本研究提出一套以「循環式光路」做為技術核心的新式雷射光學尺量測技術，用以進行精密位移及角位移量測。此套創新的「循環式雷射光學尺」導入了「薩格納克循環式光路」、「雙繞射式光路」及「多自由度光路」等設計概念，具備高穩定性及高解析度之量測能力。
依據本研究所提出的「循環式雷射光學尺」的量測原理，我們透過對稱式的光路設計使兩道偏振態相垂直(p偏振及s偏振)的光束以對稱的角度入射至反射式光柵後形成繞射，其中，兩道偏振光的第零階(0th)繞射光束(Ep0, Es0)、p偏振光的第負一階(-1st)繞射光束(Ep-1)及s偏振光的第正一階(+1st)繞射光束(Es+1)，在角稜鏡(Right-Angle Prism)之使用下，將再次反射回光柵元件形成第二次繞射，建構出「雙繞射式光路」，能使繞射光束引入雙倍的相位變化量，可達成提升系統靈敏度的目的。而後，p偏振光的第零階(0th)及第負一階(-1st)繞射光束(Ep0, Ep-1)，及s偏振光的第零階(0th)及第正一階(+1st)繞射光束(Es0, Es+1)，將分別朝著對方行進過的路徑前進，如此即可建構出著名的「薩格納克循環式光路」，因兩對偏振態相垂直的繞射光束(Ep0, Es0)及(Ep-1, Es+1)皆行經相同的光學元件，當外界環境存在擾動時，相對應的繞射光束將感受到相同的擾動量，其擾動量將於干涉訊號中相互補償，如此即可使系統具備較高之穩定性。再者，本研究藉由獨特的繞射光路設計使系統具備同時量測面內及面外位移之能力，同時亦透過側向位移分光鏡(LBS)的使用，建構出兩組同時具備面內及面外位移量測能力的偵測架構，當光柵沿著θx及θz軸向產生角位移時，透過比較兩組偵測架構所量測到的面外及面內位移變化量，即可回推待測光柵的角位移變化量(θx, θz)，使此套創新的「循環式雷射光學尺」具備位移(x, z)及角位移(θx, θz)的量測能力。
為驗證此套創新的「循環式雷射光學尺」的量測性能，我們進行了多項的實驗，包含：位移及角位移量測、隨機運動、解析度、重複性、穩定度、幾何誤差、最大量測範圍及速度極限測試等實驗，並將「循環式雷射光學尺」的量測結果與商用電容式位移計、線性光學尺及雷射干涉儀的量測結果相比較。
由實驗結果驗證，此套「循環式雷射光學尺」可在不改變光學系統架構下，同時提供四自由度位移及角位移量測資訊，其位移(x, z)及角位移(θx, θz)的量測解析度分別可達1.2 nm、1.2 nm、20.2 nrad及18.7 nrad，重複性分別可達0.2 nm、0.2 nm、0.38 nrad及0.47 nrad，此外，系統於30分鐘內的穩定度分別可控制於24 nm、24 nm、572 nrad及543 nrad，證明此套「循環式雷射光學尺」具備高解析度、高重複性、高穩定性之量測能力，日後可進一步應用於各式工程與研究領域中。

A novel laser encoder technology with “Cyclic type optical path design” for precise displacement and angular displacement measurements is proposed in this study. By incorporating the design concepts of “Sagnac Cyclic Optical Path”, “Double-diffraction Optical Path” and “Multi-DOF Optical Path”, the proposed “Cyclic Laser Encoder (CLE)” has the capability of high stability and high sensitivity.
According to measurement principle of the proposed CLE, through the symmetrical optical path design, two beams with perpendicular polarization states (p-polarized and s-polarized) are incident on a reflective grating at a symmetrical angle and then diffracted. The zeroth-order (0th) diffracted beams (Ep0, Es0) of the two polarized beams, the negative first-order (-1st) diffracted beams (Ep-1) of the p-polarized beam, and the positive first-order (+1st) diffracted beam (Es+1) of the s-polarized beam will be reflected back to the grating again to form the second diffraction under the use of a right-angle prism. In this way, a “Double-diffraction optical path” can be constructed to induce double phase variations, thereby achieving the purpose of improving the measurement sensitivity of the system. Then, the zeroth-order (0th) and the negative first-order (-1st) diffracted beams (Ep0, Ep-1) of p-polarized beam, and the zeroth-order (0th) and the positive first-order (+1st) diffracted beams (Es0, Es+1) of s-polarized beam will move towards the path traveled by the other part respectively, so that the famous “Sagnac Cyclic Optical Path” can be constructed. Meanwhile, the two pairs of diffracted beams with perpendicular polarization states (Ep0, Es0 & Ep-1, Es+1) all pass through the same optical elements. When there is disturbance in the external environment, the corresponding diffracted beams will sense the same disturbance, and the disturbance would be compensated for in the interference signal, so that the proposed system can has higher stability. The unique diffraction optical path design enables the system to simultaneously measure in-plane and out-of-plane displacements. Furthermore, two detection configurations with both in-plane and out-of-plane displacement measurement capabilities could be constructed by only using a lateral displacement beam splitter (LBS). When the grating is rotated along the θx and θz axes, by comparing the out-of-plane displacement and in-plane displacement variation measured by the two detection configurations, the angular displacements variation of the θx and θz can be obtained. Making the proposed CLE has the ability to measure displacement (x, z) and angular displacement (θx, θz) simultaneously.
In order to verify the measurement performance of this innovative CLE, we conducted a number of experiments, including: displacement and angular displacement test, random motion test, resolution test, repeatability test, stability test, geometric error test, maximum measurement range test and speed limit test. Meanwhile, the measurement results obtained by our CLE are compared with those obtained by commercial capacitive sensor, linear encoder and laser interferometer. Experimental results show that the proposed method can simultaneously provide four-degree-of-freedom displacement and angular displacement measurement information without changing the optical configuration. The measurement resolution of displacement (x, z) and angular displacement (θx, θz) can reach 1.2 nm, 1.2 nm, 20.2 nrad and 18.7 nrad while the repeatability can reach 0.2 nm, 0.2 nm, 0.38 nrad and 0.47 nrad, respectively. In addition, the stability of the system for 30 minutes can be controlled within 24 nm, 24 nm, 572 nrad and 543 nrad, which proves that our CLE has the characteristics of high resolution, high repeatability and high stability. The proposed CLE can be further applied to various engineering and research fields in the future.

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