Abstract

A grating-based interferometer for 6-DOF displacement and angle measurement is proposed in this study. The proposed interferometer is composed of three identical detection parts sharing the same light source. Each detection part utilizes three techniques: heterodyne, grating shearing, and Michelson interferometries. Displacement information in the three perpendicular directions (X, Y, Z) can be sensed simultaneously by each detection part. Furthermore, angle information (θX, θY, θZ) can be obtained by comparing the displacement measurement results between two corresponding detection parts. The feasibility and performance of the proposed grating-based interferometer are evaluated in displacement and angle measurement experiments. In comparison with the internal capacitance sensor built into the commercial piezo-stage, the measurement resolutions of the displacement and angle of our proposed interferometer are about 2 nm and 0.05 μrad.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  10. K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
    [Crossref]
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    [Crossref]
  12. 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).
    [Crossref]
  13. 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).
    [Crossref]
  14. J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
    [Crossref]
  15. 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(22), 25805–25819 (2013).
    [Crossref] [PubMed]
  16. Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
    [Crossref]
  17. A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
    [Crossref]
  18. W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
    [Crossref]
  19. 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).
    [Crossref]
  20. H. L. Hsieh, J. C. Chen, G. Lerondel, and J. Y. Lee, “Two-dimensional displacement measurement by quasi-common-optical-path heterodyne grating interferometer,” Opt. Express 19(10), 9770–9782 (2011).
    [Crossref] [PubMed]
  21. D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
    [Crossref]

2013 (2)

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).
[Crossref]

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(22), 25805–25819 (2013).
[Crossref] [PubMed]

2012 (2)

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

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).
[Crossref]

2011 (3)

J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011).
[Crossref]

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

H. L. Hsieh, J. C. Chen, G. Lerondel, and J. Y. Lee, “Two-dimensional displacement measurement by quasi-common-optical-path heterodyne grating interferometer,” Opt. Express 19(10), 9770–9782 (2011).
[Crossref] [PubMed]

2010 (1)

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

2009 (3)

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

S. F. Wang, M. H. Chiu, W. W. Chen, F. H. Kao, and R. S. Chang, “Small-displacement sensing system based on multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 48(13), 2566–2573 (2009).
[Crossref] [PubMed]

2008 (1)

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

2006 (1)

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

2005 (1)

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

2004 (1)

2003 (1)

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).
[Crossref]

1999 (1)

1998 (1)

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9(7), 1024–1030 (1998).
[Crossref]

1996 (1)

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[Crossref]

1992 (1)

A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14(3), 147–154 (1992).
[Crossref]

1985 (1)

Arai, Y.

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

Bae, E. D.

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

Bin, Z.

Bojhkov, B.

Chang, R. S.

Chang, W. Y.

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

Chen, C. D.

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[Crossref]

Chen, J. C.

Chen, J. H.

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011).
[Crossref]

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

Chen, K.

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

Chen, K. H.

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011).
[Crossref]

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

Chen, Q.

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

Chen, W. W.

Chen, Y. C.

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

Cheng, C. H.

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

Chiu, H. S.

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

Chiu, M. H.

S. F. Wang, M. H. Chiu, W. W. Chen, F. H. Kao, and R. S. Chang, “Small-displacement sensing system based on multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 48(13), 2566–2573 (2009).
[Crossref] [PubMed]

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[Crossref]

Cuifang, K.

Cunxing, C.

Demarest, F. C.

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9(7), 1024–1030 (1998).
[Crossref]

Dian, S.

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).
[Crossref]

Estler, W. T.

Fenglin, Y.

Gao, W.

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).
[Crossref]

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).
[Crossref]

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

Hosono, K.

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).
[Crossref]

Hsieh, H. L.

Ito, S.

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).
[Crossref]

Kao, F. H.

Keem, T.

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

Kim, S. W.

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

Kim, W.

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).
[Crossref]

Kimura, A.

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).
[Crossref]

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

Lai, C. W.

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

Lee, J. Y.

Lerondel, G.

Li, X.

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).
[Crossref]

Lijiang, Z.

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

Lin, D.

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

Lin, J. Y.

J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011).
[Crossref]

Lu, S. H.

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

Muto, H.

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).
[Crossref]

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

Nevièvre, M.

Oh, J. S.

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

Popov, E.

Qibo, F.

Saito, Y.

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

Sasaki, O.

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).
[Crossref]

Shi, L.

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).
[Crossref]

Shih, B. Y.

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

Shimizu, Y.

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).
[Crossref]

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).
[Crossref]

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

Shyu, L. H.

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

Su, D. C.

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[Crossref]

Suzuki, T.

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).
[Crossref]

Teimel, A.

A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14(3), 147–154 (1992).
[Crossref]

Togashi, C.

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).
[Crossref]

Tonchev, S.

Tsonev, L.

Wang, S. F.

Wu, C. C.

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

Wu, C. M.

Wu, J.

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

Wu, T. H.

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

Yan, J.

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

Yang, T. H.

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

Yin, C.

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

Yusheng, Z.

Zeng, L.

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).
[Crossref]

Appl. Opt. (4)

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

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

J. Opt. (1)

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[Crossref]

Manuf. Technol. (1)

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

Meas. Sci. Technol. (2)

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9(7), 1024–1030 (1998).
[Crossref]

Measurement (1)

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

Opt. Commun. (1)

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

Opt. Eng. (2)

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).
[Crossref]

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (1)

J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011).
[Crossref]

Precis. Eng. (4)

A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14(3), 147–154 (1992).
[Crossref]

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).
[Crossref]

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

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).
[Crossref]

Sens, Actuator A-Phys. (1)

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1

System configuration for 3-DOF displacement measurement.

Fig. 2
Fig. 2

Optical configuration for 6-DOF measurement.

Fig. 3
Fig. 3

Experimental set-up for 6-DOF measurement.

Fig. 4
Fig. 4

Measurement results for 3-DOF displacement on the X-, Y-, and Z-axes.

Fig. 5
Fig. 5

Measurement result for 3-DOF rotation angle in the θX-, θY-, and θZ-directions.

Fig. 6
Fig. 6

Measurement results for 6-DOF repeatability.

Fig. 7
Fig. 7

Measurement results for the speed limitation tests.

Fig. 8
Fig. 8

Measurement results for stability tests.

Fig. 9
Fig. 9

Sketch of the misalignment caused by the angles of yaw, pitch, and roll.

Equations (21)

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E EOM = e i( kl+ϕ ) ( e iωt/2 e iωt/2 ),
A=D=J( 0 ) E EOM ( e i ωt /2 e i ωt /2 ),B=C=J( 180 ) E EOM ( e i ωt /2 e i ωt /2 ),
ϕ qm =2mπ l q /p,
I 2 | P 2 ( 0 )( B X0 + D X+1 ) | 2 cos[ ωt+( ϕ X0 ϕ X+1 ) ]. I 3 | P 3 ( 0 )( B Y0 + A Y1 ) | 2 cos[ ωt+( ϕ Y0 ϕ Y1 ) ]
I 1 cos(ωt).
Δ Φ q =2πΔ l q /p,
Δ l q = pΔ Φ q / 2π .
E r = e i( k l r + ϕ r ) ( e iωt/2 e iωt/2 ), E m = e i( k l m + ϕ m ) ( e iωt/2 e iωt/2 ),
I 4 = | P 4 ( 45 )( E r + E m ) | 2 cos[ ωt+k( l r l m ) ].
Δ Φ Z =k( l r l m )=2Δ l Z .
Δ l Z = Δ Φ Z 2k = Δ Φ Z λ 4π ,
I 5 cos[ ωt+( ϕ X0II ϕ X+1II ) ] I 6 cos[ ωt+( ϕ Y0II ϕ Y1II ) ] I 7 cos[ ωt+k( l rII l mII ) ]
I 8 cos[ ωt+( ϕ X0Ш ϕ X+1Ш ) ] I 9 cos[ ωt+( ϕ Y0Ш ϕ Y1Ш ) ] I 10 cos[ ωt+k( l rШ l mШ ) ]
Δ θ X = tan 1 ( Δ l ZI Δ l ZII S Y ),
Δ θ Y = tan 1 ( Δ l ZII Δ l ZШ S X ),
Δ θ Z = tan 1 ( Δ l YI Δ l YII S Y ),
Δ d eY = Δ Φ Y 2π (p p yaw )=Δ d Y ( 1cos θ X ),
Δ θ eθZ =Δ θ Z Δ θ Z ' = tan 1 ( Δ l YI Δ l YII S Y ) tan 1 ( ( Δ l YI Δ l YII )cos θ X S Y ),
Δ d eX = Δ Φ X 2π (p p pitch )=Δ d X ( 1cos θ Y ),
Δ d eX =Δ d eY = Δ Φ q 2π (p p roll )= Δ Φ q 2π p( 1 1 cos θ Z )=Δ d q ( 1 1 cos θ Z ),
Δ θ eθZ =Δ θ Z Δ θ Z ' = tan 1 ( Δ l YI Δ l YII S Y ) tan 1 [ ( (Δ l YI Δ l YII ) S Y 1 cos θ Z ) ],

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