本論文提出一套新穎且架設簡易的角度量測技術-「基於Talbot 效應成像之疊紋式
角度量測系統」。本系統可分為兩部分：疊紋量測光路系統與相位分析系統。疊紋量測光路系統是將兩相隔特殊距離之雙光柵，以夾具等機構安裝於待測平台上，系統光源He-Ne Laser 搭配擴束鏡以準直光入射光柵機構，產生的疊紋影像依Talbot 效應成像於CCD。當待測平台產生角度旋轉時，裝配於一體的光柵組機構連帶產生旋轉，利用疊紋將光柵相對變化物理量放大的效果，搭配疊紋相位分析系統，便可計算待測平台旋轉角度，具有arcsec 級角度量測解析度。
本系統核心元件為兩片光柵週期介於50 ~ 200 m 之線性光柵，製作難度不高，
搭配Talbot 效應成像疊紋影像無須額外透鏡或光學元件輔助，實現高精度低成本的角度量測裝置。系統的量測解析度可藉由調整光柵間距及光柵週期來改變，具有多種組合，可依不同情況選擇適當配置。代表性的組合為200 m 光柵週期搭配1 倍Talbot距離之光柵間距(12.64 cm)，具有0.36 arcsec 角度解析度，1.06°/ arcsec 角度靈敏度，±6°量測範圍與1.36°/s 理論量測速度。

We proposed a new method based on the moiré technique and Talbot effect to measure the angular displacement. The measurement system is divided into two parts, the optical
moiré system and the phase analysis system. In the first part, the collimated He-Ne laser beam is expanded by the beam expander (BE) and is forward incident into the two linear amplitude gratings, which placed symmetrically by a centre of the precision rotation stage. These two gratings can be regarded as the a spatial modulator. After passing through the gratings, the spatial modulated light beam forms a moiré pattern and is captured by the CCD camera. By adjusting the distance D between the two gratings to meet the Talbot image distance (Zt ), the clear and high contrast moiré pattern can be obtained. When the rotation stage provides a rotation, the relative displacement between these two gratings results in the phase shifting of the moiré pattern. The phase analysis by the subfringe integration
arithmetic (SIA). Here, the period of moiré pattern is then divided into four areas, A, B, C and D for further analysis. By means of calculating the intensity integration of the four areas with SIA, the variation of phase shift of the moiré pattern can be obtained, and the rotation angle can be determined. This is the second part of this system.
Clearly, the experimental results demonstrate that the measurement range of our system can achieve ±6°. Considering the high-frequency noise, the measurement resolution of the system is about 0.36 arcsec.

[1]. J. Y. Lee and G. A. Jiang, “Displacement measurement using a wavelength phase-shifting grating interferometer,” Opt. Express 21(21), 2555325564 (2013).
[2]. S. W. Yang C. S. Lin,　and S. K. Lina, “3D surface profile measurement of unsymmetrical microstructure using Fizeau interferometric microscope,” Opt. Laser Eng. 51(4), 348357 (2013).
[3]. X. Dai, O. Sasaki, J. E. Greivenkamp, and Takamasa Suzuki, “High accuracy, wide range, rotation angle measurement by the use of two parallel interference patterns,” Appl. Opt. 36(25), 61906195 (1997).
[4]. C. C. Kuo and Y. R. Chen, “Rapid optical inspection of bubbles in the silicone rubber ,” Optik 124(13), 14801485 (2013).
[5]. X. D. Chen, D. Wei, W. Li, and Y. Deng, “Nonlinear elastodynamic behaviour analysis of high-speed spatial parallel coordinate measuring machines ,” Int. J. Adv. Robot Syst. 9(140), 16 (2012).
[6]. A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2(9), 413415 (1971).
[7]. Y. C. Lin, “Characteristics of compact-size D lens collimator for microoptics devices applications,” Microw. Opt. Techn. Let. 49(8), 19461949 (2007).
[8]. C. H. Liu H. L. Huang, and H. W. Lee, “Five-degrees-of-freedom diffractive laser encoder,” Appl. Opt. 48(14), 27672777 (2009).
[9]. J. M. Sakamoto, C. Kitano, G. M. Pacheco, and B. R. Tittmann, “High sensitivity fiber optic angular displacement sensor and its application for detection of ultrasound,” Appl. Opt. 51(20), 48414851 (2012).
[10]. S. Yokozeki, K. Patorski, and K. Ohnishi, “Collimation method using fourier imaging and moire techniques,” Appl. Opt. 14(4), 401405 (1975).
[11]. M. Ikram and G. Hussain, “Michelson interferometer for precision angle measurement,” Appl. Opt. 38(1), 113120 (1995).
[12]. P. S. Huang and Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37(28), 66366642 (1998).
[13]. J. Guo, Z. Zhu, W. Deng, and S. Shen, “Angle measurement using surface-plasmon-resonance heterodyne interferometry: a new method,” Opt. Eng. 37(11), 29983001 (1998).
[14]. S. Makinouchi, A. Watanabe, M. Takasaki, T. Ohara, J. Ong, and S.Wakui, “An evaluation of a modulated laser encoder,” Precis. Eng. 35(2), 302308 (2011).
[15]. A. Konyakhin and T. V. Turgalieva, “Three-coordinate digital autocollimator,” Opt. Techn. 80(12), 772777 (2013).
[16]. C. H. Liua and C. H. Cheng, “Development of a grating based multi-degree of freedom laser linear encoder using diffracted light,” Sens. Actuators A(181), 8793 (2012).
[17]. F. Quercioli, A. Mannoni, and B. Tiribilli, “Laser Doppler velocimetry with a compact disc pickup,” Appl. Opt. 37(25), 59325937 (1998).
[18]. Q. Li, L. Bai, S. Xue, and L. Chen, “Autofocus system for microscope,” Opt. Eng. 41(6), 12891294 (2002).
[19]. A. Khiat, F. Lamarque, C. Prelle, N. Bencheikh, and E. Dupont, “High-resolution fibre-optic sensor for angular displacement measurements,” Meas. Sci. Technol. 21(2), 110 (2010).
[20]. K. C. Fan, B. H. Liao, and F. Cheng, “Ultra-Precision Angle Measurement based on Michelson Interferometry,” CSME 34(1), 3944 (2013).
[21]. S. Saikan, K. Uchikawa, and H. Ohsawa, “Phase-modulation technique for accumulated photon echo,” Opt. Lett. 16(1), 16121628 (1991).
[22]. J. Zheng, “Optical frequency-modulated continuous-wave interferometers,” Appl. Opt. 45(12), 27232730 (2006).
[23]. O.Sasaki, C. Togashi, and T. Suzuki, “Two-dimensional rotation angle measurement using a sinusoidal phase-modulating laser diode interferometer,” Opt. Eng. 42(4), 11321136 (2003).
[24]. D. Malacara, Twyman-Green Interferometer (John Wiley & Sons.Inc, 2007)
[25]. M. H. Chiu and D. C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36(28), 71047106 (1997).
[26]. M. H. Chiu, S. F. Wang, and R. S. Chang, “Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 43(29), 54385442 (2004).
[27]. 賴律臻，「差動式疊紋自動對焦技術」，國立中央大學，碩士論文，2011
[28]. J. Guo, Z. Zhu, and W. Deng, “Small-angle measurement based on surface-plasmon resonance and the use of magneto-optical modulation,” Appl. Opt. 38(29), 65506555 (1999).
[29]. T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotation-angle measurement using an imaging method,” Opt. Eng. 40(3), 426432 (2001).
[30]. S. S. Nukala, S. S. Gorthi, and K. R. Lolla, “Novel composite coded pattern for small angle measurement using imaging method,” Proc. SPIE 6289, 1D21D11 (2006).
[31]. V. Mayer et al., “New High Resolution Optical Incremental Rotary Encoder,” presented at Integration Issues of Miniaturized Systems - MOMS, MOEMS, ICS and Electronic Components, 2008 2nd European Conference & Exhibition, Spain , pp.18 APR. 2008.
[32]. 高清芬，「旋轉編碼器之原理」，明道大學。
[33]. Lord Rayleigh, “On the manufacture and theory of diffraction-gratings,” Philos. Mag. 47(310), 8193 (1874).
[34]. H. Chandrasekaran, R. Nagarajan, and K. N. Padmanabha, “Application of moiré technique in the stress analysis of cutting tools,” Strain 14(2), 6471 (1978).
[35]. K. G. Harding and J. S. Harris, “Projection moire interferometer for vibration analysis,” Appl. Opt. 22(6), 856861 (1983).
[36]. 胡錦標，林憲陽，楊宗興，蔡奇能，謝宏榮，謝啟堂，精密光電技術，高立書局，民國八十九年。
[37]. G. Mauvoisin, F. Brémand, and A. Lagarde, “Three-dimensional shape reconstruction by phase-shifting shadow moiré,” Appl. Opt. 33(11), 21632169 (1994).
[38]. Y. B. Choi and S. W. Kim, “Phase-shifting grating projection moire´ topography,” Opt. Eng. 37(3), 10051010 (1998).
[39]. Y. Nakano, “Automatic measurements of the small angle variation using a holographic moire interferometry and a computer processing,” Proc. SPIE 673, 180183 (1986).
[40]. S. Nihommori et al, “Optical Encoder,” United States Patent, US 6,635,863 B1 (2003).
[41]. H.F. Talbot, “Facts relating to optical science,” Philos. Mag. Series 3 9(56), 401407 (1836).
[42]. Lord Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. Series 5 11(67), 196205 (1881).
[43]. Skullsinthestars, “Rolling out the (optical) carpet: the Talbot effect,” http://skullsinthestars.com/2010/03/04/rolling-out-the-optical-carpet-the-talbot-effect.
[44]. K. Creath and J. C. Wyant, Optical Shop Testing (John Wiley & Sons.Inc, 1992), Chap. 16.
[45]. M. Wang, “Subfringe integration method for automatic analysis of moire deflection tomography,” Opt. Eng. 39(10), 27262733 (2000).