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Journals >Acta Photonica Sinica >Volume 50 >Issue 7 >Page 161 > Article

Acta Photonica Sinica
Vol. 50, Issue 7, 161 (2021)

Micro-angle Measurement Method and Its Accuracy Evaluation Based on Fabry-Perot Etalon

Yuan LIU¹, Xiaoyan SHEN¹, Shinan ZHOU¹, Xubo GUO²..., Jing YU¹, Dongsheng LI¹ and Xuhui LAN¹|Show fewer author(s)

Author Affiliations

¹College of Metrology & Measurement Engineering， China Jiliang University， Hangzhou3008， China

²Department of Physics， Tsinghua University， Beijing100084， China

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DOI: 10.3788/gzxb20215007.0712004 Cite this Article

Yuan LIU, Xiaoyan SHEN, Shinan ZHOU, Xubo GUO, Jing YU, Dongsheng LI, Xuhui LAN. Micro-angle Measurement Method and Its Accuracy Evaluation Based on Fabry-Perot Etalon[J]. Acta Photonica Sinica, 2021, 50(7): 161 Copy Citation Text

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Abstract

A micro-angle measurement method based on Fabry-Perot etalon is proposed. Particularly， by calculating the displacement of center of the imaging concentric rings and the focal length of the imaging objective lens， the micro angle of the deflected mirror is obtained. The evaluation model of the micro angle uncertainty， based on the uncertainty components of the center displacement and the focal length of the imaging objective lens， is constructed. The Fabry-Perot etalon with an interval of 2 mm is selected to carry out the experimental research on the measurement of micro angles， and the data is processed. The experimental results indicate that the maximum measurement uncertainty is 0.132" and 0.045" in the range of 600" and 40"， respectively. The proposed method can provide a reference for the realization of the self-calibrated， higher-accuracy micro-angle measurement.

Keywords

Fabry-Perot etalon High accuracy Measurement method Micro-angle measurement Uncertainty analysis

k_{0} + ε = 2 d n / λ_{0}

(1)

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δ_{w} = δ / w = \sqrt[]{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2}}

(2)

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s_{δ_{w}} = \sqrt[]{{(\frac{y_{1} - y_{0}}{δ_{w}})}^{2} \times (s_{y_{1}}^{2} + s_{y_{0}}^{2}) + {(\frac{x_{1} - x_{0}}{δ_{w}})}^{2} \times (s_{x_{1}}^{2} + s_{x_{0}}^{2})}

(3)

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{(\frac{D_{i} / 2}{f / w})}^{2} = {(t a n θ_{i})}^{2} = {(\frac{k_{0} + ε}{k_{0} - i})}^{2} - 1

(4)

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f_{w} = f / w \approx \frac{D_{i}}{2} \sqrt[]{\frac{k_{0}}{2 (i + ε)}}

(5)

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\frac{U'_{p} (f_{w})}{f_{w}} = t_{p} (v_{e f f}) \times \sqrt[]{{(\frac{s_{f_{w}}}{f_{w}})}^{2} + {(\frac{u_{f a d}}{\sqrt[]{3} f_{w}})}^{2} + {(\frac{u_{w}}{\sqrt[]{3} w})}^{2}}

(6)

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α = \frac{1}{2} a r c t a n \frac{δ_{w}}{f_{w}}

(7)

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α \approx \frac{1}{2} \frac{δ_{w}}{f_{w}}

(8)

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\frac{\partial α}{\partial δ_{w}} = \frac{1}{2 f_{w}}

(9)

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\frac{\partial α}{\partial f_{w}} = - \frac{1}{2} \frac{δ_{w}}{{f_{w}}^{2}}

(10)

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\begin{array}{l} U_{p} (α) \approx c_{0} + c_{1} \cdot |α| = k_{p} \times (\frac{\partial α}{\partial δ_{w}} \sqrt[]{2} s_{x_{0}}) + k_{p} \times (\frac{\partial α}{\partial f_{w}} s_{f_{w}}) = t_{p} (v) \times (\frac{\partial α}{\partial δ_{w}} \sqrt[]{2} s_{x_{0}}) + \\ t_{p} (v_{e f f'}) \times (\frac{\partial α}{\partial f_{w}} s_{f_{w}}) = t_{p} (v) \times \frac{\sqrt[]{2} s_{x_{0}}}{2 f_{w}} + (t_{p} (v_{e f f'}) \times \frac{s_{f_{w}}}{f_{w}}) \cdot |α| \end{array}

(11)

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\begin{array}{l} U_{p} (α) \approx c_{0} + c_{1} \cdot |α| = k_{p} \times \frac{u_{c} (δ_{w})}{2 f_{w}} + k_{p} \times \frac{u_{c} (f_{w})}{f_{w}} \cdot |α| = \\ t_{p} (v_{e f f 1}) \times \frac{u_{c} (δ_{w})}{2 f_{w}} + t_{p} (v_{e f f 2}) \times \frac{u_{c} (f_{w})}{f_{w}} \cdot |α| \end{array}

(12)

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U_{p} (δ_{w}) = k_{p} \times u_{c} (δ_{w}) = t_{p} (v_{e f f 1}) \times u_{c} (δ_{w}) = t_{p} (v_{e f f 1}) \times \sqrt[]{{u_{1}}_{r e l}^{2} + {u_{2}}_{r e l}^{2} + {u_{3}}_{r e l}^{2} + {u_{4}}_{r e l}^{2}}

(13)

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υ_{e f f 1} = \frac{{(\sqrt[]{{u_{1}}_{r e l}^{2} + {u_{2}}_{r e l}^{2} + {u_{3}}_{r e l}^{2} + {u_{4}}_{r e l}^{2}})}^{4}}{\frac{{u_{1}}_{r e l}^{4}}{υ_{1}} + \frac{{u_{2}}_{r e l}^{4}}{υ_{2}} + \frac{{u_{3}}_{r e l}^{4}}{υ_{3}} + \frac{{u_{4}}_{r e l}^{4}}{υ_{4}}}

(14)

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U_{p} (f_{w}) = k_{p} \times u_{c} (f_{w}) = t_{p} (v_{e f f 2}) \times u_{c} (f_{w}) = t_{p} (v_{e f f 2}) \times \sqrt[]{{u_{5}}_{r e l}^{2} + {u_{6}}_{r e l}^{2}}

(15)

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υ_{e f f 2} = \frac{{(\sqrt[]{{u_{5}}_{r e l}^{2} + {u_{6}}_{r e l}^{2}})}^{4}}{\frac{{u_{5}}_{r e l}^{4}}{υ_{5}} + \frac{{u_{6}}_{r e l}^{4}}{υ_{6}}}

(16)

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Yuan LIU, Xiaoyan SHEN, Shinan ZHOU, Xubo GUO, Jing YU, Dongsheng LI, Xuhui LAN. Micro-angle Measurement Method and Its Accuracy Evaluation Based on Fabry-Perot Etalon[J]. Acta Photonica Sinica, 2021, 50(7): 161

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