﻿ 四自由度同步外差干涉测量系统设计
 光学仪器  2020, Vol. 42 Issue (4): 75-81 PDF

A heterodyne interferometry for simultaneous measurement of four degrees of freedom
ZHANG Shanting, GUO Hanming
School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: Aiming at the problems of limited measurement range, low accuracy and Abbe error in multi-degree-of-freedom measurement technology, a large-travel, high-precision, traceable, four- degree-of-freedom simultaneous measurement of heterodyne interferometry was proposed. The miniaturized 532 nm solid-state laser with iodine frequency stability was used, and the optical fiber coupling technology was used to spatially separate the dual-frequency light. Differential wavefront sensing technology realized the simultaneous detection of displacement, pitch and yaw, and wedge prisms are used to measure the change in straightness. The measuring instrument can suppress the non-linear error caused by dual-frequency optical aliasing, and realize the measurement with a travel range of 6 m, a displacement resolution of 0.13 nm, a pitch and yaw angle resolution of 0.026 μrad, and a straightness resolution of 14.88 nm. The measurement results are traceable.
Key words: multi-degree-of-freedom measurement    heterodyne interferometry    differential wavefront sensing    wedge-shaped prism    traceable

1 系统组成

 图 1 四自由度外差干涉测量系统 Figure 1 Measurement system for four degrees of freedom heterodyne interference.

2 测量原理 2.1 直线度测量

 $\Delta l=\frac{\lambda }{2\pi n}\Delta \phi$ (1)

 图 2 直线偏移引起直线度传感器中光路的变化 Figure 2 Change of light path in straightness sensor caused by straight line migration

 $l = d\sin\theta$ (2)

 $\Delta l = 2l({n}_{\rm{g}}-{n}_{\rm{a}})$ (3)

 $d =\dfrac{\lambda }{2\pi \sin\theta }\Delta \phi$ (4)

2.2 位移、俯仰角及偏转角测量

 $z\propto \dfrac{{\phi }_{\rm{A}}+{\phi }_{\rm{B}}+{\phi }_{\rm{C}}+{\phi }_{\rm{D}}}{4}$ (5)
 $\alpha \propto \dfrac{({\phi }_{\rm{A}}+{\phi }_{\rm{B}}) -({\phi }_{\rm{C}}+{\phi }_{\rm{D}})}{{L}_{p}}$ (6)
 $\beta \propto \dfrac{{(\phi }_{\rm{A}}+{\phi }_{\rm{C}})-({\phi }_{\rm{B}}+{\phi }_{\rm{D}})}{{L}_{y}}$ (7)

 图 3 带有倾斜测量波阵面的差分波前传感的四象限探测器示意图 Figure 3 Schematic diagram of a four-quadrant detector for differential wavefront sensing with tilt measurement wavefront.

 $z =\dfrac{\phi \lambda }{4\pi n}$ (8)

 $\alpha =\dfrac{({z}_{\rm{A}}+{z}_{\rm{B}})-({z}_{\rm{C}}+{z}_{\rm{D}})}{2h}$ (9)

 $\beta =\dfrac{({z}_{\rm{A}}+{z}_{\rm{C}})-({z}_{\rm{B}}+{z}_{\rm{D}})}{2w}$ (10)

2.3 测量行程分析

 图 4 经过楔面棱镜和楔面反射镜的光路俯视图 Figure 4 Top view of light path passing through wedge prism and wedge mirror
 \begin{aligned} \;\\ {L}_{\max}=\frac{L-{L}_{\min}}{\sin\delta } \end{aligned} (11)

 ${L}_{\min}= b + (b\sin\theta +{d}_{0})\tan(\theta -{\beta }')$ (12)

 ${n}_{{\rm{a}}}\sin\theta ={n}_{\rm{g}}\sin\beta ’$ (13)

${n}_{\rm{g}}$ = 1.5， ${n}_{\rm{a}}$ =1，得到 $\tan(\theta -\beta')\approx 0$ ，则

 ${L}_{\min}\approx b$ (14)

 ${L}_{\max}=\frac{L -b}{\sin\delta }$ (15)

$\delta$ =30'59''，L=60 mm，b=5 mm，代入式（15）得 ${L}_{\max}\approx$ 6302.6 mm。由此可见，根据所选器件尺寸搭建的测量系统，可以在6 m行程内实现四自由度的测量。

3 误差分析 3.1 系统误差分析

3.2 系统环境误差分析

3.3 阿贝误差分析

 $\Delta l = D(1-\cos\psi )= b\tan\beta (1-\cos\psi )$ (16)

 图 5 楔面棱镜偏摆引起的光程变化 Figure 5 Optical path change caused by wedge-angle prism deflection

 图 6 俯仰角引起的楔面棱镜中光路的变化 Figure 6 Optical path changes in wedge-shaped prisms caused by elevation angle
 $\Delta l = {d_0}\left( {\cos {{\alpha '}{ - 1}}} \right)$ (17)

 图 7 滚转角引起的楔面棱镜中的光路变化 Figure 7 Optical path changes in wedge prisms caused by roll angle
 $\Delta l=\Delta y\sin\theta ({n}_{\rm{g}}-{n}_{\rm{a}})= a\sin\theta (1-\cos\sigma )({n}_{\rm{g}}-{n}_{\rm{a}})$ (18)

4 结　论

 [1] 齐永岳, 赵美蓉, 林玉池. 纳米测量系统的研究现状与展望[J]. 仪器仪表学报, 2003, 24(S1): 91–94. [2] 韦丰, 陈本永, 丁启全. 大范围高精度的纳米测量现状与发展趋势[J]. 光学技术, 2005, 31(2): 302–305, 308. DOI:10.3321/j.issn:1002-1582.2005.02.044 [3] FAN Z G, HE J, ZUO B J, et al. Research on six- degree- of- freedom calibration system for wind tunnel balances with a collimated laser beam[J]. Chinese Optics Letters, 2003, 1(2): 82–84. [4] HUANG P S, LI Y. Laser measurement instrument for fast calibration of machine tools[C]//Proceedings of ASPE Annual Meeting. 1996: 644. [5] UMETSU K, FURUTNANI R, OSAWA S, et al. Geometric calibration of a coordinate measuring machine using a laser tracking system[J]. Measurement Science and Technology, 2005, 16(12): 2466–2472. DOI:10.1088/0957-0233/16/12/010 [6] LEE C B, KIM G H, LEE S K. Design and construction of a single unit multi-function optical encoder for a six-degree-of-freedom motion error measurement in an ultraprecision linear stage[J]. Measurement Science and Technology, 2011, 22(10): 105901. DOI:10.1088/0957-0233/22/10/105901 [7] LEE S W, MAYOR R, NI J. Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool[J]. Journal of Manufacturing Science and Engineering, 2005, 127(4): 857–865. DOI:10.1115/1.2035692 [8] NI J, WU S M. An on-line measurement technique for machine volumetric error compensation[J]. Journal of Manufacturing Science and Engineering, 1993, 115(1): 85–92. [9] YAN H, DUAN H Z, LI L T, et al. A dual-heterodyne laser interferometer for simultaneous measurement of linear and angular displacements[J]. Review of Scientific Instruments, 2015, 86(12): 123102. DOI:10.1063/1.4936771 [10] FAN K C, CHEN M J. A 6-degree-of-freedom measurement system for the accuracy of X-Y stages[J]. Precision Engineering, 2000, 24(1): 15–23. DOI:10.1016/S0141-6359(99)00021-5 [11] 左爱斌, 李文博, 彭月祥, 等. 调制转移光谱稳频的研究[J]. 中国激光, 2005, 32(2): 164–166. DOI:10.3321/j.issn:0258-7025.2005.02.006 [12] 贺寅竹, 赵世杰, 尉昊赟, 等. 跨尺度亚纳米分辨的可溯源外差干涉仪[J]. 物理学报, 2017, 66(6): 060601. DOI:10.7498/aps.66.060601 [13] 金涛, 刘景林, 杨卫, 等. 线性位移台直线度高精密外差干涉测量装置[J]. 光学 精密工程, 2018, 26(7): 1570–1577. [14] GROTE H, HEINZEL G, FREISE A, et al. Alignment control of GEO 600[J]. Classical and Quantum Gravity, 2004, 21(5): S441–S449. DOI:10.1088/0264-9381/21/5/009 [15] YU X Z, GILLMER S R, ELLIS J D. Beam geometry, alignment, and wavefront aberration effects on interferometric differential wavefront sensing[J]. Measurement Science and Technology, 2015, 26(12): 125203. DOI:10.1088/0957-0233/26/12/125203 [16] 刘君, 穆海华, 孙业业, 等. 激光干涉测量中的误差分析与补偿[J]. 机床与液压, 2006(9): 181–184. DOI:10.3969/j.issn.1001-3881.2006.09.064