## 3 Aberration Theory Analysis
### Telescope Stability Analysis  
In the LISA or Taiji space gravitational wave detection projects, the evaluation criteria of the telescope's stability are different from those of traditional space telescopes. Parameters such as the wavefront index and frame deformation are unified as the PSD (power spectral density) of the optical path noise. Taking LISA as an example, the PSD of the optical path noise of the telescope is required to be in the range of 0.1 mHz\(\sim\)1 Hz:  
\[PSD\leq 1\frac{\mathrm{pm}}{\sqrt{Hz}}\sqrt{1+{(\frac{f_{c}}{f})}^{2}} \tag{16}\]  
To more intuitively feel the impact of dynamic phase changes on the evaluation of telescope stability, we simulated 20,000 random samples using the Monte Carlo algorithm. Therefore, we obtained the power spectral density in the frequency range of 0.1 mHz\(\sim\)1 Hz.  
It can be seen from Figure 7 that whether it is a noise signal or its PSD, after removing the rotational symmetry aberration, the size of the noise signal can be reduced by about 5 orders of magnitude, while the power spectral density signal is nearly 6 orders of magnitude lower. Additionally, the power spectral density of the system meets the LISA requirements in the range of 0.1 mHz\(\sim\)1 Hz when the rotational symmetry aberration is eliminated. It is worth noting that the variation trend of noise and power spectral density is consistent in the two random cases, and the maximum value of power spectral density is around \(10^{-2}\). The reason may be related to the phase demodulation algorithm.  
In addition, to quantitatively analyze the degree of influence, we analyzed the power spectral density of the aberration corresponding to different aberration sizes. As shown  
Figure 7: Phase noise due to random aberrations of population size \(\lambda/30\) and its corresponding PSD. (**a**) is the phase noise signals with RSA, and (**b**) is the phase noise signals without RSA. The corresponding PSD of (**a**) is (**c**), (**b**) is (**d**).  
in Figure 8, we analyze the mean value of the power spectral density of the phase noise corresponding to the inclusion of RSA (orange) and not the inclusion of the RSA (blue) with an aberration size of \(\lambda\)/80-\(\lambda\). It can be found that as the aberration decreases, the noise level also decreases rapidly. However, compared with the completely random aberration, the noise level drops by nearly 5 orders of magnitude after removing the rotationally symmetric part, and the ratio does not change significantly with the reduction of the overall aberration. Hence, we can think that rotational symmetry aberration plays a dominant role in phase noise.  
In the field of data analysis, box plots are a good way to reflect data stability. Next, as shown in Figure 9 (box plot), we fix the overall size of the aberration and redistribute the weight of the rotationally symmetric part in the random aberration. The abscissa represents the proportion of the rotationally symmetric part of the aberration to the total aberration; the ordinate is the power spectral density. In the figure, we have removed outliers, and we kept the minimum, first quartile (Q1), second quartile (Q2), third quartile (Q3), and maximum, where Q2 is also known as the median. It is obvious that Q1, Q2, Q3, and the maximum of the phase noise shift as the ratio increases. This shows that as the ratio increases, the degree of instability of the system also increases, which is fatal to space gravitational wave telescopes.  
Figure 8: Aberration size of \(\lambda\)/80-\(\lambda\) corresponds to the PSD average value of the resulting phase noise. The blue curve corresponds to the left ordinate, and the orange curve corresponds to the right ordinate.  
Figure 9: The box plot of phase noise with an increasing RSA ratio. Here, the top edge of the blue box is Q3 (third quartile) and the bottom is Q1 (first quartile). The red line in the middle of the box is Q2 (second quartile or median). The extension cord of the box is called the whisker, and the top of the whisker is maximum and the bottom is minimum.