Abstract

A laser straightness interferometer system with rotational error compensation and simultaneous measurement of six degrees of freedom error parameters is proposed. The optical configuration of the proposed system is designed and the mathematic model for simultaneously measuring six degrees of freedom parameters of the measured object including three rotational parameters of the yaw, pitch and roll errors and three linear parameters of the horizontal straightness error, vertical straightness error and straightness error’s position is established. To address the influence of the rotational errors produced by the measuring reflector in laser straightness interferometer, the compensation method of the straightness error and its position is presented. An experimental setup was constructed and a series of experiments including separate comparison measurement of every parameter, compensation of straightness error and its position and simultaneous measurement of six degrees of freedom parameters of a precision linear stage were performed to demonstrate the feasibility of the proposed system. Experimental results show that the measurement data of the multiple degrees of freedom parameters obtained from the proposed system are in accordance with those obtained from the compared instruments and the presented compensation method can achieve good effect in eliminating the influence of rotational errors on the measurement of straightness error and its position.

© 2015 Optical Society of America

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References

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  1. O. Borisov, S. Fletcher, A. Longstaff, and A. Myers, “Performance evaluation of a new taut wire system or straightness measurement of machine tools,” Precis. Eng. 38(3), 492–498 (2014).
    [Crossref]
  2. 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]
  3. P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage,” Precis. Eng. 35(3), 424–430 (2011).
    [Crossref]
  4. J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. 38(3), 578–588 (2014).
    [Crossref]
  5. J. J. Kroll, “Six degree of freedom optical sensor for dynamic measurement of linear axes,” PhD thesis, University of North Carolina, Charlotte (2003).
  6. J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” J. Eng. Ind. 114(3), 362–369 (1992).
  7. 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(1), 761–765 (2006).
    [Crossref]
  8. I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
    [Crossref]
  9. K. C. Fan and Y. Zhao, “A laser straightness measurement system using optical fiber and modulation techniques,” Int. J. Mach. Tools Manuf. 40(14), 2073–2081 (2000).
    [Crossref]
  10. C. H. Liu, J. H. Chen, and Y. F. Teng, “Development of a straightness measurement and compensation system with multiple right-angle reflectors and a lead zirconate titanate-based compensation stage,” Rev. Sci. Instrum. 80(11), 115105 (2009).
    [Crossref] [PubMed]
  11. A. Kimura, W. Gao, and L. J. 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(5), 054005 (2010).
    [Crossref]
  12. C. H. Liu and C. H. Cheng, “Development of a grating based multi-degree-of-freedom laser linear encoder using diffracted light,” Sen. Actuators A. 181(7), 87–93 (2012).
    [Crossref]
  13. L. J. Zhu, L. Li, J. H. Liu, and Z. H. Zhang, “A method for measuring the guideway straightness error based on polarized interference principle,” Int. J. Mach. Tools Manuf. 49(3–4), 285–290 (2009).
    [Crossref]
  14. P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precis. Eng. 35(4), 686–692 (2011).
    [Crossref]
  15. K. C. Fan and M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
    [Crossref]
  16. R. R. Baldwin, “Interferometer system for measuring straightness and roll,” U.S. Patent, 3790284 (1974).
  17. D. R. McMurtry and R. J. Chaney, “Straightness interferometer system,” U.S. Patent, 5026163 (1991).
  18. S. T. Lin, “A laser interferometer for measuring straightness,” Opt. Laser Technol. 33(3), 195–199 (2001).
    [Crossref]
  19. Q. H. Chen, D. J. Lin, J. Wu, J. Q. Yan, and C. Y. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
    [Crossref]
  20. B. Chen, E. Zhang, L. Yan, C. Li, W. Tang, and Q. Feng, “A laser interferometer for measuring straightness and its position based on heterodyne interferometry,” Rev. Sci. Instrum. 80(11), 115113 (2009).
    [Crossref] [PubMed]
  21. S. T. Lin, S. L. Yeh, C. S. Chiu, and M. S. Huang, “A calibrator based on the use of low-coherent light source straightness interferometer and compensation method,” Opt. Express 19(22), 21929–21937 (2011).
    [Crossref] [PubMed]
  22. Y. S. Zhai, Q. B. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44(4), 839–843 (2012).
    [Crossref]
  23. R. C. Quenelle, “Nonlinearity in interferometer measurements,” Hewlett Pachard J. 34, 10 (1983).
  24. D. J. Loner, B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Annals-Manufac, Technol. 52(1), 439–442 (2003).
  25. C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
    [Crossref]
  26. W. M. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30(3), 337–346 (2006).
    [Crossref]

2014 (2)

O. Borisov, S. Fletcher, A. Longstaff, and A. Myers, “Performance evaluation of a new taut wire system or straightness measurement of machine tools,” Precis. Eng. 38(3), 492–498 (2014).
[Crossref]

J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. 38(3), 578–588 (2014).
[Crossref]

2013 (1)

2012 (2)

C. H. Liu and C. H. Cheng, “Development of a grating based multi-degree-of-freedom laser linear encoder using diffracted light,” Sen. Actuators A. 181(7), 87–93 (2012).
[Crossref]

Y. S. Zhai, Q. B. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44(4), 839–843 (2012).
[Crossref]

2011 (3)

S. T. Lin, S. L. Yeh, C. S. Chiu, and M. S. Huang, “A calibrator based on the use of low-coherent light source straightness interferometer and compensation method,” Opt. Express 19(22), 21929–21937 (2011).
[Crossref] [PubMed]

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precis. Eng. 35(4), 686–692 (2011).
[Crossref]

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage,” Precis. Eng. 35(3), 424–430 (2011).
[Crossref]

2010 (1)

A. Kimura, W. Gao, and L. J. 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(5), 054005 (2010).
[Crossref]

2009 (4)

B. Chen, E. Zhang, L. Yan, C. Li, W. Tang, and Q. Feng, “A laser interferometer for measuring straightness and its position based on heterodyne interferometry,” Rev. Sci. Instrum. 80(11), 115113 (2009).
[Crossref] [PubMed]

L. J. Zhu, L. Li, J. H. Liu, and Z. H. Zhang, “A method for measuring the guideway straightness error based on polarized interference principle,” Int. J. Mach. Tools Manuf. 49(3–4), 285–290 (2009).
[Crossref]

I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
[Crossref]

C. H. Liu, J. H. Chen, and Y. F. Teng, “Development of a straightness measurement and compensation system with multiple right-angle reflectors and a lead zirconate titanate-based compensation stage,” Rev. Sci. Instrum. 80(11), 115105 (2009).
[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(1), 761–765 (2006).
[Crossref]

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

2005 (1)

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

2003 (1)

D. J. Loner, B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Annals-Manufac, Technol. 52(1), 439–442 (2003).

2001 (1)

S. T. Lin, “A laser interferometer for measuring straightness,” Opt. Laser Technol. 33(3), 195–199 (2001).
[Crossref]

2000 (2)

K. C. Fan and M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

K. C. Fan and Y. Zhao, “A laser straightness measurement system using optical fiber and modulation techniques,” Int. J. Mach. Tools Manuf. 40(14), 2073–2081 (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]

1992 (1)

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” J. Eng. Ind. 114(3), 362–369 (1992).

1983 (1)

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

Bin, Z.

Borisov, O.

O. Borisov, S. Fletcher, A. Longstaff, and A. Myers, “Performance evaluation of a new taut wire system or straightness measurement of machine tools,” Precis. Eng. 38(3), 492–498 (2014).
[Crossref]

Büchner, H.-J.

I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
[Crossref]

Chaves-Jacob, J.

J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. 38(3), 578–588 (2014).
[Crossref]

Chen, B.

B. Chen, E. Zhang, L. Yan, C. Li, W. Tang, and Q. Feng, “A laser interferometer for measuring straightness and its position based on heterodyne interferometry,” Rev. Sci. Instrum. 80(11), 115113 (2009).
[Crossref] [PubMed]

Chen, J. H.

C. H. Liu, J. H. Chen, and Y. F. Teng, “Development of a straightness measurement and compensation system with multiple right-angle reflectors and a lead zirconate titanate-based compensation stage,” Rev. Sci. Instrum. 80(11), 115105 (2009).
[Crossref] [PubMed]

Chen, M. J.

K. C. Fan and M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

Chen, Q. H.

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

Cheng, C. H.

C. H. Liu and C. H. Cheng, “Development of a grating based multi-degree-of-freedom laser linear encoder using diffracted light,” Sen. Actuators A. 181(7), 87–93 (2012).
[Crossref]

Chiu, C. S.

Cosijns, S. J. A. G.

D. J. Loner, B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Annals-Manufac, Technol. 52(1), 439–442 (2003).

Cuifang, K.

Cunxing, C.

Fan, K. C.

K. C. Fan and M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

K. C. Fan and Y. Zhao, “A laser straightness measurement system using optical fiber and modulation techniques,” Int. J. Mach. Tools Manuf. 40(14), 2073–2081 (2000).
[Crossref]

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(1), 761–765 (2006).
[Crossref]

Feng, Q.

B. Chen, E. Zhang, L. Yan, C. Li, W. Tang, and Q. Feng, “A laser interferometer for measuring straightness and its position based on heterodyne interferometry,” Rev. Sci. Instrum. 80(11), 115113 (2009).
[Crossref] [PubMed]

Feng, Q. B.

Y. S. Zhai, Q. B. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44(4), 839–843 (2012).
[Crossref]

Fenglin, Y.

Fletcher, S.

J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. 38(3), 578–588 (2014).
[Crossref]

O. Borisov, S. Fletcher, A. Longstaff, and A. Myers, “Performance evaluation of a new taut wire system or straightness measurement of machine tools,” Precis. Eng. 38(3), 492–498 (2014).
[Crossref]

Flore, J.

J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. 38(3), 578–588 (2014).
[Crossref]

Gao, W.

A. Kimura, W. Gao, and L. J. 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(5), 054005 (2010).
[Crossref]

Haitjema, H.

D. J. Loner, B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Annals-Manufac, Technol. 52(1), 439–442 (2003).

Hou, W. M.

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

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(1), 761–765 (2006).
[Crossref]

Hsu, 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(1), 761–765 (2006).
[Crossref]

Huang, M. S.

Huang, P. S.

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” J. Eng. Ind. 114(3), 362–369 (1992).

Jäger, G.

I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
[Crossref]

Jywe, W. Y.

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(1), 761–765 (2006).
[Crossref]

Kimura, A.

A. Kimura, W. Gao, and L. J. 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(5), 054005 (2010).
[Crossref]

Knarren, B. A. W. H.

D. J. Loner, B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Annals-Manufac, Technol. 52(1), 439–442 (2003).

Li, C.

B. Chen, E. Zhang, L. Yan, C. Li, W. Tang, and Q. Feng, “A laser interferometer for measuring straightness and its position based on heterodyne interferometry,” Rev. Sci. Instrum. 80(11), 115113 (2009).
[Crossref] [PubMed]

Li, L.

L. J. Zhu, L. Li, J. H. Liu, and Z. H. Zhang, “A method for measuring the guideway straightness error based on polarized interference principle,” Int. J. Mach. Tools Manuf. 49(3–4), 285–290 (2009).
[Crossref]

Lin, D. J.

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

Lin, S. T.

Linares, J. M.

J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. 38(3), 578–588 (2014).
[Crossref]

Liu, C. H.

C. H. Liu and C. H. Cheng, “Development of a grating based multi-degree-of-freedom laser linear encoder using diffracted light,” Sen. Actuators A. 181(7), 87–93 (2012).
[Crossref]

C. H. Liu, J. H. Chen, and Y. F. Teng, “Development of a straightness measurement and compensation system with multiple right-angle reflectors and a lead zirconate titanate-based compensation stage,” Rev. Sci. Instrum. 80(11), 115105 (2009).
[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(1), 761–765 (2006).
[Crossref]

Liu, J. H.

L. J. Zhu, L. Li, J. H. Liu, and Z. H. Zhang, “A method for measuring the guideway straightness error based on polarized interference principle,” Int. J. Mach. Tools Manuf. 49(3–4), 285–290 (2009).
[Crossref]

Loner, D. J.

D. J. Loner, B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Annals-Manufac, Technol. 52(1), 439–442 (2003).

Longstaff, A.

O. Borisov, S. Fletcher, A. Longstaff, and A. Myers, “Performance evaluation of a new taut wire system or straightness measurement of machine tools,” Precis. Eng. 38(3), 492–498 (2014).
[Crossref]

J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. 38(3), 578–588 (2014).
[Crossref]

Myers, A.

O. Borisov, S. Fletcher, A. Longstaff, and A. Myers, “Performance evaluation of a new taut wire system or straightness measurement of machine tools,” Precis. Eng. 38(3), 492–498 (2014).
[Crossref]

Ni, J.

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” J. Eng. Ind. 114(3), 362–369 (1992).

Osawa, S.

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage,” Precis. Eng. 35(3), 424–430 (2011).
[Crossref]

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precis. Eng. 35(4), 686–692 (2011).
[Crossref]

Qibo, F.

Quenelle, R. C.

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

Rahneberg, I.

I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
[Crossref]

Sato, O.

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage,” Precis. Eng. 35(3), 424–430 (2011).
[Crossref]

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precis. Eng. 35(4), 686–692 (2011).
[Crossref]

Schallakans, P. H. J.

D. J. Loner, B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Annals-Manufac, Technol. 52(1), 439–442 (2003).

Schwenke, H.

J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. 38(3), 578–588 (2014).
[Crossref]

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(1), 761–765 (2006).
[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(1), 761–765 (2006).
[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(1), 761–765 (2006).
[Crossref]

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]

Takahashi, S.

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage,” Precis. Eng. 35(3), 424–430 (2011).
[Crossref]

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precis. Eng. 35(4), 686–692 (2011).
[Crossref]

Takamasu, K.

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precis. Eng. 35(4), 686–692 (2011).
[Crossref]

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage,” Precis. Eng. 35(3), 424–430 (2011).
[Crossref]

Takamura, T.

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage,” Precis. Eng. 35(3), 424–430 (2011).
[Crossref]

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precis. Eng. 35(4), 686–692 (2011).
[Crossref]

Takatsuji, T.

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precis. Eng. 35(4), 686–692 (2011).
[Crossref]

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage,” Precis. Eng. 35(3), 424–430 (2011).
[Crossref]

Tang, W.

B. Chen, E. Zhang, L. Yan, C. Li, W. Tang, and Q. Feng, “A laser interferometer for measuring straightness and its position based on heterodyne interferometry,” Rev. Sci. Instrum. 80(11), 115113 (2009).
[Crossref] [PubMed]

Teng, Y. F.

C. H. Liu, J. H. Chen, and Y. F. Teng, “Development of a straightness measurement and compensation system with multiple right-angle reflectors and a lead zirconate titanate-based compensation stage,” Rev. Sci. Instrum. 80(11), 115105 (2009).
[Crossref] [PubMed]

Uhlmann, E.

J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. 38(3), 578–588 (2014).
[Crossref]

Wintering, J.

J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. 38(3), 578–588 (2014).
[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]

Wu, J.

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

Wu, S. M.

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” J. Eng. Ind. 114(3), 362–369 (1992).

Yan, J. Q.

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

Yan, L.

B. Chen, E. Zhang, L. Yan, C. Li, W. Tang, and Q. Feng, “A laser interferometer for measuring straightness and its position based on heterodyne interferometry,” Rev. Sci. Instrum. 80(11), 115113 (2009).
[Crossref] [PubMed]

Yang, P.

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precis. Eng. 35(4), 686–692 (2011).
[Crossref]

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage,” Precis. Eng. 35(3), 424–430 (2011).
[Crossref]

Yeh, S. L.

Yin, C. Y.

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

Yusheng, Z.

Zeng, L. J.

A. Kimura, W. Gao, and L. J. 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(5), 054005 (2010).
[Crossref]

Zhai, Y. S.

Y. S. Zhai, Q. B. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44(4), 839–843 (2012).
[Crossref]

Zhang, B.

Y. S. Zhai, Q. B. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44(4), 839–843 (2012).
[Crossref]

Zhang, E.

B. Chen, E. Zhang, L. Yan, C. Li, W. Tang, and Q. Feng, “A laser interferometer for measuring straightness and its position based on heterodyne interferometry,” Rev. Sci. Instrum. 80(11), 115113 (2009).
[Crossref] [PubMed]

Zhang, Z. H.

L. J. Zhu, L. Li, J. H. Liu, and Z. H. Zhang, “A method for measuring the guideway straightness error based on polarized interference principle,” Int. J. Mach. Tools Manuf. 49(3–4), 285–290 (2009).
[Crossref]

Zhao, Y.

K. C. Fan and Y. Zhao, “A laser straightness measurement system using optical fiber and modulation techniques,” Int. J. Mach. Tools Manuf. 40(14), 2073–2081 (2000).
[Crossref]

Zhu, L. J.

L. J. Zhu, L. Li, J. H. Liu, and Z. H. Zhang, “A method for measuring the guideway straightness error based on polarized interference principle,” Int. J. Mach. Tools Manuf. 49(3–4), 285–290 (2009).
[Crossref]

CIRP Annals-Manufac, Technol. (1)

D. J. Loner, B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Annals-Manufac, Technol. 52(1), 439–442 (2003).

Hewlett Pachard J. (1)

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

Int. J. Mach. Tools Manuf. (2)

K. C. Fan and Y. Zhao, “A laser straightness measurement system using optical fiber and modulation techniques,” Int. J. Mach. Tools Manuf. 40(14), 2073–2081 (2000).
[Crossref]

L. J. Zhu, L. Li, J. H. Liu, and Z. H. Zhang, “A method for measuring the guideway straightness error based on polarized interference principle,” Int. J. Mach. Tools Manuf. 49(3–4), 285–290 (2009).
[Crossref]

J. Eng. Ind. (1)

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” J. Eng. Ind. 114(3), 362–369 (1992).

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(1), 761–765 (2006).
[Crossref]

Meas. Sci. Technol. (3)

A. Kimura, W. Gao, and L. J. 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(5), 054005 (2010).
[Crossref]

Q. H. Chen, D. J. Lin, J. Wu, J. Q. Yan, and C. Y. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[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 (2)

Opt. Laser Technol. (2)

S. T. Lin, “A laser interferometer for measuring straightness,” Opt. Laser Technol. 33(3), 195–199 (2001).
[Crossref]

Y. S. Zhai, Q. B. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44(4), 839–843 (2012).
[Crossref]

Precis. Eng. (6)

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precis. Eng. 35(4), 686–692 (2011).
[Crossref]

K. C. Fan and M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage,” Precis. Eng. 35(3), 424–430 (2011).
[Crossref]

J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. 38(3), 578–588 (2014).
[Crossref]

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

O. Borisov, S. Fletcher, A. Longstaff, and A. Myers, “Performance evaluation of a new taut wire system or straightness measurement of machine tools,” Precis. Eng. 38(3), 492–498 (2014).
[Crossref]

Proc. SPIE (1)

I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
[Crossref]

Rev. Sci. Instrum. (2)

C. H. Liu, J. H. Chen, and Y. F. Teng, “Development of a straightness measurement and compensation system with multiple right-angle reflectors and a lead zirconate titanate-based compensation stage,” Rev. Sci. Instrum. 80(11), 115105 (2009).
[Crossref] [PubMed]

B. Chen, E. Zhang, L. Yan, C. Li, W. Tang, and Q. Feng, “A laser interferometer for measuring straightness and its position based on heterodyne interferometry,” Rev. Sci. Instrum. 80(11), 115113 (2009).
[Crossref] [PubMed]

Sen. Actuators A. (1)

C. H. Liu and C. H. Cheng, “Development of a grating based multi-degree-of-freedom laser linear encoder using diffracted light,” Sen. Actuators A. 181(7), 87–93 (2012).
[Crossref]

Other (3)

J. J. Kroll, “Six degree of freedom optical sensor for dynamic measurement of linear axes,” PhD thesis, University of North Carolina, Charlotte (2003).

R. R. Baldwin, “Interferometer system for measuring straightness and roll,” U.S. Patent, 3790284 (1974).

D. R. McMurtry and R. J. Chaney, “Straightness interferometer system,” U.S. Patent, 5026163 (1991).

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Figures (18)

Fig. 1
Fig. 1

Schematic of the laser straightness interferometer system with simultaneous measurement of six degrees of freedom error parameters.

Fig. 2
Fig. 2

Schematic of six degrees of freedom motion errors of a moving stage.

Fig. 3
Fig. 3

The coordinate definition of the analytical model of the proposed system.

Fig. 4
Fig. 4

Schematic of the incident (emergent) points on the upper and down right-angle prisms without rotational error.

Fig. 5
Fig. 5

The coordinate definition of the analytical model of the upper and down right-angle prisms.

Fig. 6
Fig. 6

The influence of the horizontal straightness error on L1 and L2.

Fig. 7
Fig. 7

The projection of the three vectors onto the plane o u x u y u .

Fig. 8
Fig. 8

Schematic of the influence of the pitch error on the optical paths L1 and L2 outside RR.

Fig. 9
Fig. 9

The experimental setup.

Fig. 10
Fig. 10

Experimental results of measuring yaw and pitch errors.

Fig. 11
Fig. 11

Experimental result of measuring roll error.

Fig. 12
Fig. 12

Experimental result of measuring horizontal straightness error.

Fig. 13
Fig. 13

Experimental result of measuring vertical straightness error without compensation.

Fig. 14
Fig. 14

Experimental result of measuring vertical straightness error with compensation.

Fig. 15
Fig. 15

Experimental result of measuring straightness error’s position.

Fig. 16
Fig. 16

Simultaneous measurement and repeatability experimental results.

Fig. 17
Fig. 17

Schematic of the polarization state of the emergent beams from RR.

Fig. 18
Fig. 18

Simulation of the influence of the polarization state change on the measurement signal.

Tables (2)

Tables Icon

Table 1 Repeatability results of simultaneously measuring six degrees of freedom error parameters

Tables Icon

Table 2 Uncertainty results of six degrees of freedom error parameters

Equations (41)

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

{ α= Δ x PSD 2f β= Δ y PSD 2f
P ui :{ x ui = H 1 cosθ+H y ui = D 2 w z ui = H 1 sinθB
P di :{ x di = H 2 cosθ+H y di = D 2 w z di = H 2 sinθB
P uo :{ x uo = H 1 cosθ+H y uo = D 2 +w z uo = H 1 sinθB
P do :{ x do = H 2 cosθ+H y do = D 2 +w z do = H 2 sinθB
T U M =[ 2 2 (θ+γβ) 2 2 (θγβ) 1 HBβ 2 2 (1+α) 2 2 (1+α) γ Hγ+Bα 2 2 (1α) 2 2 (1+α) βθ HβB 0 0 0 1 ]=[ R U M P U M 0 1 ]
[ x ui M , y ui M , z ui M ,1] T = T U M [ x ui U , y ui U , z ui U ,1] T
{ x ui U = 2 2 [(H+ H 1 cosθ)γ+Bα+( D 2 +w)+ L 2 (1+α)] y ui U = 2 2 [(H+ H 1 cosθ)γ+Bα+( D 2 +w)+ L 2 (1+α)] z ui U = H 1 +Bβ( D 2 +w)γ
I ui = R M U I 0 = [ 2 2 (α1), 2 2 (α1), β] T
I 1 = [ 2 2 ( α n 1), 2 2 ( α n 1), β n ] T
M=[ 12 N x U 2 2 N x U N y U 2 N x U N z U 2 N x U N y U 12 N y U 2 2 N y U N z U 2 N x U N z U 2 N y U N z U 12 N z U 2 ]
I uo = [ 2 2 (α+1), 2 2 (α+1), β] T
{ x uo M =H+ H 1 cosθ(D+2w)γ β n L y uo M =2(H+ H 1 cosθ)γ+2Bα+( D 2 +w)+L(α α n ) z uo M =(H+ H 1 cosθ)β α n L H 1 θB+( D 2 +w)α
I 3 R = R U M I uo = [θ2β, 0, 1] T
{ x do M =H H 2 cosθ(D+2w)γ β n L y do M =2(H H 2 cosθ)γ+2Bα+( D 2 +w)+L(α α n ) z do M =(H H 2 cosθ)β α n L H 2 θB+( D 2 +w)α
I 7 R = R D M I do = [θ2β, 0, 1] T
URP: { Δ x uo M =(D+2w)γ β n L Δ y uo M =2(H+ H 1 cosθ)γ+2Bα+L(α α n ) Δ z uo M =(H+ H 1 cosθ)β α n L+( D 2 +w)α
DRP: { Δ x do M =(D+2w)γ β n L Δ y do M =2(H H 2 cosθ)γ+2Bα+L(α α n ) Δ z do M =(H H 2 cosθ)β α n L+( D 2 +w)α
URP: Δ x up =( s 0 +s+ z uo M )tan(θ+2β)( s 0 +s+ z uo )tanθ2( s 0 +sB)β
DRP: Δ x dn =( s 0 +s+ z do )tanθ( s 0 +s+ z do M )tan(θ2β)2( s 0 +sB)β
{ Δ x QD1 =Δ y uo M 2w=2(H+ H 1 cosθ)γ2BαL(α α n )2w Δ y QD1 =Δ x uo M Δ x up 2 s QW1 β=(D+2w)γ β n L2( s 0 +sB+ s QW1 )β
{ Δ x QD2 =Δ y do M +2w=2(H H 2 cosθ)γ+2Bα+L(α α n )+2w Δ y QD2 =Δ x do M Δ x dn 2 s QW2 β=(D+2w)γ β n L2( s 0 +sB+ s QW2 )β
γ= Δ x QD1 +Δ x QD2 2( H 1 + H 2 )cosθ = Δ x QD1 +Δ x QD2 2( s 0 +sB)sin2θ
w= Δ x QD2 Δ x QD1 4 + γ( H 2 H 1 )cosθ 2 HγBα L 2 (α α n ) Δ x QD2 Δ x QD1 4 HγBα L 2 (α α n )
{ α= Δ x PSD 2f β= Δ y PSD 2f γ= Δ x QD1 +Δ x QD2 2( s 0 +sB)sin2θ w= Δ x QD2 Δ x QD1 4 HγBα L 2 (α α n ) Δh= L 1 L 2 2sinθ s= L 1 + L 2 2cosθ
L in =nL
β'= β 1 = β 2 = β 3 =arcsin( β n )
L 1 in = Lin 1 +Lin 2 +Lin 3 cosβ' = nL cos β cos α
cos α = 1 ( α n ) 2
L 1 in = nL n 2 β 2 n 2 + α 2 nL
L 2 in nL
ΔL 1 out =( P D ¯ + C P 1 ¯ )2 PD ¯ =2(H+ H 1 )β α n L
ΔL 2 out =( P F ¯ + E P 1 ¯ )2 PF ¯ =2(H H 2 )β α n L
L 1 = 2 L 1 (ΔL 1 in +ΔL 1 out ) 2 = L 1 (H+ H 1 )β α 2n L
L 2 = 2 L 2 ' (ΔL 2 in ' +ΔL 2 out ' ) 2 = L 2 ' (H H 2 )β α 2n L
Δh= L 1 L 2 2sinθ = L 1 L 2 2sinθ ( s 0 +sB)β =Δ h ( s 0 +sB)β
s= L 1 + L 2 2cosθ = L 1 + L 2 2cosθ +ΔhβHβ α 2n L = s +ΔhβHβ α 2n L s ' Hβ α 2n L
{ δα= δΔ x PSD 2f δβ= δΔ y PSD 2f δγ= ( δΔ x QD1 2( s 0 +sB)sin2θ ) 2 + ( δΔ x QD2 2( s 0 +sB)sin2θ ) 2 + [ (Δ x QD2 +Δ x QD2 )δΔs 2 ( s 0 +sB) 2 sin2θ ] 2 δw= ( δΔ x QD1 4 ) 2 + ( δΔ x QD1 4 ) 2 + (Hδγ) 2 + [(B L 2 + 1 n )δα] 2 δΔh= ( δ L 1 2sinθ ) 2 + ( δ L 2 2sinθ ) 2 + [( s 0 +sB)δβ] 2 + (βδs) 2 δs= ( δ L 1 2cosθ ) 2 + ( δ L 2 2cosθ ) 2 + (Hδβ) 2 + ( L 2n δα) 2
E x = E f 2 cos(2π f 2 t+ ϕ 0 f 2 )
E y =sin ε 1 E x f 1 cos(2π f 1 t+ ϕ 1 )+cos ε 1 E y f 1 cos(2π f 1 t+ ϕ 1 + δ 1 ) +cos ε 2 E x f 2 cos(2π f 2 t+ ϕ 2 )+sin ε 2 E y f 2 cos(2π f 2 t+ ϕ 2 + δ 2 )
I= 1 2 [ cos ε 1 E f 2 E y f 1 cos(Δωt+ ϕ 1 - ϕ 0 f 2 + δ 1 )+sin ε 1 E f 2 E x f 1 cos(Δωt+ ϕ 1 - ϕ 0 f 2 ) +sin ε 1 cos ε 2 E x f 1 E x f 2 cos(Δωt+ ϕ 1 ϕ 2 ) +sin ε 1 sin ε 2 E x f 1 E y f 2 cos(Δωt+ ϕ 1 ϕ 2 δ 2 ) +cos ε 1 cos ε 2 E y f 1 E x f 2 cos(Δωt+ ϕ 1 ϕ 2 + δ 1 ) +cos ε 1 sin ε 2 E y f 1 E y f 2 cos(Δωt+ ϕ 1 ϕ 2 + δ 1 - δ 2 ) ]

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