Research on rotational coupling of test mass interferometer based on laser heterodyne interferometry
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摘要:
空间引力波探测采用激光外差干涉进行百万公里臂长间的测试质量微小位移波动检测，要求干涉系统在毫赫兹频段达到皮米级测量精度。干涉仪中测试质量的转动会通过转动-转动与转动-平动两类误差耦合共同限制系统灵敏度，本文旨在采取先抑制转动-转动耦合，再抑制转动-平动耦合的策略系统研究这两类误差耦合的耦合原理，建立耦合误差模型，并进行耦合误差消减。本文利用了激光外差干涉与波前传感技术，搭建了测试质量干涉仪系统，实现了位移与转角的高灵敏度测量和噪声分析；通过实验标定了偏摆镜与探测器之间的坐标变换关系，并将偏摆镜旋转至最小耦合角度以使偏摆镜与探测器之间的坐标系尽量重合，实现了转动-转动耦合的抑制；通过几何关系建立了光学模型及实验对参数进行标定，建立了实时补偿系统，实现了对转动-平动耦合实现动态抑制。经过抑制，转动-转动的耦合系数达到了约12.5 mrad/rad；转动-平动的耦合误差在时域消减了约90%，在频域上了降低约一个量级，为空间引力波探测干涉仪的多自由度解耦及噪声抑制奠定了理论和实验基础。
Abstract:Space-based gravitational wave detection uses laser heterodyne interferometry to measure picometer-level displacement fluctuations of test masses separated by millions of kilometers. The interferometric system must achieve picometer-level accuracy in the millihertz frequency band. In the interferometer, test-mass rotation limits system sensitivity through two types of coupling errors: rotation–rotation coupling and rotation–translation coupling. This paper systematically investigates the mechanisms of these two errors and adopts a sequential suppression strategy: rotation–rotation coupling is first suppressed, followed by rotation–translation coupling.A test-mass interferometer is developed based on laser heterodyne interferometry and wavefront sensing, enabling high-sensitivity displacement and angular measurement as well as noise analysis. The coordinate transformation between the steering mirror and the detector is experimentally calibrated. The steering mirror is then rotated to the minimum-coupling angle, aligning the two coordinate systems and suppressing rotation–rotation coupling. An optical model is further established based on geometric relationships, and its parameters are experimentally calibrated. A real-time compensation system is developed to dynamically suppress rotation–translation coupling.After suppression, the rotation–rotation coupling coefficient is approximately 12.5 mrad/rad. The rotation–translation coupling error is reduced by about 90% in the time domain and by approximately one order of magnitude in the frequency domain. These results provide a theoretical and experimental foundation for multi-degree-of-freedom decoupling and noise suppression in interferometers for space-based gravitational wave detection.
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图 2 高精度相位解调原理图（ADC：模数转换器；LUT：相位查找表；PA：相位累加器；PIR：相位积分寄存器；PI Controller：比例积分控制器；NCO：数控振荡器;f：频率）
Figure 2. Schematic diagram of high-precision phase demodulation (ADC: Analog-to-Digital Converter; LUT: Phase Look-Up Table; PA: Phase Accumulator; PIR: Phase Integrator Register; PI Controller: Proportional-Integral Controller; NCO: Numerically Controlled Oscillator; f: Frequency)
图 3 激光外差干涉测量系统布局（OA：光衰减器；50/50FS: 50/50 光纤分束器；AOM：声光移频器；FC：光纤准直器；IOP：干涉光路；TM:测试质量；QPD：四象限光电探测器；PD：光电探测器；PDS：相位解调系统；PC: 个人计算机）
Figure 3. Layout of the laser heterodyne interferometry measurement system (OA: Optical Attenuator; 50/50FS: 50/50 Fiber Splitter; AOM: Acousto-Optic Frequency Shifter; FC: Fiber Collimator; IOP: Interferometric Optical Path; TM: Test Mass; QPD: Quadrant Photodetector; PD: Photodetector; PDS: Phase Demodulation System; PC: Personal Computer)
图 4 干涉光路设计图(P：偏振器;BS：分束器；HWP：二分之一波片；PBS：偏振分束器；QWP：四分之一波片；M：反射镜；QPD：四象限光电探测器；PD：光电探测器)
Figure 4. Interferometric optical layout (P: Polarizer; BS: Beam Splitter; HWP: Half Wave Plate; PBS: Polarizing Beam Splitter; QWP: Quarter Wave Plate; M: Mirror; QPD: Quadrant Photodetector; PD: Photodetector)
图 5 测试质量安装在微动位移台上的干涉仪搭建
Figure 5. The test mass mounted on a micro-motion stage in the interferometer setup.
图 6 干涉光路平动读出。（a）量程；（b）分辨率3 nm和1 nm
Figure 6. Translation readout of the interferometric optical path. (a) Measurement range; (b) resolution of 3 nm and 1 nm.
图 7 干涉光路平动读出噪声谱密度结果及相位计电子学本底噪声
Figure 7. Spectral density of translational readout noise from the interferometric optical path and electronic baseline noise of the phase meter
图 8 偏摆镜安装姿态与初始角度
Figure 8. Mounting configuration and initial angle of the steering mirror.
图 9 干涉光路中绕x轴和绕y轴转动的相位信号
Figure 9. Phase signals for rotations about the x-axis and y-axis in the interferometric optical path
图 10 干涉光路绕x轴和绕y轴转动的分辨率
Figure 10. Rotational resolution about the x-axis and y-axis in the interferometric optical path
图 11 干涉光路中测试质量绕x轴和y轴的转动读出噪声谱密度结果
Figure 11. Readout noise spectral density for rotations about the x-axis and y-axis in the interferometer.
图 12 转动-转动耦合平均值耦合量随偏摆镜旋转刻度变化关系
Figure 12. The relationship between the mean tilt-to-tilt coupling and the steering mirror rotation scale.
图 14 转动-平动耦合系数标定。(a)绕x轴转动的转动角度
$ \theta $ 和耦合误差$ \text{δL} $ ；(b)绕y轴转动的转动角度$ \theta $ 和耦合误差$ \text{δL} $ Figure 14. Tilt-to-length coupling coefficient calibration. (a) Rotation angle θ and Coupling error δL about the x-axis; (b) Rotation angle θ and coupling error δL about the y-axis
图 15 时域补偿效果。(a)绕x轴转动补偿前后对比；(b)绕y轴转动补偿前后对比
Figure 15. Time-domain compensation effects. (a) Comparisonbefore and after compensation forx-axisrotation; (b) Comparison before and after compensation for y-axis rotation
图 16 频域补偿效果 (a)绕x轴转动补偿前后对比；(b)绕y轴转动补偿前后对比
Figure 16. Frequency-domain compensation effects.(a)Comparison before and after compensation for x-axis rotation; (b) Comparison before and after compensation for y-axis rotation
表 1 偏摆镜转动角度与四象限探测器输出关系
Table 1. Relationship between Tip-Tilt Mirror Angle and Quadrant Detector Output
Steering mirror angle
(θx,θy) (μrad)Quadrant photodetector angle output
(dwsx,dwsy) (μrad)(θx1,θy1) = (0,0) (dwsx1,dwsy2) = (0,0) (θx2,θy2) = (400,0) (dwsx2,dwsy2) = (400,13) (θx3,θy3) = (0,400) (dwsx3,dwsy3) = (-13,400) 
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