Abstract

A grating interferometer based on the wavelength-modulated phase-shifting method for displacement measurements is proposed. A laser beam with sequential phase shifting can be accomplished using a wavelength-modulated light passing through an unequal-path-length optical configuration. The optical phase of the moving grating is measured by the wavelength-modulated phase-shifting technique and the proposed time-domain quadrature detection method. The displacement of the grating is determined by the grating interferometry theorem with the measured phase variation. Experimental results reveal that the proposed method can detect a displacement up to a large distance of 1 mm and displacement variation down to the nanometer range.

© 2013 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]

2013 (2)

C. C. Wu, C. C. Hsu, J. Y. Lee, Y. Z. Chen, and J. S. Yang, “Littrow-type self-aligned laser encoder with high tolerance using double diffractions,” Opt. Commun.297, 89–97 (2013).
[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]

2012 (3)

J. Y. Lee, M. P. Lu, K. Y. Lin, and S. H. Huang, “Measurement of in-plane displacement by wavelength-modulated heterodyne speckle interferometry,” Appl. Opt.51(8), 1095–1100 (2012).
[CrossRef] [PubMed]

A. Kimura, W. Gao, W. J. 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]

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,” Meas.45(6), 1510–1514 (2012).
[CrossRef]

2011 (2)

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]

J. Y. Lee and M. P. Lu, “Optical heterodyne grating shearing interferometry for long-range positioning applications,” Opt. Commun.284(3), 857–862 (2011).
[CrossRef]

2010 (1)

A. Kimura, W. Gao, and L. Zeng, “Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder,” Meas. Sci. Technol.21, 054005 (2010).

2009 (3)

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]

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]

2008 (3)

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. Actuators A Phys.141(1), 217–223 (2008).
[CrossRef]

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” J. Appl. Phys.47, 1833–1837 (2008).

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun.281(9), 2582–2589 (2008).
[CrossRef]

2007 (1)

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A Phys.137(1), 185–191 (2007).
[CrossRef]

2005 (1)

C. F. Kao, C. C. Chang, and M. H. Lu, “Double-diffraction planar encoder by conjugate optics,” Opt. Eng.44, 023063 (2005).

2004 (2)

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys.110(1-3), 294–300 (2004).
[CrossRef]

C. M. Wu, “Heterodyne interferometric system with sub-nanometer accuracy for measurement of straightness,” Appl. Opt.43(19), 3812–3816 (2004).
[CrossRef] [PubMed]

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]

1995 (1)

1994 (1)

1992 (1)

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

1988 (1)

R. J. Moffat, “Describing the uncertainties in experimental results,” Exp. Therm. Fluid Sci.1(1), 3–17 (1988).
[CrossRef]

1985 (1)

Arguel, P.

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys.110(1-3), 294–300 (2004).
[CrossRef]

Bojhkov, B.

Bonnefont, S.

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys.110(1-3), 294–300 (2004).
[CrossRef]

Chang, C. C.

C. F. Kao, C. C. Chang, and M. H. Lu, “Double-diffraction planar encoder by conjugate optics,” Opt. Eng.44, 023063 (2005).

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, H. Y.

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun.281(9), 2582–2589 (2008).
[CrossRef]

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A Phys.137(1), 185–191 (2007).
[CrossRef]

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,” Meas.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. 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]

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, 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,” Meas.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, 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,” Meas.45(6), 1510–1514 (2012).
[CrossRef]

Chen, Y. Z.

C. C. Wu, C. C. Hsu, J. Y. Lee, Y. Z. Chen, and J. S. Yang, “Littrow-type self-aligned laser encoder with high tolerance using double diffractions,” Opt. Commun.297, 89–97 (2013).
[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,” Meas.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. Actuators A Phys.141(1), 217–223 (2008).
[CrossRef]

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.

Fan, K. C.

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” J. Appl. Phys.47, 1833–1837 (2008).

Fourment, S.

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys.110(1-3), 294–300 (2004).
[CrossRef]

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. J. 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, and L. Zeng, “Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder,” Meas. Sci. Technol.21, 054005 (2010).

Hosono, K.

A. Kimura, W. Gao, W. J. 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]

Hsu, C. C.

C. C. Wu, C. C. Hsu, J. Y. Lee, Y. Z. Chen, and J. S. Yang, “Littrow-type self-aligned laser encoder with high tolerance using double diffractions,” Opt. Commun.297, 89–97 (2013).
[CrossRef]

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun.281(9), 2582–2589 (2008).
[CrossRef]

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A Phys.137(1), 185–191 (2007).
[CrossRef]

Huang, S. H.

Ishii, Y.

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]

Jay, J.

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys.110(1-3), 294–300 (2004).
[CrossRef]

Jourlin, Y.

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys.110(1-3), 294–300 (2004).
[CrossRef]

Kao, C. F.

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” J. Appl. Phys.47, 1833–1837 (2008).

C. F. Kao, C. C. Chang, and M. H. Lu, “Double-diffraction planar encoder by conjugate optics,” Opt. Eng.44, 023063 (2005).

Kao, F. H.

Kim, W. J.

A. Kimura, W. Gao, W. J. 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. J. 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, and L. Zeng, “Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder,” Meas. Sci. Technol.21, 054005 (2010).

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. Actuators A Phys.141(1), 217–223 (2008).
[CrossRef]

Lee, J. Y.

C. C. Wu, C. C. Hsu, J. Y. Lee, Y. Z. Chen, and J. S. Yang, “Littrow-type self-aligned laser encoder with high tolerance using double diffractions,” Opt. Commun.297, 89–97 (2013).
[CrossRef]

J. Y. Lee, M. P. Lu, K. Y. Lin, and S. H. Huang, “Measurement of in-plane displacement by wavelength-modulated heterodyne speckle interferometry,” Appl. Opt.51(8), 1095–1100 (2012).
[CrossRef] [PubMed]

J. Y. Lee and M. P. Lu, “Optical heterodyne grating shearing interferometry for long-range positioning applications,” Opt. Commun.284(3), 857–862 (2011).
[CrossRef]

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun.281(9), 2582–2589 (2008).
[CrossRef]

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A Phys.137(1), 185–191 (2007).
[CrossRef]

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]

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]

Lin, K. Y.

Lozes, F.

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys.110(1-3), 294–300 (2004).
[CrossRef]

Lu, M. H.

C. F. Kao, C. C. Chang, and M. H. Lu, “Double-diffraction planar encoder by conjugate optics,” Opt. Eng.44, 023063 (2005).

Lu, M. P.

J. Y. Lee, M. P. Lu, K. Y. Lin, and S. H. Huang, “Measurement of in-plane displacement by wavelength-modulated heterodyne speckle interferometry,” Appl. Opt.51(8), 1095–1100 (2012).
[CrossRef] [PubMed]

J. Y. Lee and M. P. Lu, “Optical heterodyne grating shearing interferometry for long-range positioning applications,” Opt. Commun.284(3), 857–862 (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]

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” J. Appl. Phys.47, 1833–1837 (2008).

Moffat, R. J.

R. J. Moffat, “Describing the uncertainties in experimental results,” Exp. Therm. Fluid Sci.1(1), 3–17 (1988).
[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]

Nevièvre, M.

Noullet, J. L.

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys.110(1-3), 294–300 (2004).
[CrossRef]

Onodera, R.

Parriaux, O.

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys.110(1-3), 294–300 (2004).
[CrossRef]

Popov, E.

Sarrabayrouse, G.

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys.110(1-3), 294–300 (2004).
[CrossRef]

Shen, H. M.

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” J. Appl. Phys.47, 1833–1837 (2008).

Shi, L.

A. Kimura, W. Gao, W. J. 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. Actuators 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. J. 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]

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. Actuators A Phys.141(1), 217–223 (2008).
[CrossRef]

Teimel, A.

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

Tonchev, S.

Tsonev, L.

Wang, S. F.

Weng, H. F.

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun.281(9), 2582–2589 (2008).
[CrossRef]

Wu, C. C.

C. C. Wu, C. C. Hsu, J. Y. Lee, Y. Z. Chen, and J. S. Yang, “Littrow-type self-aligned laser encoder with high tolerance using double diffractions,” Opt. Commun.297, 89–97 (2013).
[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]

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun.281(9), 2582–2589 (2008).
[CrossRef]

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A Phys.137(1), 185–191 (2007).
[CrossRef]

Wu, C. M.

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. Actuators A Phys.141(1), 217–223 (2008).
[CrossRef]

Yang, J. S.

C. C. Wu, C. C. Hsu, J. Y. Lee, Y. Z. Chen, and J. S. Yang, “Littrow-type self-aligned laser encoder with high tolerance using double diffractions,” Opt. Commun.297, 89–97 (2013).
[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]

Zeng, L.

A. Kimura, W. Gao, W. J. 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, and L. Zeng, “Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder,” Meas. Sci. Technol.21, 054005 (2010).

Appl. Opt. (6)

Exp. Therm. Fluid Sci. (1)

R. J. Moffat, “Describing the uncertainties in experimental results,” Exp. Therm. Fluid Sci.1(1), 3–17 (1988).
[CrossRef]

J. Appl. Phys. (1)

C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” J. Appl. Phys.47, 1833–1837 (2008).

Meas. (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,” Meas.45(6), 1510–1514 (2012).
[CrossRef]

Meas. Sci. Technol. (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]

Opt. Commun. (4)

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]

C. C. Wu, C. C. Hsu, J. Y. Lee, Y. Z. Chen, and J. S. Yang, “Littrow-type self-aligned laser encoder with high tolerance using double diffractions,” Opt. Commun.297, 89–97 (2013).
[CrossRef]

J. Y. Lee and M. P. Lu, “Optical heterodyne grating shearing interferometry for long-range positioning applications,” Opt. Commun.284(3), 857–862 (2011).
[CrossRef]

C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun.281(9), 2582–2589 (2008).
[CrossRef]

Opt. Eng. (2)

C. F. Kao, C. C. Chang, and M. H. Lu, “Double-diffraction planar encoder by conjugate optics,” Opt. Eng.44, 023063 (2005).

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. 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]

Opt. Lett. (1)

Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder (1)

A. Kimura, W. Gao, and L. Zeng, “Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder,” Meas. Sci. Technol.21, 054005 (2010).

Precis. Eng. (3)

A. Kimura, W. Gao, W. J. 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]

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

Sens. Actuators A Phys. (3)

S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys.110(1-3), 294–300 (2004).
[CrossRef]

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. Actuators A Phys.141(1), 217–223 (2008).
[CrossRef]

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A Phys.137(1), 185–191 (2007).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the wavelength phase-shifting grating interferometer. The diffraction beams can be reflected by (a) the mirrors or (b) the corner cube retro-reflectors. These reflected beams are diffracted by the grating G again, and interfere with each other. FG: Function Generator, LD: Laser Diode, BS: Beam Splitter, G: Grating, M: Mirror, C: Corner Cube Retro-reflectors, PZT: Piezoelectric actuators, PD: Photodetector, PC: Personal Computer.

Fig. 2
Fig. 2

Simulated interference signal intensity which is a function of the injection current.

Fig. 3
Fig. 3

The intensity of the interference signals for the injection current ia (upper curve) and wavelength λ1 (lower curve) for the injection current iab and wavelength λ2.

Fig. 4
Fig. 4

Lissajous patterns of (a) the original (I1 and I2), and (b) modified (I'1 and I'2) interference signals.

Fig. 5
Fig. 5

Measurement results for a long displacement of ~1 mm. Red curves: measured displacements, blue curves: measured by the encoder. Curves are offset by a few seconds for convenience of observation.

Fig. 6
Fig. 6

Measurement results for forward and backward displacement with amplitudes of about 50, 20, 10 and 5 μm.

Fig. 7
Fig. 7

Measurement results for the step-wise motion with step of 50 and 25 nm.

Fig. 8
Fig. 8

Measurement results for the step-wise motion with steps of 10 and 5 nm.

Fig. 9
Fig. 9

(a) Interference signals and (b) phase noises, including high- and low-frequency noises.

Fig. 10
Fig. 10

Periodic nonlinearity error.

Equations (27)

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E ±1 exp( i 2π λ l ±1 ±i ϕ g ).
E ' ±1 exp( i 2π λ 2 l ±1 ±i2 ϕ g ).
I | E ' +1 +E ' 1 | 2 =1+cos( 2πΔl /λ +ϕ ),
ϕ=4 ϕ g =8πΔx/Λ,
Δx=( Λ/8π )ϕ.
I( t )S( t )[ 1+Vcos( 2πΔl / λ( t ) +ϕ ) ],
I 1 S 1 [ 1+Vcos( 2πΔl / λ 1 +ϕ ) ], 0<t<T/2 ,
I 2 S 2 [ 1+Vcos( 2πΔl / λ 2 +ϕ ) ], T/2 <t<T,
I 2 S 2 [ 1+Vcos( 2π λ 1 Δl 2π λ 1 2 ΔλΔl+ϕ ) ]= S 2 [ 1+Vsin( 2π λ 1 Δl+ϕ ) ],
I( i )=( S 0 + m s i )[ 1+Vcos( 2πΔl / ( λ 1 + m λ i ) +ϕ ) ],
I 1min = S 1 ( 1V ),
I 1max = S 1 ( 1+V ).
S 1 =( I 1min + I 1max )/2.
S 2 =( I 2min + I 2max )/2.
I 1 =( I 1 S 1 )/ S 1 =Vcos( 2πΔl / λ 1 +ϕ ),
I 2 =( I 2 S 2 )/ S 2 =Vsin( 2πΔl / λ 1 +ϕ ).
ϕ= tan 1 ( I 2 / I 1 )2πΔl/ λ 1 .
e= I 1 2 + I 2 2 .
s= dϕ dΔx = 8π Λ .
dϕ= ( ϕ I 1 d I 1 ) 2 + ( ϕ I 2 d I 2 ) 2 = 1 I 1 2 + I 2 2 ( I 2 d I 1 ) 2 + ( I 1 d I 2 ) 2 .
dϕ= [ ( I 2 S 2 )d I 1 ] 2 + [ ( I 1 S 1 )d I 2 ] 2 S 2 ( I 1 S 1 ) 2 / S 1 + S 1 ( I 2 S 2 ) 2 / S 2 .
dϕ= dI SV .
I 1err = V 1err cos( ϕ )+ S 1err ,
I 2err = V 2err sin( ϕ+ε )+ S 2err ,
δϕ= ϕ err ϕ= tan 1 [ V 2err sin( ϕ+ε )+ S 2err V 1err cos( ϕ )+ S 1err ]ϕ.
ε= 8π Λ u 2f .
dϕ dt = 8π Λ dΔx dt = 8π Λ uπ f ODR .

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