# The Influence of On-Orbit Micro-Vibration on Space Gravitational Wave Detection

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## Abstract

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## 1. Introduction

^{1/2}in the low-frequency range from 0.1 mHz to 1 Hz [7]. To achieve this level of sensitivity, every component of a gravitational wave detector must meet stringent specifications. In fact, the normal operation of satellite payloads requires a very quiet environment, and there are many disturbances inside the spacecraft under orbital conditions [8,9,10]. The micro-vibration of the spacecraft caused by the in-orbit working state of the spacecraft reaction wheel assembly, gyroscope, etc., will also change the wavefront of the beam emitted or received by the telescope [11,12,13]. High-sensitivity precision equipment, such as micro-vibration in space gravitational wave detection, may not only have a resonance effect on the telescope structure, but also, as found in the analysis of this paper, low-frequency vibrations are directly coupled to the errors of gravitational wave measurements. High-frequency vibrations may even directly affect the phase demodulation accuracy of the optical phase-locked loop and have a more significant impact on the gravitational wave measurement.

## 2. Vibration Coupling Model

#### 2.1. Heterodyne Interference Model

#### 2.2. The Principle of Optical Phase-Locked Loop

- After the incident light enters the Costas Loop, the optical signal is converted into an electrical signal.
- The sine and cosine signals are multiplied by the incident signal through the NCO (numerical oscillator). Although in actual detection, the heterodyne frequency is unknown due to the existence of the Doppler effect, there is a subsequent feedback mechanism so that the final NCO emits the same sine–cosine signal as the heterodyne frequency. However, for the convenience of this research, the feedback processing process has nothing to do with phase demodulation, and we assume that the heterodyne frequency is known. Then, we obtain:

- (3)
- When the frequency of the NCO is consistent with the heterodyne frequency, we obtain the phase formula of the beam:$$\mathsf{\Delta}\phi =-\frac{S}{C}$$

#### 2.3. The Influence of Micro-Vibration on Phase Demodulation

## 3. Numerical Simulation

^{−5}s, which can complete the integration of the heterodyne interference signal in multiple cycles. Finally, we couple 20 kinds of vibration noise with amplitude 10

^{−12}m and frequency $0.1f\sim 2f$ into the interference model, and the phase measurement is completed within 100 sampling times. The results are shown in Figure 4.

^{−6}rad; converting this to optical path noise is about 6.98 × 10

^{−13}m. This value is close to the amplitude of the input vibration noise, while the vibration noise of other frequencies is much lower than this value, indicating that most of the vibration noise is isolated by the phase-locked loop, but the multiplier part is retained.

## 4. Simulation

## 5. Discussion

^{−8}m, the large optical path error caused by the frequency is close to its amplitude, while the optical path difference caused by other frequencies is much smaller than the amplitude.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Bian, L.; Cai, R.-G.; Cao, S.; Cao, Z.; Gao, H.; Guo, Z.-K.; Lee, K.; Li, D.; Liu, J.; Lu, Y. The gravitational-wave physics II: Progress. Sci. China Phys. Mech. Astron.
**2021**, 64, 120401. [Google Scholar] [CrossRef] - Zheng, K.; Xu, M. Design and Thermal Stability Analysis of Swing Micro-Mirror Structure for Gravitational Wave Observatory in Space. Machines
**2021**, 9, 104. [Google Scholar] [CrossRef] - Jennrich, O. LISA technology and instrumentation. Class. Quantum Gravity
**2009**, 26, 153001. [Google Scholar] [CrossRef] [Green Version] - Chen, Z.; Leng, R.; Yan, C.; Fang, C.; Wang, Z. Analysis of Telescope Wavefront Aberration and Optical Path Stability in Space Gravitational Wave Detection. Appl. Sci.
**2022**, 12, 12697. [Google Scholar] [CrossRef] - Dong, Y.; Liu, H.; Luo, Z.; Li, Y.; Jin, G. A comprehensive simulation of weak-light phase-locking for space-borne gravitational wave antenna. Sci. China Technol. Sci.
**2016**, 59, 730–737. [Google Scholar] [CrossRef] [Green Version] - Luo, Z.; Guo, Z.; Jin, G.; Wu, Y.; Hu, W. A brief analysis to Taiji: Science and technology. Results Phys.
**2020**, 16, 102918. [Google Scholar] [CrossRef] - Wang, Z.; Yu, T.; Zhao, Y.; Luo, Z.; Sha, W.; Fang, C.; Wang, Y.; Wang, S.; Qi, K.; Wang, Y.; et al. Research on Telescope TTL Coupling Noise in Intersatellite Laser Interferometry. Photonic Sens.
**2019**, 10, 265–274. [Google Scholar] [CrossRef] [Green Version] - Zhang, L.; Wang, W.; Shi, Y. Development of a magnetorheological damper of the micro-vibration using fuzzy PID algorithm. Arab. J. Sci. Eng.
**2019**, 44, 2763–2773. [Google Scholar] [CrossRef] - Zhou, W.; Li, D. Experimental research on a vibration isolation platform for momentum wheel assembly. J. Sound Vib.
**2013**, 332, 1157–1171. [Google Scholar] [CrossRef] - Yu, Y.; Gong, X.; Zhang, L.; Jia, H.; Xuan, M. Full-Closed-Loop Time-Domain Integrated Modeling Method of Optical Satellite Flywheel Micro-Vibration. Appl. Sci.
**2021**, 11, 1328. [Google Scholar] [CrossRef] - Wang, H.; Wang, W.; Wang, X.; Zou, G.-y.; Li, G.; Fan, X. Space camera image degradation induced by satellite micro-vibration. Acta Photon. Sin.
**2013**, 42, 1212–1217. [Google Scholar] [CrossRef] - Pang, S.-W.; Yang, L.; Qu, G. New development of micro-vibration integrated modeling and assessment technology for high performance spacecraft. Struct. Environ. Eng.
**2007**, 34, 1–9. [Google Scholar] - Jiao, X.; Zhang, J.; Li, W.; Wang, Y.; Ma, W.; Zhao, Y. Advances in spacecraft micro-vibration suppression methods. Prog. Aerosp. Sci.
**2023**, 138, 100898. [Google Scholar] [CrossRef] - De Groot, P.J. Vibration in phase-shifting interferometry. JOSA A
**1995**, 12, 354–365. [Google Scholar] [CrossRef] [Green Version] - Deck, L.L. Suppressing phase errors from vibration in phase-shifting interferometry. Appl. Opt.
**2009**, 48, 3948–3960. [Google Scholar] [CrossRef] [PubMed] - Liu, Q.; Wang, Y.; He, J.; Ji, F. Modified three-step iterative algorithm for phase-shifting interferometry in the presence of vibration. Appl. Opt.
**2015**, 54, 5833–5841. [Google Scholar] [CrossRef] [PubMed] - Smythe, R.; Moore, R. Instantaneous phase measuring interferometry. Opt. Eng.
**1984**, 23, 361–364. [Google Scholar] [CrossRef] - Zhao, Y.; Shen, J.; Fang, C.; Liu, H.; Wang, Z.; Luo, Z. Tilt-to-length noise coupled by wavefront errors in the interfering beams for the space measurement of gravitational waves. Opt. Express
**2020**, 28, 25545–25561. [Google Scholar] [CrossRef] - Zhao, Y.; Shen, J.; Fang, C.; Wang, Z.; Gao, R.; Sha, W. Far-field optical path noise coupled with the pointing jitter in the space measurement of gravitational waves. Appl. Opt.
**2021**, 60, 438–444. [Google Scholar] [CrossRef] - Zhao, Y.; Wang, Z.; Li, Y.; Fang, C.; Liu, H.; Gao, H. Method to Remove Tilt-to-Length Coupling Caused by Interference of Flat-Top Beam and Gaussian Beam. Appl. Sci.
**2019**, 9, 4112. [Google Scholar] [CrossRef] [Green Version] - Sasso, C.P.; Mana, G.; Mottini, S. Coupling of wavefront errors and jitter in the LISA interferometer: Far-field propagation. Class. Quantum Gravity
**2018**, 35, 185013. [Google Scholar] [CrossRef] - Han, S.; Tong, J.; Wang, Z.; Yu, T.; Sui, Y. Simulation system of a laser heterodyne interference signal for space gravitational wave detection. Infrared Laser Eng.
**2022**, 51, 20210572. [Google Scholar] - Wang, Z.; Yu, T.; Sui, Y.; Wang, Z. Beat-Notes Acquisition of Laser Heterodyne Interference Signal for Space Gravitational Wave Detection. Sensors
**2023**, 23, 3124. [Google Scholar] [CrossRef] - Meshksar, N.; Mehmet, M.; Isleif, K.S.; Heinzel, G. Applying Differential Wave-Front Sensing and Differential Power Sensing for Simultaneous Precise and Wide-Range Test-Mass Rotation Measurements. Sensors
**2020**, 21, 164. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Schematic diagram of the far-field beam received by the gravitational wave telescope. The laser beam is expanded by the telescope and sent out to the detector after being received at the other end. The arrows in the figure indicate the direction of propagation of the beam.

**Figure 2.**Schematic diagram of Costas Loop demodulation algorithm. The incident light is converted into an electrical signal via a phase meter, and the NCO sends out a heterodyne frequency signal and multiplies and integrates the interference signal to obtain a phase signal.

**Figure 3.**Schematic diagram of the principle of QPD, which is composed of A–D four photoelectric detectors, and the width of each connected slit is about 1 um.

**Figure 4.**The amplitude is 10

^{−12}m, and the frequency distribution is between 0.1f and 2f. The phase noise generated by 20 kinds of vibration noise.

**Figure 5.**Influence of vibration noise in the test frequency band of gravitational waves in space on the phase measurement. The red line is the phase effect brought on by the vibration noise of 0.1 MHz, 1 MHz, 10 MHz, 100 MHz, and 1 Hz. They almost overlap; the blue line is the time distribution of the vibration noise itself.

**Figure 6.**Experimental diagram. PC1 inputs the signal parameters into the signal generation circuit to generate an analog interference signal, and then sends the signal to the OPLL circuit, and PC2 analyzes and receives the phase signal in real time.

**Figure 8.**Experimental results diagram, the upper part is f ≠ 2 MHz and the lower part is f = 2 MHz.

Amplitude (m) | f =2 MHz | f ≠ 2 MHz | ||
---|---|---|---|---|

Phase (Rad) | Optical Path (m) | Phase (Rad) | Optical Path (m) | |

10^{−12} | 4.12045 × 10^{−6} | 6.98114 × 10^{−13} | −3.5877 × 10^{−10} | −6.07852 × 10^{−17} |

10^{−11} | 4.12029 × 10^{−5} | 6.98087 × 10^{−12} | −3.5306 × 10^{−9} | −5.98178 × 10^{−16} |

10^{−10} | 4.11873 × 10^{−4} | 6.97823 × 10^{−11} | −2.9720 × 10^{−8} | −5.03536 × 10^{−15} |

10^{−9} | 4.10321 × 10^{−3} | 6.95194 × 10^{−10} | 2.6232 × 10^{−7} | 4.44440 × 10^{−14} |

10^{−8} | 3.95824 × 10^{−2} | 6.70632 × 10^{−9} | 5.8774 × 10^{−5} | 9.95789 × 10^{−12} |

10^{−7} | 0.321447 | 5.44617 × 10^{−8} | 0.0093 | 1.57567 × 10^{−9} |

10^{−6} | −0.443667 | −7.51691 × 10^{−8} | −0.3666 | −6.21118 × 10^{−8} |

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**MDPI and ACS Style**

Chen, Z.; Fang, C.; Wang, Z.; Yan, C.; Wang, Z.
The Influence of On-Orbit Micro-Vibration on Space Gravitational Wave Detection. *Photonics* **2023**, *10*, 908.
https://doi.org/10.3390/photonics10080908

**AMA Style**

Chen Z, Fang C, Wang Z, Yan C, Wang Z.
The Influence of On-Orbit Micro-Vibration on Space Gravitational Wave Detection. *Photonics*. 2023; 10(8):908.
https://doi.org/10.3390/photonics10080908

**Chicago/Turabian Style**

Chen, Zhiwei, Chao Fang, Zhenpeng Wang, Changxiang Yan, and Zhi Wang.
2023. "The Influence of On-Orbit Micro-Vibration on Space Gravitational Wave Detection" *Photonics* 10, no. 8: 908.
https://doi.org/10.3390/photonics10080908