannot be obtained through the inversion of the real part and the imaginary part in the formula according to the complex amplitude integration in the theoretical simulation. At the same time, due to the Doppler effect between satellites [19], the frequency difference between the local Gaussian beam and the far-field beam is about \(\pm 15\) MHz. Therefore, the method we use is the quadrature demodulation algorithm shown in Figure 3 below. The time-varying optical signal acquired on the phase meter can be written as:  
\[I=\overline{I}[1+c\cos(\omega_{het}t+\Delta\varphi)] \tag{13}\]  
where \(\overline{I}=\frac{1}{2}(I_{max}+I_{\text{min}})\) is the average intensity of the resulting light field, \(c=\frac{I_{max}-I_{\text{min}}}{I_{max}+I_{\text{min}}}\) is the signal contrast, \(\omega_{het}=\omega_{gauss}-\omega_{flat}=2\pi(f_{1}-f_{2})\) is Heterodyne frequency, and \(\Delta\varphi\) is the phase difference between the two beams. The subsequent demodulation steps are as follows:  
1. The received interferometric light signal is converted into an electrical signal by a phase meter. 2. We use the NCO (numerical oscillator) to send out sine and cosine signals and multiply the interference signal. Then, we get Formula (11). It should be noted here that in actual detection, the heterodyne frequency is unknown due to the existence of the Doppler effect, and a subsequent feedback mechanism makes the final NCO send out the same sine and cosine signals as the heterodyne frequency.