本研究提出一套以「多次繞射」為設計概念進行系統開發的「三繞射式雙光柵干涉儀」，用以進行精密位移及旋轉角度量測。此套「三繞射式雙光柵干涉儀」具備外差干涉術、光柵干涉術、三繞射式光路及共面偵測式光路等技術優勢，使系統具高靈敏度、高解析度、高穩定度及五自由度(x, z, θx, θy, θz)的量測能力。
於系統設計時，透過電光調變器與氦氖雷射來產生外差光源，用以將待測物理訊息（如：位移）載在特定頻率，藉此降低外界之低頻振動對量測結果所造成之影響。而後，透過Wollaston prism元件及透鏡的搭配，使兩道偏振態互相垂直的外差光束(p偏振光束及s偏振光束)聚焦於第一片光柵後形成第一次繞射，其中p偏振光束之第正一階繞射光束與s偏振光束之第負一階繞射光束將相互平行且完全重合，而後此兩道重合的繞射光束在穿過一分光鏡後形成穿透及反射，反射後的兩道重合繞射光束經過一檢偏器後由光感測器所接收，形成第一組干涉訊號，用以量測第一片光柵之位移量。於此同時，穿透後的兩道重合繞射光束將正向入射第二片光柵後產生第二次繞射，接著透過平面鏡的使用使兩道經過二次繞射的重合繞射光束沿原路徑反射並再次入射第二片光柵形成第三次繞射，藉由三次的繞射可引入三倍的相位變化，用以提升干涉儀系統之靈敏度，此兩道經過三次繞射後的重合繞射光束在穿過檢偏器後由光感測器所接收，形成第二組干涉訊號，用以量測第二片光柵之位移量。當兩片光柵皆沿著面內方向移動時，藉由量測兩組干涉訊號的變化即可回推待測光柵的面內位移量；當兩片光柵沿著面外方向移動時，p偏振光束之第正一階繞射光束與s偏振光束之第負一階繞射光束於光柵上的偵測點位置會於光柵面內方向產生側向偏移，將面內側向偏移量與光源入射角進行正切運算後即可回推出光柵的面外位移量；當兩片光柵產生θy軸向之旋轉角度時，兩片光柵的面內位移量將有所差異，透過比較兩組干涉訊號之變化（即比較兩個面內位移量）即可回推θy軸向之旋轉角度變化量，即具備三自由度位移及旋轉角之量測能力。此外，為了延伸系統的量測能力，藉由側向位移分光鏡的加入產生第二道平行外差光束，在共用所有光學元件的情況下建立第二組偵測架構，形成所謂的共面偵測式光路架構，使每一片待測光柵上下方均形成偵測點，第二組偵測架構同樣亦具備三自由度位移及旋轉角之量測能力，當光柵沿θz軸向產生旋轉角度時，透過比較兩組偵測架構於面內位移量即可推得θz軸向之旋轉角度變化量；當光柵沿θx軸向產生旋轉角度時，亦同樣利用比較兩組偵測架構於面外位移量的差異，即可回推θx軸向之旋轉角度變化量，如此即可使此套干涉儀具備五自由度(x, z, θx, θy, θz)的量測能力。
由實驗結果可知，本研究所提出的「三繞射式雙光柵干涉儀」於面內、面外位移(x, z)及三個旋轉軸(θx, θy, θz)之量測解析度分別約可達0.7 nm、1.5 nm、20.2 nrad、14.6 nrad及24.6 nrad；最大量測範圍分別可達50 mm、0.8 mm及800 μrad；系統於10分鐘內的穩定度分別優於0.9 nm、4.6 nm、1.8 nrad、3.9 nrad及9.1 nrad，證明此套干涉儀系統具備優異的量測能力，未來可廣泛應用於各種需提供精密位移或旋轉角度量測資訊的場合中。

This research proposes a set of “triple-diffraction type interferometer based on two gratings” with the design concept of “multi-diffraction” for precise displacement and rotation angle measurement. The proposed system has the technical advantages of heterodyne interferometry, grating interferometry, triple-diffraction optical path and coplanar detection optical path so that the system has capabilities of high sensitivity, high resolution, high stability and five degree-of-freedom measurement.
In the system design, the heterodyne light source is generated by electro-optic modulator and He-Ne laser, which is used to carry the measurement information (such as displacement) on a specific frequency, thereby reducing the influence of external low-frequency vibration on the measurement results. Then through the combination of Wollaston prism and lens, two heterodyne beams with mutually perpendicular polarization states (p-polarized beam and s-polarized beam) are focused on the first grating to form the first diffraction, in which the positive first-order diffraction beam of p-polarized beam and the negative first-order diffraction beam of s-polarized beam will be parallel and completely coincide with each other. Then, the two diffraction beams are passed through a beam splitter to form penetration and reflection beams, and the reflected beams are passed through an analyzer and received by the detector, which is used to measure the displacement of the first grating. Meanwhile, the penetrating beams are incident to the second grating to produce the second diffraction. Then, by the use of mirrors, the two beams are reflected to the second grating again along the original path to form the third diffraction. Through the third diffraction, three times of phase variation can be introduced to improve the sensitivity of the system and the interference signals are used to measure the displacement of the second grating. When the gratings are moved in in-plane direction, the displacement can be obtained by measuring the two interference signals. When the gratings are moved along the out-of-plane-direction, the position of the detection point of the positive first-order diffraction beam of the p-polarization beam and the negative first-order diffraction beam of the s-polarization beam on the grating will produce a lateral offset in the in-plane direction of the grating. After the in-plane offset is tangent to the incident angle of the light source, the out of plane displacement of the grating can be deduced. When the two gratings produce the rotation angle of the θy axis, the in-plane displacement of the gratings will be different. By comparing the difference of the two signals (ie, comparing the two in-plane displacements), the rotation angle of the θy axis can be obtained. In addition, in order to extend the measurement capability of the system, a second parallel heterodyne beam is generated by adding a lateral beam splitter and a second detection configuration is established when all optical elements are shared to form coplanar detection optical configuration. In other words, detection points are formed above and below each gratingand the second detection configuration also has the ability to measure three degree-of-freedom displacement and rotation angle. When the gratings are moved along θz axis, the rotation angle can be obtained by comparing the in-plane displacement of the two detection configurations. When the gratings are moved along θx axis, the rotation angle can be obtained by comparing the out-of-plane displacement of the two detection configurations.
It can be known from the experimental results that the resolutions for displacement and rotation angle are approximately 0.7 nm, 1.5 nm, 20.2 nrad, 14.6 nrad and 24.6 nrad, the maximum range can reach 50 mm, 0.8 mm and 800 μrad, the stabilities are better than 0.9 nm, 4.6 nm, 1.8 nrad, 3.9 nrad and 9.1 nrad in ten minutes, proving that the proposed system has excellent measurement capability and can be widely applied in the fields of precision measurement and automatic optical inspection.

[1] A. A. Michelson, “The relative motion of the Earth and the Luminiferous ether,” Am. J. Sci. 22(128), 120 (1881).
[2] J. Lee, H. Yoon and T. H. Yoon, “High-resolution parallel multipass laser interferometer with an interference fringe spacing of 15 nm,” Opt. Commun. 284(5), 1118-1122 (2011).
[3] J. Cui, Z. He, Y. Jiu, J. Tan and T. Sun, “Homodyne laser interferometer involving minimal quadrature phase error to obtain subnanometer nonlinearity,” Appl. Opt. 55, 7086-7092 (2016).
[4] V. C. P. Kumar, C. Joenathan, A. Ganesan and U. Somasundram, “Increasing the sensitivity for tilt measurement using a cyclic interferometer with multiple reflections,” Opt. Eng. 55(8), 084103 (2016).
[5] S. H. Eang and K. Cho, “Balanced-path homodyne I/Q-interferometer scheme with very simple optical arrangement using a polarizing beam displacer,” Opt. Express 25, 8237-8244 (2017).
[6] K. N. Joo, J. D. Ellis, E. S. Buice, J. W. Spronck and R. H. M. Schmidt, “High resolution heterodyne interferometer without detectable periodic nonlinearity,” Opt. Express 18, 1159-1165 (2010).
[7] Y. Park and K. Cho, “Heterodyne interferometer scheme using a double pass in an acousto-optic modulator,” Opt. Lett. 36, 331-333 (2011).
[8] J. Qi, Z. Wang, J. H. Huang, B. Yu, J. Gao and S. Donati, “Enhancing the sensitivity of roll-angle measurement with a novel interferometric configuration based on waveplates and folding mirror,” Rev. Sci. Instrum. 87(3), 036106 (2016).
[9] H. L. Hsieh, J. Y. Lee, L. Y. Chen and Y. Yang, “Development of an angular displacement measurement technique through birefringence heterodyne interferometry,” Opt. Express 24, 6802-6813 (2016).
[10] J. Qi, Z. Wang, J. Huang and J. Gao, “Resolution-enhanced heterodyne laser interferometer with differential configuration for roll angle measurement,” Opt. Express 26, 9634-9644 (2018).
[11] H. Fu, Y. Wang, P. Hu, J. Tan and Z. Fan, “Nonlinear Errors Resulting from Ghost Reflection and Its Coupling with Optical Mixing in Heterodyne Laser Interferometers,” Sensors 18(3), 758 (2018).
[12] Y. Le, S. Jin and T. Jin, “Simple heterodyne interferometer using a polarizing beam displacer,” Opt. Lett. 46, 2015-2018 (2021).
[13] C. C. Wu, C. Y. Cheng and Z. Y. Yang, “Optical homodyne common-path grating interferometer with sub-nanometer displacement resolution,” Proc. SPIE 7791, 779105 (2010).
[14] Chang-geng Wang, “Autocollimation form dual-frequency laser diffraction grating interferometer design and analysis,” IEEE Xplore 148-150 (2010).
[15] H. L. Hsieh, J. Y. Lee, W. T. Wu, J. C. Chen, R. Deturche and G. Lerondel, “Quasi-common-optical-path heterodyne grating interferometer for displacement measurement,” Meas. Sci. 21, 115304 (2010).
[16] J. Y. Lee and G. A. Jiang, “Displacement measurement using a wavelength-phase-shifting grating interferometer,” Opt. Express 21, 25553-25564 (2013).
[17] J. Deng, C. Zhou, X. Yan, C. Wei and Y. Lu, “Research on a grating interferometer with high optical subdivision based on quasi-Littrow configuration,” Proc. SPIE 10022, 1002213 (2016).
[18] J. Guan, P. Kochert, C. Weichert, R. Koning, L. Siaudinyte and J. Flugge, “A differential interferometric heterodyne encoder with 30 picometer periodic nonlinearity and sub-nanometer stability,” Precis. Eng. 50, 114-118 (2017).
[19] X. Xing, D. Chang, P. Hu and J. Tan, “Spatially separated heterodyne grating interferometer for eliminating periodic nonlinear errors,” Opt. Express 25(25), 31384 (2017).
[20] J. Deng, X. Yan, C. Wei, Y. Lu, M. Li, X. Xiang, W. Jia and C. Zhou, “Eightfold optical encoder with high-density grating,” Appl. Opt. 57, 2366-2375 (2018).
[21] 詹翔安，「四自由度共路徑式線性光學尺之開發」，碩士論文，國立臺灣科技大學，2019。
[22] C. H. Liu and C. H. Cheng, “Development of a grating based multi-degree-of-freedom laser linear encoder using diffracted light,” Sens. Actuator A Phys. 181, 87-93 (2012).
[23] A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576-585 (2012).
[24] X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771-781 (2013).
[25] F. Qibo, Z. Bin, C. Cunxing, K. Cuifang, Z. Yusheng and Y. Fenglin, “Development of a simple system for simultaneously measuring 6DOF geometric motion errors of a linear guide,” Opt. Express 21, 25805-25819 (2013).
[26] C. Lin, S. Yan, D. Ding and G. Wang, “Two-dimensional diagonal-based heterodyne grating interferometer with enhanced signal-to-noise ratio and optical subdivision,” Opt. Eng. 57(6), 064102 (2018).
[27] 陳暐，「六自由度Wollaston雷射光學尺」，碩士論文，國立臺灣科技大學，2016。
[28] S. H. Chen, H. L. Hsieh, H. Y. Chen and Y. X. Lai, “Development of a double-diffraction grating interferometer for measurements of displacement and angle,” Proc. SPIE 11352, 113521K (2020).
[29] J. Williamson, H. Martin and X. Jiang, “High resolution position measurement from dispersed reference interferometry using template matching,” Opt. Express 24, 10103-10114 (2016).
[30] A. J. Fleming, “A review of nanometer resolution position sensors: Operation and performance,” Sens. Actuators A Phys. 190, 106-126(2013).
[31] Q. Lu, C.Wang, J. Bai, K. Wang, W. Lian, S. Lou, X. Jiao, G. Yang, “Subnanometer resolution displacement sensor based on a grating interferometric cavity with intensity compensation and phase modulation,” Appl. Opt. 54, 4188-4196 (2015).
[32] Bobroff, N. “Recent advances in displacement measuring interferometry,” Meas. Sci. Techn. 4, 907(1993).
[33] H. L. Hsieh, S. W. Pan, “Development of a grating-based interferometer for six-degree-of-freedom displacement and angle measurements,” Opt. Express 23, 2451-2465 (2015).
[34] W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. L. Chen, X. D. Lu, W. Knappf, A. Weckenmanng, W. T. Estlerh, H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann Manuf. Techn. 64, 773-796 (2015).
[35] J. Y. Lee, C. T. Hung, “Wavelength-modulated standing-wave interferometry for small out-of-plane displacement measure-ment,” Opt. Laser Techn. 124, 105967 (2020).
[36] H. L. Hsieh, W. Chen, “Heterodyne Wollaston laser encoder for measurement of in-plane displacement,” Opt. Express 24, 8693-8707 (2016).
[37] L. Chassagne, M. Wakim, S. Xu, S. Topcu, P. Ruaux, P. Juncar, Y. Alayli, “A 2D nano-positioning system with sub-nanometric repeatability over the millimetre displacement range,” Meas. Sci. Technol. 18, 11 (2007).
[38] 盧洺霈，「反射式準共光程外差光柵干涉儀應用於長行程精密定位技術之研究」，碩士論文，國立中央大學，2009。
[39] 陳應誠，「電光晶體調制外差式干涉儀與其非線性誤差之消減」，碩士論文，國立清華大學，1994。