Abstract

A laser heterodyne interferometer for simultaneous measuring displacement and angle based on the Faraday effect is proposed. The optical configuration of the proposed interferometer is designed and the mathematic model for measuring displacement and angle is established. The influences of the translational, lateral and rotational movements of the measuring reflector on displacement and angle measurement are analyzed in detail. The experimental setup based on the proposed interferometer was constructed and a series of experiments of angle comparison and simultaneous measuring displacement and angle were performed to verify the feasibility of the proposed interferometer for precision displacement and angle measurement.

© 2014 Optical Society of America

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References

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  1. J. Lazar, O. Cip, M. Cizek, J. Hrabina, M. Sery, and P. Klapetek, “Laser interferometric measuring system for positioning in nanometrology,” WSEAS Trans. Cir. Syst. 9(10), 660–669 (2010).
  2. B. Y. Chen, L. P. Yan, X. G. Yao, T. Yang, D. C. Li, W. J. Dong, C. R. Li, and W. H. Tang, “Development of a laser synthetic wavelength interferometer for large displacement measurement with nanometer accuracy,” Opt. Express 18(3), 3000–3010 (2010).
    [Crossref] [PubMed]
  3. V. Korpelainen and A. Lassila, “Calibration of a commercial AFM: traceability for a coordinate system,” Meas. Sci. Technol. 18(2), 395–403 (2007).
    [Crossref]
  4. S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127(4), 857–865 (2005).
    [Crossref]
  5. C. H. Menq, J. H. Zhang, and J. Shi, “Design and development of an interferometer with improved angular tolerance and its application to x-y theta measurement,” Rev. Sci. Instrum. 71(12), 4633–4638 (2000).
    [Crossref]
  6. H. Bosse and G. Wilkening, “Developments at PTB in nanometrology for support of the semiconductor industry,” Meas. Sci. Technol. 16(11), 2155–2166 (2005).
    [Crossref]
  7. W. Jywe, T. H. Hsu, and C. H. Liu, “Non-bar, an optical calibration system for five-axis CNC machine tools,” Int. J. Mach. Tools Manuf. 59, 16–23 (2012).
    [Crossref]
  8. W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. 48, 761–765 (2006).
    [Crossref]
  9. C. F. Kuang, E. Hong, and J. Ni, “A high-precision five-degree-of-freedom measurement system based on laser collimator and interferometry techniques,” Rev. Sci. Instrum. 78(9), 095105 (2007).
    [Crossref] [PubMed]
  10. W. S. Park, H. S. Cho, Y. K. Byun, and N. Y. Park, “Measurement of 6-DOF displacement of rigid bodies through splitting a laser beam: experimental investigation,” Proc. SPIE 4190, 80–91 (2001).
    [Crossref]
  11. C. J. Chen, P. D. Lin, and W. Y. Jywe, “An optoelectronic measurement system for measuring 6-degree-of-freedom motion error of rotary parts,” Opt. Express 15(22), 14601–14617 (2007).
    [Crossref] [PubMed]
  12. X. H. 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).
    [Crossref]
  13. Q. B. Feng, Optical Measurement Techniques and Applications (Tsinghua University, 2008), Chap.2.
  14. R. C. Quenelle, “Nonlinearity in interferometer measurements,” Hewlett Packard J. 34, 10 (1983).
  15. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
    [Crossref]
  16. C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
    [Crossref]
  17. W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30(3), 337–346 (2006).
    [Crossref]
  18. 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(2), 1159–1165 (2010).
    [Crossref] [PubMed]
  19. E. Z. Zhang, B. Y. Chen, L. P. Yan, T. Yang, Q. Hao, W. J. Dong, and C. R. Li, “Laser heterodyne interferometric signal processing method based on rising edge locking with high frequency clock signal,” Opt. Express 21(4), 4638–4652 (2013).
    [Crossref] [PubMed]

2013 (2)

X. H. 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).
[Crossref]

E. Z. Zhang, B. Y. Chen, L. P. Yan, T. Yang, Q. Hao, W. J. Dong, and C. R. Li, “Laser heterodyne interferometric signal processing method based on rising edge locking with high frequency clock signal,” Opt. Express 21(4), 4638–4652 (2013).
[Crossref] [PubMed]

2012 (1)

W. Jywe, T. H. Hsu, and C. H. Liu, “Non-bar, an optical calibration system for five-axis CNC machine tools,” Int. J. Mach. Tools Manuf. 59, 16–23 (2012).
[Crossref]

2010 (3)

2007 (3)

C. J. Chen, P. D. Lin, and W. Y. Jywe, “An optoelectronic measurement system for measuring 6-degree-of-freedom motion error of rotary parts,” Opt. Express 15(22), 14601–14617 (2007).
[Crossref] [PubMed]

V. Korpelainen and A. Lassila, “Calibration of a commercial AFM: traceability for a coordinate system,” Meas. Sci. Technol. 18(2), 395–403 (2007).
[Crossref]

C. F. Kuang, E. Hong, and J. Ni, “A high-precision five-degree-of-freedom measurement system based on laser collimator and interferometry techniques,” Rev. Sci. Instrum. 78(9), 095105 (2007).
[Crossref] [PubMed]

2006 (2)

W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. 48, 761–765 (2006).
[Crossref]

W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30(3), 337–346 (2006).
[Crossref]

2005 (2)

H. Bosse and G. Wilkening, “Developments at PTB in nanometrology for support of the semiconductor industry,” Meas. Sci. Technol. 16(11), 2155–2166 (2005).
[Crossref]

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127(4), 857–865 (2005).
[Crossref]

2001 (1)

W. S. Park, H. S. Cho, Y. K. Byun, and N. Y. Park, “Measurement of 6-DOF displacement of rigid bodies through splitting a laser beam: experimental investigation,” Proc. SPIE 4190, 80–91 (2001).
[Crossref]

2000 (1)

C. H. Menq, J. H. Zhang, and J. Shi, “Design and development of an interferometer with improved angular tolerance and its application to x-y theta measurement,” Rev. Sci. Instrum. 71(12), 4633–4638 (2000).
[Crossref]

1996 (1)

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

1993 (1)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[Crossref]

1983 (1)

R. C. Quenelle, “Nonlinearity in interferometer measurements,” Hewlett Packard J. 34, 10 (1983).

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[Crossref]

Bosse, H.

H. Bosse and G. Wilkening, “Developments at PTB in nanometrology for support of the semiconductor industry,” Meas. Sci. Technol. 16(11), 2155–2166 (2005).
[Crossref]

Buice, E. S.

Byun, Y. K.

W. S. Park, H. S. Cho, Y. K. Byun, and N. Y. Park, “Measurement of 6-DOF displacement of rigid bodies through splitting a laser beam: experimental investigation,” Proc. SPIE 4190, 80–91 (2001).
[Crossref]

Chen, B. Y.

Chen, C. J.

Cho, H. S.

W. S. Park, H. S. Cho, Y. K. Byun, and N. Y. Park, “Measurement of 6-DOF displacement of rigid bodies through splitting a laser beam: experimental investigation,” Proc. SPIE 4190, 80–91 (2001).
[Crossref]

Cip, O.

J. Lazar, O. Cip, M. Cizek, J. Hrabina, M. Sery, and P. Klapetek, “Laser interferometric measuring system for positioning in nanometrology,” WSEAS Trans. Cir. Syst. 9(10), 660–669 (2010).

Cizek, M.

J. Lazar, O. Cip, M. Cizek, J. Hrabina, M. Sery, and P. Klapetek, “Laser interferometric measuring system for positioning in nanometrology,” WSEAS Trans. Cir. Syst. 9(10), 660–669 (2010).

Dian, S.

X. H. 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).
[Crossref]

Dong, W. J.

Ellis, J. D.

Fang, T. H.

W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. 48, 761–765 (2006).
[Crossref]

Gao, W.

X. H. 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).
[Crossref]

Hao, Q.

Hong, E.

C. F. Kuang, E. Hong, and J. Ni, “A high-precision five-degree-of-freedom measurement system based on laser collimator and interferometry techniques,” Rev. Sci. Instrum. 78(9), 095105 (2007).
[Crossref] [PubMed]

Hou, W.

W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30(3), 337–346 (2006).
[Crossref]

Hrabina, J.

J. Lazar, O. Cip, M. Cizek, J. Hrabina, M. Sery, and P. Klapetek, “Laser interferometric measuring system for positioning in nanometrology,” WSEAS Trans. Cir. Syst. 9(10), 660–669 (2010).

Hsieh, C. C.

W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. 48, 761–765 (2006).
[Crossref]

Hsu, T. H.

W. Jywe, T. H. Hsu, and C. H. Liu, “Non-bar, an optical calibration system for five-axis CNC machine tools,” Int. J. Mach. Tools Manuf. 59, 16–23 (2012).
[Crossref]

W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. 48, 761–765 (2006).
[Crossref]

Ito, S.

X. H. 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).
[Crossref]

Joo, K. N.

Jywe, W.

W. Jywe, T. H. Hsu, and C. H. Liu, “Non-bar, an optical calibration system for five-axis CNC machine tools,” Int. J. Mach. Tools Manuf. 59, 16–23 (2012).
[Crossref]

Jywe, W. Y.

C. J. Chen, P. D. Lin, and W. Y. Jywe, “An optoelectronic measurement system for measuring 6-degree-of-freedom motion error of rotary parts,” Opt. Express 15(22), 14601–14617 (2007).
[Crossref] [PubMed]

W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. 48, 761–765 (2006).
[Crossref]

Klapetek, P.

J. Lazar, O. Cip, M. Cizek, J. Hrabina, M. Sery, and P. Klapetek, “Laser interferometric measuring system for positioning in nanometrology,” WSEAS Trans. Cir. Syst. 9(10), 660–669 (2010).

Korpelainen, V.

V. Korpelainen and A. Lassila, “Calibration of a commercial AFM: traceability for a coordinate system,” Meas. Sci. Technol. 18(2), 395–403 (2007).
[Crossref]

Kuang, C. F.

C. F. Kuang, E. Hong, and J. Ni, “A high-precision five-degree-of-freedom measurement system based on laser collimator and interferometry techniques,” Rev. Sci. Instrum. 78(9), 095105 (2007).
[Crossref] [PubMed]

Lassila, A.

V. Korpelainen and A. Lassila, “Calibration of a commercial AFM: traceability for a coordinate system,” Meas. Sci. Technol. 18(2), 395–403 (2007).
[Crossref]

Lazar, J.

J. Lazar, O. Cip, M. Cizek, J. Hrabina, M. Sery, and P. Klapetek, “Laser interferometric measuring system for positioning in nanometrology,” WSEAS Trans. Cir. Syst. 9(10), 660–669 (2010).

Lee, S. W.

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127(4), 857–865 (2005).
[Crossref]

Li, C. R.

Li, D. C.

Li, X. H.

X. H. 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).
[Crossref]

Lin, P. D.

Liu, C. H.

W. Jywe, T. H. Hsu, and C. H. Liu, “Non-bar, an optical calibration system for five-axis CNC machine tools,” Int. J. Mach. Tools Manuf. 59, 16–23 (2012).
[Crossref]

W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. 48, 761–765 (2006).
[Crossref]

Mayor, R.

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127(4), 857–865 (2005).
[Crossref]

Menq, C. H.

C. H. Menq, J. H. Zhang, and J. Shi, “Design and development of an interferometer with improved angular tolerance and its application to x-y theta measurement,” Rev. Sci. Instrum. 71(12), 4633–4638 (2000).
[Crossref]

Muto, H.

X. H. 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).
[Crossref]

Ni, J.

C. F. Kuang, E. Hong, and J. Ni, “A high-precision five-degree-of-freedom measurement system based on laser collimator and interferometry techniques,” Rev. Sci. Instrum. 78(9), 095105 (2007).
[Crossref] [PubMed]

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127(4), 857–865 (2005).
[Crossref]

Park, N. Y.

W. S. Park, H. S. Cho, Y. K. Byun, and N. Y. Park, “Measurement of 6-DOF displacement of rigid bodies through splitting a laser beam: experimental investigation,” Proc. SPIE 4190, 80–91 (2001).
[Crossref]

Park, W. S.

W. S. Park, H. S. Cho, Y. K. Byun, and N. Y. Park, “Measurement of 6-DOF displacement of rigid bodies through splitting a laser beam: experimental investigation,” Proc. SPIE 4190, 80–91 (2001).
[Crossref]

Quenelle, R. C.

R. C. Quenelle, “Nonlinearity in interferometer measurements,” Hewlett Packard J. 34, 10 (1983).

Schmidt, R. H. M.

Sery, M.

J. Lazar, O. Cip, M. Cizek, J. Hrabina, M. Sery, and P. Klapetek, “Laser interferometric measuring system for positioning in nanometrology,” WSEAS Trans. Cir. Syst. 9(10), 660–669 (2010).

Sheu, Y. H.

W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. 48, 761–765 (2006).
[Crossref]

Shi, J.

C. H. Menq, J. H. Zhang, and J. Shi, “Design and development of an interferometer with improved angular tolerance and its application to x-y theta measurement,” Rev. Sci. Instrum. 71(12), 4633–4638 (2000).
[Crossref]

Shien, W. H.

W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. 48, 761–765 (2006).
[Crossref]

Shimizu, Y.

X. H. 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).
[Crossref]

Shyu, L. H.

W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. 48, 761–765 (2006).
[Crossref]

Spronck, J. W.

Su, C. S.

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

Tang, W. H.

Wilkening, G.

H. Bosse and G. Wilkening, “Developments at PTB in nanometrology for support of the semiconductor industry,” Meas. Sci. Technol. 16(11), 2155–2166 (2005).
[Crossref]

Wu, C. M.

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

Yan, L. P.

Yang, T.

Yao, X. G.

Zhang, E. Z.

Zhang, J. H.

C. H. Menq, J. H. Zhang, and J. Shi, “Design and development of an interferometer with improved angular tolerance and its application to x-y theta measurement,” Rev. Sci. Instrum. 71(12), 4633–4638 (2000).
[Crossref]

Hewlett Packard J. (1)

R. C. Quenelle, “Nonlinearity in interferometer measurements,” Hewlett Packard J. 34, 10 (1983).

Int. J. Mach. Tools Manuf. (1)

W. Jywe, T. H. Hsu, and C. H. Liu, “Non-bar, an optical calibration system for five-axis CNC machine tools,” Int. J. Mach. Tools Manuf. 59, 16–23 (2012).
[Crossref]

J. Manuf. Sci. Eng. (1)

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127(4), 857–865 (2005).
[Crossref]

J. Phys. Conf. Ser. (1)

W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. 48, 761–765 (2006).
[Crossref]

Meas. Sci. Technol. (4)

V. Korpelainen and A. Lassila, “Calibration of a commercial AFM: traceability for a coordinate system,” Meas. Sci. Technol. 18(2), 395–403 (2007).
[Crossref]

H. Bosse and G. Wilkening, “Developments at PTB in nanometrology for support of the semiconductor industry,” Meas. Sci. Technol. 16(11), 2155–2166 (2005).
[Crossref]

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[Crossref]

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

Opt. Express (4)

Precis. Eng. (2)

X. H. 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).
[Crossref]

W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30(3), 337–346 (2006).
[Crossref]

Proc. SPIE (1)

W. S. Park, H. S. Cho, Y. K. Byun, and N. Y. Park, “Measurement of 6-DOF displacement of rigid bodies through splitting a laser beam: experimental investigation,” Proc. SPIE 4190, 80–91 (2001).
[Crossref]

Rev. Sci. Instrum. (2)

C. H. Menq, J. H. Zhang, and J. Shi, “Design and development of an interferometer with improved angular tolerance and its application to x-y theta measurement,” Rev. Sci. Instrum. 71(12), 4633–4638 (2000).
[Crossref]

C. F. Kuang, E. Hong, and J. Ni, “A high-precision five-degree-of-freedom measurement system based on laser collimator and interferometry techniques,” Rev. Sci. Instrum. 78(9), 095105 (2007).
[Crossref] [PubMed]

WSEAS Trans. Cir. Syst. (1)

J. Lazar, O. Cip, M. Cizek, J. Hrabina, M. Sery, and P. Klapetek, “Laser interferometric measuring system for positioning in nanometrology,” WSEAS Trans. Cir. Syst. 9(10), 660–669 (2010).

Other (1)

Q. B. Feng, Optical Measurement Techniques and Applications (Tsinghua University, 2008), Chap.2.

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Figures (13)

Fig. 1
Fig. 1 Schematic of laser heterodyne interferometer for simultaneous measuring displacement and angle based on the Faraday effect.
Fig. 2
Fig. 2 Schematic for simultaneous measuring displacement and angle.
Fig. 3
Fig. 3 Schematic of movement along the moving axis.
Fig. 4
Fig. 4 Schematic of movement perpendicular to the moving axis.
Fig. 5
Fig. 5 Schematic of rotational movement around arbitrary center.
Fig. 6
Fig. 6 Schematic of rotational center on the moving axis.
Fig. 7
Fig. 7 Schematic of return beams with a rotational angle error of FR.
Fig. 8
Fig. 8 Simulation of the influence of rotational angle error of FR.
Fig. 9
Fig. 9 Experimental setup.
Fig. 10
Fig. 10 Angle measurement experiment with the step of 1°.
Fig. 11
Fig. 11 Angle measurement experiment with the step of 0.0001°.
Fig. 12
Fig. 12 Experimental results in millimeter range.
Fig. 13
Fig. 13 Experimental results in micrometer range.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

Δ f ( θ ) = 2 n H 1 - ( n sin θ n ) 2 2 n H
l 1 = n l 2 n Δ L 1 + 2 Δ f ( θ ) = n l 4 n L sin θ 4 n H tan θ sin θ + 2 Δ f ( θ )
l 2 = n l + 2 n Δ L 2 + 2 Δ f ( θ ) = n l + 4 n L sin θ 4 n H tan θ sin θ + 2 Δ f ( θ )
l = l 2 + l 1 + 8 n H sin θ tan [ arc sin ( n sin θ n ) ] 4 Δ f ( θ ) 8 n
θ = arc sin ( ( l 2 l 1 ) 4 n S )
l = l 1 + l 2 + 8 n H sin θ tan [ arc sin ( n sin θ n ) ] 4 Δ f ( θ ) 8 n + d [ sin ( α + θ ) sin α ]
l = l 1 + l 2 + 8 n H sin θ tan [ arc sin ( n sin θ n ) ] 4 Δ f ( θ ) 8 n d [ 1 - cos θ ]
[ E RB E TB y ] = [ E 1 cos ( 2 π f 1 t + φ 1 ) cos 2 α E 2 cos ( 2 π f 2 t + φ 2 ) ]
I 1 2 cos 2 α E 1 E 2 cos ( 2 π Δ f t + Δ φ )

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