## 5 Discussion  
Telescopes are one of the important components in space gravitational wave detection missions. Both LISA and the Taiji mission put forward more stringent requirements for telescope indicators. However, as the space environment changes, the telescope will generate additional dynamic aberration, and the aberration will generate phase noise through the interference of the double beams in the gravitational wave measurement, thereby reducing the stability of the telescope.  
In this article, we first built the interference model between the far-field beam and the local beam in space gravitational wave detection and theoretically deduced the inversion method of its phase; then, we used the general form of the Zernike polynomial to deduce the aberration form that plays a dominant role in the phase effect. At the same time, to better simulate the phase demodulation method in the actual situation, we deduced the phase demodulation method on the four-quadrant detector.  
In terms of simulation, we first simulated the effects of different sizes of spherical aberration and coma (common aberration) on the phase and preliminarily found that the RSA plays a dominant role in the phase noise. Therefore, a Monte Carlo random algorithm was used to simulate the case where the fixed aberration size is \(\lambda/60\), and the influence of the RSA on the phase demodulation was compared with and without the exclusion of the RSA. The result was consistent with the theoretical derivation. In demodulated phase noise, the variance aberration is about 8 orders of magnitude.  
To further analyze the influence of the rotationally symmetric part of the aberration on the stability of the telescope, we took the LISA requirements for the stability of the telescope as an example, established a correlation model between the aberration and the phase noise power spectral density, and simulated the noise power spectral density corresponding to aberrations of different sizes. Then, it was further analyzed that when the aberration size is fixed at \(\lambda/30\), the rotationally symmetric part accounts for different proportions of the corresponding power spectral density, which finally shows that the stability of the system can meet the requirements when the proportion is less than 1%.  
In general, the above analysis and conclusions point out that for the detection of gravitational waves in space, the additional dynamic aberration caused by the telescope's environmental influence will greatly affect the stability of the telescope, and the rotational symmetry of the aberration plays an important role. Therefore, when designing a telescope, the design of its structure and the allocation of tolerances should minimize the generation of rotational symmetry aberrations, which has guiding significance for the design of optical systems in LISA or Taiji missions.  
Conceptualization, Z.C.; methodology, Z.C.; software, Z.C. and R.L.; validation, Z.C. and R.L.; formal analysis, Z.C.; investigation, Z.C.; writing--original draft preparation, Z.C.; writing--review and editing, Z.W. and C.F.; visualization, C.Y.; resources, C.Y.; supervision, Z.W. and C.F.; funding acquisition, Z.W. and C.F. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number 62075214, and the National Key R&D Program of China, grant number 2020 YFC2200104.  
The data and the source code are publicly available on \"[[https://github.com/DrChenZW/code_AS.git](https://github.com/DrChenZW/code_AS.git)]([https://github.com/DrChenZW/code_AS.git](https://github.com/DrChenZW/code_AS.git))\" (accessed on 9 December 2022).  
The authors declare no conflict of interest.