Abstract

An autonomous method for calibrating the reference flat surface of an interferometer is proposed with the uncertainty analysis. The method consists of three phases; the first step is multiple rotating shifts of a specimen, the second is a linear shift, and the last is multiple rotating shifts again. The profile of the reference flat surface is basically determined by the linear shift. The linear shift errors that occurred during the linear shift are identified by the rotating shifts. The rotating shift errors caused by the rotating shifts can be compensated and the residual uncertainty can be reduced in proportion to the square root of the number of rotating shifts per one revolution. Finally, the uncertainty analysis is carried out in detail.

© 2012 Optical Society of America

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References

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  1. F. T. Farago and M. A. Curtis, Handbook of Dimensional Measurement (Industrial, 1994).
  2. R. Thalmann, “Straightness and alignment,” in Handbook of Optical Metrology, T. Yoshizawa, ed. (CRC, 2009), Sec. 17, pp. 411–421.
  3. D. Malacara, Handbook of Optical Shop Testing (Wiley, 1992).
  4. W. T. Estler, “Calibration and use of optical straightedges in the metrology of precision machines,” Opt. Eng. 24, 372–379 (1985).
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    [CrossRef]
  6. Kuroda Precision Industries, “Measuring system: Nanometoro 300TT,” http://www.kuroda-precision.co.jp/products/sokutei/nano/sokutei08.htm .
  7. W. Gao, J. Yokoyama, and K. Kiyono, “Straightness measurement of cylinder by multi-probe method,” Precis. Eng. 28, 279–289 (2002).
    [CrossRef]
  8. J. Yamaguchi, “Measurement of straight motion accuracy using the improved sequential three-point method,” J. Jpn. Soc. Precis. Eng. 59, 773–778 (1993) [in Japanese].
    [CrossRef]
  9. T. Kume, K. Enami, Y. Higashi, and K. Ueno, “Zero difference in straightness using 3-point method,” J. Jpn. Soc. Precis. Eng. 75, 657–662 (2009) [in Japanese].
    [CrossRef]
  10. I. Fujimoto, K. Nishimura, and Y. Pyun, “Autonomous calibration method of the zero-difference without using a standard gauge for a straightness-measuring machine,” Precis. Eng. 35, 153–163 (2011).
    [CrossRef]
  11. I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyoun, “A study on the relationship between measurement uncertainty and the size of the disc gauge used to calibrate a straightness measuring system,” Meas. Sci. Technol. 22, 125201 (2011).
    [CrossRef]
  12. I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyun, “A Technique to Measure the Flatness of next-generation 450 mm wafers using a three-point method with an autonomous calibration function,” Precis. Eng. 36, 270–280 (2012).
    [CrossRef]
  13. T. Takatsuji, N. Ueki, K. Hibino, S. Osawa, and T. Kurosawa, “Japanese ultimate flatness interferometer and its preliminary experiment,” Proc. SPIE 4401, 83–90 (2001).
    [CrossRef]
  14. U. Griesmann, Q. Wang, M. Tricard, P. Dumas, and C. Hill, “Manufacture and metrology of 300 mm silicon wafers with ultra-low thickness variations,” in Proceedings of Characterization and Metrology for Nanoelectronics International Conference on Frontiers of Characterization and Metrology, p. 931 (2007).
  15. B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, and P. S. Fairman, “Calibration of a 300 mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39, 5161–5171 (2000).
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    [CrossRef]
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  21. K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40, 1637–1648 (2001).
    [CrossRef]
  22. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856–5865 (2006).
    [CrossRef]
  23. U. Griesmann, Q. Wang, and J. Soons, “A comparison of three-flat tests,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper OFMC3.
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    [CrossRef]
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    [CrossRef]
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  29. I. Fujimoto and T. Lee, “Self-calibration algorithm of systematic errors for interferometer,” J. Kor. Soc. Precis. Eng. 22(5), 63–71 (2005) [in Korean].
  30. International Organization for Standardization, “Guide to the expression of uncertainty in measurement” (ISO, 1995).

2012 (1)

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyun, “A Technique to Measure the Flatness of next-generation 450 mm wafers using a three-point method with an autonomous calibration function,” Precis. Eng. 36, 270–280 (2012).
[CrossRef]

2011 (2)

I. Fujimoto, K. Nishimura, and Y. Pyun, “Autonomous calibration method of the zero-difference without using a standard gauge for a straightness-measuring machine,” Precis. Eng. 35, 153–163 (2011).
[CrossRef]

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyoun, “A study on the relationship between measurement uncertainty and the size of the disc gauge used to calibrate a straightness measuring system,” Meas. Sci. Technol. 22, 125201 (2011).
[CrossRef]

2009 (1)

T. Kume, K. Enami, Y. Higashi, and K. Ueno, “Zero difference in straightness using 3-point method,” J. Jpn. Soc. Precis. Eng. 75, 657–662 (2009) [in Japanese].
[CrossRef]

2007 (1)

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601 (2007).
[CrossRef]

2006 (1)

2005 (1)

I. Fujimoto and T. Lee, “Self-calibration algorithm of systematic errors for interferometer,” J. Kor. Soc. Precis. Eng. 22(5), 63–71 (2005) [in Korean].

2003 (1)

2002 (1)

W. Gao, J. Yokoyama, and K. Kiyono, “Straightness measurement of cylinder by multi-probe method,” Precis. Eng. 28, 279–289 (2002).
[CrossRef]

2001 (3)

T. Takatsuji, N. Ueki, K. Hibino, S. Osawa, and T. Kurosawa, “Japanese ultimate flatness interferometer and its preliminary experiment,” Proc. SPIE 4401, 83–90 (2001).
[CrossRef]

M. F. Küchel, “A new approach to solve the three flat problem,” Optik 112, 381–391 (2001).
[CrossRef]

K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40, 1637–1648 (2001).
[CrossRef]

2000 (1)

1998 (1)

S. Kiyono, W. Gao, and H. Seio, “Theoretical study on absolute measurement method of surface shape by interferograms,” J. Jpn. Soc. Precis. Eng. 64, 1137–1145 (1998) [in Japanese].

1997 (1)

R. Mercier, M. Lamare, P. Picart, and J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from the best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

1996 (1)

1993 (2)

C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993).
[CrossRef]

J. Yamaguchi, “Measurement of straight motion accuracy using the improved sequential three-point method,” J. Jpn. Soc. Precis. Eng. 59, 773–778 (1993) [in Japanese].
[CrossRef]

1992 (1)

T. Ito, T. Hinaji, and O. Horiuchi, “High precision flatness measurement by combing two-orientation method and radial shift method,” J. Jpn. Soc. Precis. Eng. 58, 883–886 (1992) [in Japanese].

1989 (1)

D. Moore, “Design considerations in multi-probe roundness measurement,” J Phys: E. Sci. Instrum. 22, 339–343 (1989).
[CrossRef]

1985 (1)

W. T. Estler, “Calibration and use of optical straightedges in the metrology of precision machines,” Opt. Eng. 24, 372–379 (1985).

1984 (1)

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).

1967 (1)

Ai, C.

Curtis, M. A.

F. T. Farago and M. A. Curtis, Handbook of Dimensional Measurement (Industrial, 1994).

Dumas, P.

U. Griesmann, Q. Wang, M. Tricard, P. Dumas, and C. Hill, “Manufacture and metrology of 300 mm silicon wafers with ultra-low thickness variations,” in Proceedings of Characterization and Metrology for Nanoelectronics International Conference on Frontiers of Characterization and Metrology, p. 931 (2007).

Enami, K.

T. Kume, K. Enami, Y. Higashi, and K. Ueno, “Zero difference in straightness using 3-point method,” J. Jpn. Soc. Precis. Eng. 75, 657–662 (2009) [in Japanese].
[CrossRef]

Estler, W. T.

W. T. Estler, “Calibration and use of optical straightedges in the metrology of precision machines,” Opt. Eng. 24, 372–379 (1985).

Evans, C. J.

Fairman, P. S.

Farago, F. T.

F. T. Farago and M. A. Curtis, Handbook of Dimensional Measurement (Industrial, 1994).

Farrant, D. I.

Forbes, G.

Freischlad, K. R.

Fritz, B. S.

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).

Fujimoto, I.

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyun, “A Technique to Measure the Flatness of next-generation 450 mm wafers using a three-point method with an autonomous calibration function,” Precis. Eng. 36, 270–280 (2012).
[CrossRef]

I. Fujimoto, K. Nishimura, and Y. Pyun, “Autonomous calibration method of the zero-difference without using a standard gauge for a straightness-measuring machine,” Precis. Eng. 35, 153–163 (2011).
[CrossRef]

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyoun, “A study on the relationship between measurement uncertainty and the size of the disc gauge used to calibrate a straightness measuring system,” Meas. Sci. Technol. 22, 125201 (2011).
[CrossRef]

I. Fujimoto and T. Lee, “Self-calibration algorithm of systematic errors for interferometer,” J. Kor. Soc. Precis. Eng. 22(5), 63–71 (2005) [in Korean].

Gao, W.

W. Gao, J. Yokoyama, and K. Kiyono, “Straightness measurement of cylinder by multi-probe method,” Precis. Eng. 28, 279–289 (2002).
[CrossRef]

S. Kiyono, W. Gao, and H. Seio, “Theoretical study on absolute measurement method of surface shape by interferograms,” J. Jpn. Soc. Precis. Eng. 64, 1137–1145 (1998) [in Japanese].

Griesmann, U.

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601 (2007).
[CrossRef]

U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856–5865 (2006).
[CrossRef]

U. Griesmann, Q. Wang, M. Tricard, P. Dumas, and C. Hill, “Manufacture and metrology of 300 mm silicon wafers with ultra-low thickness variations,” in Proceedings of Characterization and Metrology for Nanoelectronics International Conference on Frontiers of Characterization and Metrology, p. 931 (2007).

U. Griesmann, Q. Wang, and J. Soons, “A comparison of three-flat tests,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper OFMC3.

Hibino, K.

T. Takatsuji, N. Ueki, K. Hibino, S. Osawa, and T. Kurosawa, “Japanese ultimate flatness interferometer and its preliminary experiment,” Proc. SPIE 4401, 83–90 (2001).
[CrossRef]

Higashi, Y.

T. Kume, K. Enami, Y. Higashi, and K. Ueno, “Zero difference in straightness using 3-point method,” J. Jpn. Soc. Precis. Eng. 75, 657–662 (2009) [in Japanese].
[CrossRef]

Hill, C.

U. Griesmann, Q. Wang, M. Tricard, P. Dumas, and C. Hill, “Manufacture and metrology of 300 mm silicon wafers with ultra-low thickness variations,” in Proceedings of Characterization and Metrology for Nanoelectronics International Conference on Frontiers of Characterization and Metrology, p. 931 (2007).

Hinaji, T.

T. Ito, T. Hinaji, and O. Horiuchi, “High precision flatness measurement by combing two-orientation method and radial shift method,” J. Jpn. Soc. Precis. Eng. 58, 883–886 (1992) [in Japanese].

Horiuchi, O.

T. Ito, T. Hinaji, and O. Horiuchi, “High precision flatness measurement by combing two-orientation method and radial shift method,” J. Jpn. Soc. Precis. Eng. 58, 883–886 (1992) [in Japanese].

Ito, T.

T. Ito, T. Hinaji, and O. Horiuchi, “High precision flatness measurement by combing two-orientation method and radial shift method,” J. Jpn. Soc. Precis. Eng. 58, 883–886 (1992) [in Japanese].

Iwasaki, Y.

Iwata, K.

Kestner, R. N.

Kiyono, K.

W. Gao, J. Yokoyama, and K. Kiyono, “Straightness measurement of cylinder by multi-probe method,” Precis. Eng. 28, 279–289 (2002).
[CrossRef]

Kiyono, S.

S. Kiyono, W. Gao, and H. Seio, “Theoretical study on absolute measurement method of surface shape by interferograms,” J. Jpn. Soc. Precis. Eng. 64, 1137–1145 (1998) [in Japanese].

Küchel, M. F.

M. F. Küchel, “A new approach to solve the three flat problem,” Optik 112, 381–391 (2001).
[CrossRef]

Kume, T.

T. Kume, K. Enami, Y. Higashi, and K. Ueno, “Zero difference in straightness using 3-point method,” J. Jpn. Soc. Precis. Eng. 75, 657–662 (2009) [in Japanese].
[CrossRef]

Kurosawa, T.

T. Takatsuji, N. Ueki, K. Hibino, S. Osawa, and T. Kurosawa, “Japanese ultimate flatness interferometer and its preliminary experiment,” Proc. SPIE 4401, 83–90 (2001).
[CrossRef]

Lamare, M.

R. Mercier, M. Lamare, P. Picart, and J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from the best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

Lee, T.

I. Fujimoto and T. Lee, “Self-calibration algorithm of systematic errors for interferometer,” J. Kor. Soc. Precis. Eng. 22(5), 63–71 (2005) [in Korean].

Malacara, D.

D. Malacara, Handbook of Optical Shop Testing (Wiley, 1992).

Marioge, J. P.

R. Mercier, M. Lamare, P. Picart, and J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from the best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

Mercier, R.

R. Mercier, M. Lamare, P. Picart, and J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from the best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

Moore, D.

D. Moore, “Design considerations in multi-probe roundness measurement,” J Phys: E. Sci. Instrum. 22, 339–343 (1989).
[CrossRef]

Nishimura, K.

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyun, “A Technique to Measure the Flatness of next-generation 450 mm wafers using a three-point method with an autonomous calibration function,” Precis. Eng. 36, 270–280 (2012).
[CrossRef]

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyoun, “A study on the relationship between measurement uncertainty and the size of the disc gauge used to calibrate a straightness measuring system,” Meas. Sci. Technol. 22, 125201 (2011).
[CrossRef]

I. Fujimoto, K. Nishimura, and Y. Pyun, “Autonomous calibration method of the zero-difference without using a standard gauge for a straightness-measuring machine,” Precis. Eng. 35, 153–163 (2011).
[CrossRef]

Oreb, B. F.

Osawa, S.

T. Takatsuji, N. Ueki, K. Hibino, S. Osawa, and T. Kurosawa, “Japanese ultimate flatness interferometer and its preliminary experiment,” Proc. SPIE 4401, 83–90 (2001).
[CrossRef]

Picart, P.

R. Mercier, M. Lamare, P. Picart, and J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from the best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

Pyoun, Y.

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyoun, “A study on the relationship between measurement uncertainty and the size of the disc gauge used to calibrate a straightness measuring system,” Meas. Sci. Technol. 22, 125201 (2011).
[CrossRef]

Pyun, Y.

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyun, “A Technique to Measure the Flatness of next-generation 450 mm wafers using a three-point method with an autonomous calibration function,” Precis. Eng. 36, 270–280 (2012).
[CrossRef]

I. Fujimoto, K. Nishimura, and Y. Pyun, “Autonomous calibration method of the zero-difference without using a standard gauge for a straightness-measuring machine,” Precis. Eng. 35, 153–163 (2011).
[CrossRef]

Schulz, G.

Schwider, J.

Seio, H.

S. Kiyono, W. Gao, and H. Seio, “Theoretical study on absolute measurement method of surface shape by interferograms,” J. Jpn. Soc. Precis. Eng. 64, 1137–1145 (1998) [in Japanese].

Sonosaki, A.

Soons, J.

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601 (2007).
[CrossRef]

U. Griesmann, Q. Wang, and J. Soons, “A comparison of three-flat tests,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper OFMC3.

Takatsuji, T.

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyun, “A Technique to Measure the Flatness of next-generation 450 mm wafers using a three-point method with an autonomous calibration function,” Precis. Eng. 36, 270–280 (2012).
[CrossRef]

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyoun, “A study on the relationship between measurement uncertainty and the size of the disc gauge used to calibrate a straightness measuring system,” Meas. Sci. Technol. 22, 125201 (2011).
[CrossRef]

T. Takatsuji, N. Ueki, K. Hibino, S. Osawa, and T. Kurosawa, “Japanese ultimate flatness interferometer and its preliminary experiment,” Proc. SPIE 4401, 83–90 (2001).
[CrossRef]

Thalmann, R.

R. Thalmann, “Straightness and alignment,” in Handbook of Optical Metrology, T. Yoshizawa, ed. (CRC, 2009), Sec. 17, pp. 411–421.

Tricard, M.

U. Griesmann, Q. Wang, M. Tricard, P. Dumas, and C. Hill, “Manufacture and metrology of 300 mm silicon wafers with ultra-low thickness variations,” in Proceedings of Characterization and Metrology for Nanoelectronics International Conference on Frontiers of Characterization and Metrology, p. 931 (2007).

Ueki, N.

T. Takatsuji, N. Ueki, K. Hibino, S. Osawa, and T. Kurosawa, “Japanese ultimate flatness interferometer and its preliminary experiment,” Proc. SPIE 4401, 83–90 (2001).
[CrossRef]

Ueno, K.

T. Kume, K. Enami, Y. Higashi, and K. Ueno, “Zero difference in straightness using 3-point method,” J. Jpn. Soc. Precis. Eng. 75, 657–662 (2009) [in Japanese].
[CrossRef]

Walsh, C. J.

Wang, Q.

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601 (2007).
[CrossRef]

U. Griesmann, Q. Wang, M. Tricard, P. Dumas, and C. Hill, “Manufacture and metrology of 300 mm silicon wafers with ultra-low thickness variations,” in Proceedings of Characterization and Metrology for Nanoelectronics International Conference on Frontiers of Characterization and Metrology, p. 931 (2007).

U. Griesmann, Q. Wang, and J. Soons, “A comparison of three-flat tests,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper OFMC3.

Wyant, J. C.

Yamaguchi, J.

J. Yamaguchi, “Measurement of straight motion accuracy using the improved sequential three-point method,” J. Jpn. Soc. Precis. Eng. 59, 773–778 (1993) [in Japanese].
[CrossRef]

Yokoyama, J.

W. Gao, J. Yokoyama, and K. Kiyono, “Straightness measurement of cylinder by multi-probe method,” Precis. Eng. 28, 279–289 (2002).
[CrossRef]

Appl. Opt. (7)

J Phys: E. Sci. Instrum. (1)

D. Moore, “Design considerations in multi-probe roundness measurement,” J Phys: E. Sci. Instrum. 22, 339–343 (1989).
[CrossRef]

J. Jpn. Soc. Precis. Eng. (4)

J. Yamaguchi, “Measurement of straight motion accuracy using the improved sequential three-point method,” J. Jpn. Soc. Precis. Eng. 59, 773–778 (1993) [in Japanese].
[CrossRef]

T. Kume, K. Enami, Y. Higashi, and K. Ueno, “Zero difference in straightness using 3-point method,” J. Jpn. Soc. Precis. Eng. 75, 657–662 (2009) [in Japanese].
[CrossRef]

S. Kiyono, W. Gao, and H. Seio, “Theoretical study on absolute measurement method of surface shape by interferograms,” J. Jpn. Soc. Precis. Eng. 64, 1137–1145 (1998) [in Japanese].

T. Ito, T. Hinaji, and O. Horiuchi, “High precision flatness measurement by combing two-orientation method and radial shift method,” J. Jpn. Soc. Precis. Eng. 58, 883–886 (1992) [in Japanese].

J. Kor. Soc. Precis. Eng. (1)

I. Fujimoto and T. Lee, “Self-calibration algorithm of systematic errors for interferometer,” J. Kor. Soc. Precis. Eng. 22(5), 63–71 (2005) [in Korean].

Meas. Sci. Technol. (1)

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyoun, “A study on the relationship between measurement uncertainty and the size of the disc gauge used to calibrate a straightness measuring system,” Meas. Sci. Technol. 22, 125201 (2011).
[CrossRef]

Opt. Eng. (3)

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).

W. T. Estler, “Calibration and use of optical straightedges in the metrology of precision machines,” Opt. Eng. 24, 372–379 (1985).

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601 (2007).
[CrossRef]

Optik (1)

M. F. Küchel, “A new approach to solve the three flat problem,” Optik 112, 381–391 (2001).
[CrossRef]

Precis. Eng. (3)

I. Fujimoto, K. Nishimura, T. Takatsuji, and Y. Pyun, “A Technique to Measure the Flatness of next-generation 450 mm wafers using a three-point method with an autonomous calibration function,” Precis. Eng. 36, 270–280 (2012).
[CrossRef]

W. Gao, J. Yokoyama, and K. Kiyono, “Straightness measurement of cylinder by multi-probe method,” Precis. Eng. 28, 279–289 (2002).
[CrossRef]

I. Fujimoto, K. Nishimura, and Y. Pyun, “Autonomous calibration method of the zero-difference without using a standard gauge for a straightness-measuring machine,” Precis. Eng. 35, 153–163 (2011).
[CrossRef]

Proc. SPIE (1)

T. Takatsuji, N. Ueki, K. Hibino, S. Osawa, and T. Kurosawa, “Japanese ultimate flatness interferometer and its preliminary experiment,” Proc. SPIE 4401, 83–90 (2001).
[CrossRef]

Pure Appl. Opt. (1)

R. Mercier, M. Lamare, P. Picart, and J. P. Marioge, “Two-flat method for bi-dimensional measurement of absolute departure from the best sphere,” Pure Appl. Opt. 6, 117–126 (1997).
[CrossRef]

Other (7)

U. Griesmann, Q. Wang, and J. Soons, “A comparison of three-flat tests,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper OFMC3.

U. Griesmann, Q. Wang, M. Tricard, P. Dumas, and C. Hill, “Manufacture and metrology of 300 mm silicon wafers with ultra-low thickness variations,” in Proceedings of Characterization and Metrology for Nanoelectronics International Conference on Frontiers of Characterization and Metrology, p. 931 (2007).

Kuroda Precision Industries, “Measuring system: Nanometoro 300TT,” http://www.kuroda-precision.co.jp/products/sokutei/nano/sokutei08.htm .

F. T. Farago and M. A. Curtis, Handbook of Dimensional Measurement (Industrial, 1994).

R. Thalmann, “Straightness and alignment,” in Handbook of Optical Metrology, T. Yoshizawa, ed. (CRC, 2009), Sec. 17, pp. 411–421.

D. Malacara, Handbook of Optical Shop Testing (Wiley, 1992).

International Organization for Standardization, “Guide to the expression of uncertainty in measurement” (ISO, 1995).

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Figures (7)

Fig. 1.
Fig. 1.

Section forms which express the specimen.

Fig. 2.
Fig. 2.

Errors caused by rotating shift and linear shift of the specimen: (a) rotating shift errors and (b) linear shift errors.

Fig. 3.
Fig. 3.

Rotating shift of the specimen before and after its linear shift.

Fig. 4.
Fig. 4.

System configuration for calibrating the reference flat surface of interferometer.

Fig. 5.
Fig. 5.

Definition of each measuring data.

Fig. 6.
Fig. 6.

Relationship between the standard uncertainty of the measured profile of the specimen and the uncertainty of the repeatability of the tilt errors when the measuring areas are 60, 120, 180, 240, and 300mmφ, and the number of the rotating shifts per one revolution nφ is 36.

Fig. 7.
Fig. 7.

Relationship between the expanded uncertainty U of the calibrated reference flat surface and the number of the rotating shifts per one revolution nφ when the measuring area is 300mmφ, and the uncertainties of the repeatability of the tilt errors are the uniform distributions of ±1, ±2, ±3, ±4, and ±5μrad.

Equations (95)

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z(x,y)=k=1nak(y)xk+a0(y),
a0(y)=k=0nbkyk.
z¯(x,y)=k=2nak(y)xk+(a1(y)a1(0))x+k=2nbkyk.
I(ϕ,x,y)=η(ϕ)x+ς(ϕ)y+τ(ϕ),
Iα(ϕ,x,y)=ηα(ϕ)x+ςα(ϕ)y+τα(ϕ).
η(0)=ς(0)=0,τ(0)=0
ξ(α,x,y)=p(α)x+r(α)y+g(α),
p(0)=r(0)=0,g(0)=0
02πη(ϕ)dϕ=02πηα(ϕ)dϕ,
02πς(ϕ)dϕ=02πςα(ϕ)dϕ,
η(ϕ)=η(ϕ)E[η(ϕ)]=η(ϕ)η¯(ϕ),
ηα(ϕ)=ηα(ϕ)E[ηα(ϕ)]ηα(ϕ)η¯(ϕ)p(α),
ς(ϕ)=ς(ϕ)E[ς(ϕ)]=ς(ϕ)ς¯(ϕ),
ςα(ϕ)=ςα(ϕ)E[ςα(ϕ)]ςα(ϕ)ς¯(ϕ)r(α).
z˜(0,x,y)=z(x,y)+ε(x,y)
z˜(0,l,θ)=z(l,θ)+ε(l,θ).
z˜(ϕ,l,θ)=z(l,θϕ)+ε(l,θ)+η(ϕ)lcos(θ)+ς(ϕ)lsin(θ)+τ(ϕ).
z˜(α,x,y)=z(xα,y)+ε(x,y)+p(α)x+r(α)y+g(α),
z˜α(0,l,θ)=zα(l,θ)+εα(l,θ)+p(α)lcos(θ)+r(α)lsin(θ)+g(α).
z˜α(ϕ,l,θ)=zα(l,θϕ)+εα(l,θ)+p(α)lcos(θϕ)+r(α)lsin(θϕ)+g(α)+ηα(ϕ)lcos(θ)+ςα(ϕ)lsin(θ)+τα(ϕ).
z˜(ϕ,l,θ)z˜(0,l,θ)=z(l,θϕ)z(l,θ)+η(ϕ)lcos(θ)+ς(ϕ)lsin(θ)+τ(ϕ).
zϕ(l,θ)=z(l,θϕ)
z˜(ϕ)(l,θ)=z˜(ϕ,l,θ).
(f,g)=1πl02πf(l,θ)g(l,θ)dθ.
(z˜(ϕ)z˜(0),cos)=(zϕ,cos)(z,cos)+η(ϕ)l(cos,cos)+ς(ϕ)l(sin,cos)+τ(ϕ)(1,cos).
(z˜(ϕ)z˜(0),cos)=(zϕ,cos)(z,cos)+η(ϕ).
(z˜(ϕ)z˜(0),sin)=(zϕ,sin)(z,sin)+ς(ϕ).
z˜α(ϕ,l,θ)z˜α(0,l,θ)=zα(l,θϕ)zα(l,θ)+p(α)l(cos(θϕ)cos(θ))+r(α)l(sin(θϕ)sin(θ))+ηα(ϕ)lcos(θ)+ςα(ϕ)lsin(θ)+τα(ϕ).
(zα(ϕ)zα(0),cos)=(zαϕ,cos)(zα,cos)+p(α)(cosϕ1)r(α)sinϕ+ηα(ϕ),
(z˜α(ϕ)z˜α(0),sin)=(zαϕ,sin)(zα,sin)+p(α)sinϕ+r(α)(cosϕ1)+ςα(ϕ).
(z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),cos)=p(α)(cosϕ1)r(α)sinϕ+(ηα(ϕ)η(ϕ)).
(z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),sin)=p(α)sinϕ+r(α)(cosϕ1)+(ςα(ϕ)ς(ϕ)).
(h,1)ϕ=12π02πh(l,ϕ)dϕ.
p(α)=1(12π)((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),cos),1)ϕ+1(12π)((ηαη),1)ϕ,
r(α)=1(12π)((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),sin),1)ϕ+1(12π)((ςας),1)ϕ,
z˜(α,x,y)z˜(0,x,y)=z(xα,y)z(x,y)+p(α)(xα)+r(α)y+g(α)
z˜(α,x,y)z˜(0,x,y)=k=3nak(y)((xα)kxk)+c(α,y)x+d(α,y),
c(α,y)=2αa2(y)+p(α),
d(α,y)=α2a2(y)αa1(y)αp(α)+r(α)y+g(α).
M(α,y)=A(α)u(α,y),
A(α)=((x1α)nx1n(x1α)n1x1n1(x1α)3x13(xkα)nxkn(xkα)n1xkn1(xkα)3xk3(xmα)nxmn(xmα)n1xmn1(xmα)3xm3x1xkxm111),
M(α,y)=(z˜(α,x1,y)z˜(0,x1,y)z˜(α,xk,y)z˜(0,xk,y)z˜(α,xm1,y)z˜(0,xm1,y)z˜(α,xm,y)z˜(α,xm,y)),
u(α,y)=(an(y),,a3(y),c(α,y),d(α,y))T.
u(α,y)=(A(α,y)TA(α,y))1A(α,y)TM(α,y).
z˜(π/2,l,θ)z˜(0,l,θ)=k=2n((ak(0)bk))yk+(a1(0)b1+ς(π/2))y+τ(π/2),
hk=ak(0)bk(k=2,3,,n)
a2(y)=p(α)c(α,y)2α.
b2=a2(0)h2.
a1(y)x=(r(α)y+α2(a2(y)a2(0))(d(α,y)d(α,0))+a1(0))αx.
a1(y)x+b1y=(r(α)y+α2(a2(y)a2(0))(d(α,y)d(α,0)))αx+(a1(0)x+b1y).
z¯(x,y)=k=2nak(y)xk+k=2nbkyk+(r(α)y+α2(a2(y)a2(0))(d(α,y)d(α,0)))αx.
z˜(0,x,y)=z¯(x,y)+ε¯(x,y)+(ax+by)
a=k=1n[(s=1nys2)xk+(s=1nxsys)yk]z(0,xk,yk)k=1nxk2k=1nyk2k=1nxkyk,
b=k=1n[(s=1nxsys)xk+(k=1nxk2)yk]z(0,xk,yk)k=1nxk2k=1nyk2k=1nxkyk.
ε¯(x,y)=z˜(0,x,y)z¯(x,y)(ax+by).
δε¯(x,y)=(δtilt(x,y)2+δacc(x,y)2+δax+by2+3δmes2)1/2,
δtilt(x,y)=12(2π1)α(δ((ηαη),1)ϕ2(x2+y2)2+4δ((ςας),1)ϕ2x2y2)1/2,
δacc(x,y)=2α{x2mx+y2my+1(2π1)2nϕnθ2l2[(x2+y2)2q=1nθcos(θq)2+4y2x2q=1nθsin(θq)2]}1/2δmes,
δax+by={k=1n[(s=1nys2)xk+(s=1nxsys)yk]2x2+k=1n[(s=1nxsys)xk+(k=1nxk2)yk]2y2}1/2(k=1nxk2k=1nyk2k=1nxkyk)δmes.
δz˜α(ϕ,l,θ)=δz˜α(0,l,θ)=δz˜(ϕ,l,θ)=δz˜(0,l,θ)=δmes.
δ((ηαη),1)ϕ=δ((ςας),1)ϕ=δηαςαηςϕ.
δtilt(x,y)=[12(2π1)α((x2+y2)2+4x2y2)1/2]δηαςαηςϕ.
δtilt(r)=maxx2+y2rδtilt(x,y)=r2(2π1)αδηαςαηςϕ.
δacc(x,y)=2{1m+1(2π1)2nϕnθ2[q=1nθcos(θq)2+sin22θq=1nθsin(θq)2]}1/2(rα)δmes,
δacc(r)=maxx2+y2rδacc(x,y)=2{1m+2π2(2π1)2nϕnθ2}1/2(rα)δmes,
δacc(r)=2{1m+2π2(2π1)2nϕnθ2}1/2(rα)δmes,
(xk,yk)=(((ks1)sr,0)(1k2s+1)(0,(k3s2)sr)(2s+2k4s+2)((k5s3)sr,(k5s3)sr)(4s+3k6s+3)((k7s4)sr,(k7s4)sr)(6s+4k8s+4),
δax+by2=r2k=1nxk2δmes2,
δax+by=[1+s2(s+1)(2s+1)2]1/2δmes.
δ((ηαη),1)ϕ=δ((ςας),1)ϕ=(tan(lμrad)3)1nϕ=tan(1.0μrad)63.
δtilt(r)=r2(2π1)αδηαςαηςϕ=13.5nm.
δacc(r)=2{1m+2π2(2π1)2nϕnθ2}1/2(rα)δmes=0.07nm.
δax+by=(s2(s+1)(2s+1)2)1/2δmes=0.112nm.
U=2[δtilt(r)2+δacc(r)2+δax+by2+3δmes2]1/2(k=2).
U2[δtilt(r)2+3δmes2]1/2=2[r4(12π)2α2tan2(lμrad)3nϕ+3δmes2]1/2nϕ23δmes(k=2)
z(x,y)xz˜(α,x,y)z˜(0,x,y)αp(α)x+r(α)yα+αp(α)g(α)α
z(x,y)=0x[z˜(α,s,y)z˜(0,s,y)α]dsp(α)x2/2+r(α)yxααp(α)+g(α)αx+z(0,y)=xmxαk=1mx[z˜(α,xk,y)z˜(0,xk,y)]+12(12π)((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),cos),1)ϕx2α+1(12π)((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),sin),1)ϕyxα12(12π)((ηαη),1)ϕx2α1(12π)((ςας),1)ϕyxα+αp(α)g(α)αx+z(0,y).
z(0,y)=z(y,0)+(z˜(0,l,π/2)z˜(π/2,l,π/2))+ς(π/2)y+τ(π/2)=ymyαk=1my[z˜(α,yk,0)z˜(0,yk,0)]+12(12π)((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),cos),1)ϕy2α12(12π)((ηαη),1)ϕy2α+αp(α)g(α)αy+(z˜(0,l,π/2)z˜(π/2,l,π/2))+ς(π/2)y+τ(π/2),
z˜(0,l,π/2)=z˜(0,0,y),z˜(π/2,l,π/2)=z˜(π/2,0,y).
z¯(x,y)=xmxαk=1mx[z˜(α,xk,y)z˜(0,xk,y)]+ymyαk=1my[z˜(α,yk,0)z˜(0,yk,0)]+12(12π)((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),cos),1)ϕx2+y2α+1(12π)((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),sin),1)ϕyxα+(z˜(0,0,y)z˜(π/2,0,y))12(12π)((ηαη),1)ϕx2+y2α1(12π)((ςας),1)ϕyxα.
δtilt(x,y)=12(2π1)α(δ((ηαη),1)ϕ2(x2+y2)2+4δ((ςας),1)ϕ2x2y2)1/2.
δk=1mx[z˜(α,xk,y)z˜(0,xk,y)]=2mxδmes,δk=1my[z˜(α,yk,0)z˜(0,yk,0)]=2myδmes.
((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),w),1)ϕ=12π2l0ϕ2πdϕ0θ2π(z˜α(ϕ,l,θ)z˜α(0,l,θ)z˜(ϕ,l,θ)+z˜(0,l,θ))w(θ)dθ,
δ((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),w),1)ϕ={E[((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),w),1)ϕE[((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),w),1)ϕ]]2}1/22nϕ1/2nθl(q=1nθw(θq)2)1/2δmes.
12(12π)((z˜α(ϕ)z˜α(0)z˜(ϕ)+z˜(0),cos),1)ϕx2+y2α+1(12π)((z˜α(φ)z˜α(0)z˜(ϕ)+z˜(0),sin),1)ϕyxα
1(2π1)nϕ1/2nθlα[(x2+y2)2q=1nθcos(θq)2+4y2x2q=1nθsin(θq)2]1/2δmes.
δacc(x,y)=2α{x2mx+y2my+1(2π1)2nϕnθ2l2[(x2+y2)2q=1nθcos(θq)2+4y2x2q=1nθsin(θq)2]}1/2δmes.
δz¯(x,y)=(δtilt(x,y)2+δacc(x,y)2+2δmes2)1/2.
ε¯(x,y)=(z˜(0,x,y)z¯(x,y))(ax+by).
δε¯(x,y)=(δmes2+δz¯(x,y)2+δax+by2)1/2.
δax+by=(x2δa2+y2δb2)1/2.
δa={k=1n[(s=1nys2)xk+(s=1nxsys)yk]2}1/2(k=1nxk2k=1nyk2k=1nxkyk)δmes,
δb={k=1n[(s=1nxsys)xk+(k=1nxk2)yk]2}1/2(k=1nxk2k=1nyk2k=1nxkyk)δmes.
(δmes2+δax+by2)1/2.
δε¯(x,y)=(δtilt(x,y)2+δacc(x,y)2+δax+by2+3δmes2)1/2.

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