Abstract

We propose to use thin films to provide a drastic improvement of measurement sensitivity in the recently developed small-angle measurement method, namely, angle measurement based on the internal-reflection effect. By designing the thin films (single layer or multiple layers) so that they provide an antireflection effect in the vicinity of the critical angle, we show that the sensitivity of angle measurement can be increased exponentially with the increase of the number of thin-film layers. This method provides a new means of designing angle sensors with increased sensitivities without having to increase the number of reflections and therefore the physical size and the required fabrication accuracy of the reflection prisms. We describe the design of the thin films for this particular application and the analysis of measurement sensitivity and range as determined by the material and the number of layers of the thin films. Selection of the optimal initial angle for high linearity performance is also discussed.

© 1999 Optical Society of America

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References

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  1. J. Rohlin, “An interferometer for precision angle measurement,” Appl. Opt. 2, 762–763 (1963).
    [CrossRef]
  2. D. Malacara, O. Harris, “Interferometric measurement of angles,” Appl. Opt. 9, 1630–1633 (1970).
    [CrossRef] [PubMed]
  3. G. D. Chapman, “Interferometric angular measurement,” Appl. Opt. 13, 1646–1651 (1974).
    [CrossRef] [PubMed]
  4. R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. 34, 3–13 (1983).
  5. P. Shi, E. Stijns, “New optical methods for measuring small-angle rotations,” Appl. Opt. 27, 4342–4344 (1988).
    [CrossRef] [PubMed]
  6. T. Takano, S. Yonehara, “Basic investigations on an angle measurement system using a laser,” IEEE Trans. Aerosp. Electron. Syst. 26, 657–662 (1990).
    [CrossRef]
  7. P. Shi, E. Stijns, “Improving the linearity of the Michelson interferometric angular measurement by a parameter compensation method,” Appl. Opt. 32, 44–51 (1993).
    [CrossRef] [PubMed]
  8. P. R. Yoder, E. R. Schlesinger, J. L. Chickvary, “Active annular-beam laser autocollimator system,” Appl. Opt. 14, 1890–1895 (1975).
    [CrossRef] [PubMed]
  9. L. D. Hutcheson, “Practical electro-optic deflection measurements system,” Opt. Eng. 15, 61–63 (1976).
    [CrossRef]
  10. A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Prec. Eng. 4, 5–8 (1982).
    [CrossRef]
  11. F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
    [CrossRef]
  12. G. G. Luther, R. D. Deslattes, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747–750 (1984).
    [CrossRef]
  13. W. Duis, J. Trede, G.-J. Ulbrich, M. Mross, “Design and performance of a high resolution, high accuracy automatic autocollimator,” in Precision Engineering and Optomechanics, D. Vukobratovich, ed., Proc. SPIE1167, 297–304 (1989).
    [CrossRef]
  14. P. S. Huang, S. Kiyono, O. Kamada, “Angle measurement based on the internal-reflection effect: a new method,” Appl. Opt. 31, 6047–6055 (1992).
    [CrossRef] [PubMed]
  15. P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect and the use of right-angle prisms,” Appl. Opt. 34, 4976–4981 (1995).
    [CrossRef] [PubMed]
  16. P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35, 2239–2241 (1996).
    [CrossRef] [PubMed]
  17. P. S. Huang, Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37, 6636–6642 (1998).
    [CrossRef]
  18. P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect and its application in measurement of geometric errors of machine tools,” in Proceedings of the Eighth Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1993), pp. 350–353.
  19. S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” Int. J. Jpn. Soc. Prec. Eng. 27, 373–378 (1993).
  20. P. S. Huang, X. Xu, “Optical probe for surface profiling of aspherical mirrors,” in Proceedings of the Tenth Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1995), pp. 112–115.
  21. P. S. Huang, Y. Li, “Laser measurement instrument for fast calibration of machine tools,” in Proceedings of the Eleventh Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1996), pp. 644–647.
  22. S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” Int. J. Jpn. Soc. Prec. Eng. 30, 349–350 (1996).
  23. M.-H. Chiu, D.-C. Su, “Angle measurement using total-internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750–1753 (1997).
    [CrossRef]
  24. M.-H. Chiu, D.-C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36, 7104–7106 (1997).
    [CrossRef]
  25. W. Zhou, L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37, 5957–5963 (1998).
    [CrossRef]
  26. W. Zhou, L. Cai, “An angular displacement interferometer based on total internal reflection,” Meas. Sci. Technol. 9, 1647–1652 (1998).
    [CrossRef]
  27. S. Zhang, S. Kiyono, Y. Uda, “Nanoradian angle sensor and in situ self-calibration,” Appl. Opt. 37, 4154–4159 (1998).
    [CrossRef]
  28. K. Kato, T. Musha, K. Ito, “Method and apparatus for detecting focussing error signal of objective lens,” U.S. Patent No.4,505,584, (19March1985).
  29. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991), Chap. 4, p. 46.

1998 (4)

1997 (2)

M.-H. Chiu, D.-C. Su, “Angle measurement using total-internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750–1753 (1997).
[CrossRef]

M.-H. Chiu, D.-C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36, 7104–7106 (1997).
[CrossRef]

1996 (2)

S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” Int. J. Jpn. Soc. Prec. Eng. 30, 349–350 (1996).

P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35, 2239–2241 (1996).
[CrossRef] [PubMed]

1995 (1)

1993 (2)

S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” Int. J. Jpn. Soc. Prec. Eng. 27, 373–378 (1993).

P. Shi, E. Stijns, “Improving the linearity of the Michelson interferometric angular measurement by a parameter compensation method,” Appl. Opt. 32, 44–51 (1993).
[CrossRef] [PubMed]

1992 (1)

1990 (1)

T. Takano, S. Yonehara, “Basic investigations on an angle measurement system using a laser,” IEEE Trans. Aerosp. Electron. Syst. 26, 657–662 (1990).
[CrossRef]

1988 (1)

1984 (1)

G. G. Luther, R. D. Deslattes, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747–750 (1984).
[CrossRef]

1983 (2)

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. 34, 3–13 (1983).

1982 (1)

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Prec. Eng. 4, 5–8 (1982).
[CrossRef]

1976 (1)

L. D. Hutcheson, “Practical electro-optic deflection measurements system,” Opt. Eng. 15, 61–63 (1976).
[CrossRef]

1975 (1)

1974 (1)

1970 (1)

1963 (1)

Cai, L.

W. Zhou, L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37, 5957–5963 (1998).
[CrossRef]

W. Zhou, L. Cai, “An angular displacement interferometer based on total internal reflection,” Meas. Sci. Technol. 9, 1647–1652 (1998).
[CrossRef]

Chapman, G. D.

Chickvary, J. L.

Chiu, M.-H.

M.-H. Chiu, D.-C. Su, “Angle measurement using total-internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750–1753 (1997).
[CrossRef]

M.-H. Chiu, D.-C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36, 7104–7106 (1997).
[CrossRef]

Deslattes, R. D.

G. G. Luther, R. D. Deslattes, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747–750 (1984).
[CrossRef]

Duis, W.

W. Duis, J. Trede, G.-J. Ulbrich, M. Mross, “Design and performance of a high resolution, high accuracy automatic autocollimator,” in Precision Engineering and Optomechanics, D. Vukobratovich, ed., Proc. SPIE1167, 297–304 (1989).
[CrossRef]

Ennos, A. E.

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Prec. Eng. 4, 5–8 (1982).
[CrossRef]

Harris, O.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991), Chap. 4, p. 46.

Huang, P. S.

P. S. Huang, Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37, 6636–6642 (1998).
[CrossRef]

P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35, 2239–2241 (1996).
[CrossRef] [PubMed]

P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect and the use of right-angle prisms,” Appl. Opt. 34, 4976–4981 (1995).
[CrossRef] [PubMed]

P. S. Huang, S. Kiyono, O. Kamada, “Angle measurement based on the internal-reflection effect: a new method,” Appl. Opt. 31, 6047–6055 (1992).
[CrossRef] [PubMed]

P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect and its application in measurement of geometric errors of machine tools,” in Proceedings of the Eighth Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1993), pp. 350–353.

P. S. Huang, Y. Li, “Laser measurement instrument for fast calibration of machine tools,” in Proceedings of the Eleventh Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1996), pp. 644–647.

P. S. Huang, X. Xu, “Optical probe for surface profiling of aspherical mirrors,” in Proceedings of the Tenth Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1995), pp. 112–115.

Hutcheson, L. D.

L. D. Hutcheson, “Practical electro-optic deflection measurements system,” Opt. Eng. 15, 61–63 (1976).
[CrossRef]

Ito, K.

K. Kato, T. Musha, K. Ito, “Method and apparatus for detecting focussing error signal of objective lens,” U.S. Patent No.4,505,584, (19March1985).

Kamada, O.

Kato, K.

K. Kato, T. Musha, K. Ito, “Method and apparatus for detecting focussing error signal of objective lens,” U.S. Patent No.4,505,584, (19March1985).

Kiyono, S.

S. Zhang, S. Kiyono, Y. Uda, “Nanoradian angle sensor and in situ self-calibration,” Appl. Opt. 37, 4154–4159 (1998).
[CrossRef]

S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” Int. J. Jpn. Soc. Prec. Eng. 30, 349–350 (1996).

S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” Int. J. Jpn. Soc. Prec. Eng. 27, 373–378 (1993).

P. S. Huang, S. Kiyono, O. Kamada, “Angle measurement based on the internal-reflection effect: a new method,” Appl. Opt. 31, 6047–6055 (1992).
[CrossRef] [PubMed]

Li, Y.

P. S. Huang, Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37, 6636–6642 (1998).
[CrossRef]

P. S. Huang, Y. Li, “Laser measurement instrument for fast calibration of machine tools,” in Proceedings of the Eleventh Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1996), pp. 644–647.

Luther, G. G.

G. G. Luther, R. D. Deslattes, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747–750 (1984).
[CrossRef]

Malacara, D.

Mito, M.

S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” Int. J. Jpn. Soc. Prec. Eng. 30, 349–350 (1996).

Mross, M.

W. Duis, J. Trede, G.-J. Ulbrich, M. Mross, “Design and performance of a high resolution, high accuracy automatic autocollimator,” in Precision Engineering and Optomechanics, D. Vukobratovich, ed., Proc. SPIE1167, 297–304 (1989).
[CrossRef]

Musha, T.

K. Kato, T. Musha, K. Ito, “Method and apparatus for detecting focussing error signal of objective lens,” U.S. Patent No.4,505,584, (19March1985).

Ni, J.

P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35, 2239–2241 (1996).
[CrossRef] [PubMed]

P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect and the use of right-angle prisms,” Appl. Opt. 34, 4976–4981 (1995).
[CrossRef] [PubMed]

P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect and its application in measurement of geometric errors of machine tools,” in Proceedings of the Eighth Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1993), pp. 350–353.

Quenelle, R. C.

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. 34, 3–13 (1983).

Rohlin, J.

Sato, H.

S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” Int. J. Jpn. Soc. Prec. Eng. 27, 373–378 (1993).

Schlesinger, E. R.

Schuda, F. J.

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

Shan, X.

S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” Int. J. Jpn. Soc. Prec. Eng. 27, 373–378 (1993).

Shi, P.

Stijns, E.

Su, D.-C.

M.-H. Chiu, D.-C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36, 7104–7106 (1997).
[CrossRef]

M.-H. Chiu, D.-C. Su, “Angle measurement using total-internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750–1753 (1997).
[CrossRef]

Takano, T.

T. Takano, S. Yonehara, “Basic investigations on an angle measurement system using a laser,” IEEE Trans. Aerosp. Electron. Syst. 26, 657–662 (1990).
[CrossRef]

Trede, J.

W. Duis, J. Trede, G.-J. Ulbrich, M. Mross, “Design and performance of a high resolution, high accuracy automatic autocollimator,” in Precision Engineering and Optomechanics, D. Vukobratovich, ed., Proc. SPIE1167, 297–304 (1989).
[CrossRef]

Uda, Y.

S. Zhang, S. Kiyono, Y. Uda, “Nanoradian angle sensor and in situ self-calibration,” Appl. Opt. 37, 4154–4159 (1998).
[CrossRef]

S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” Int. J. Jpn. Soc. Prec. Eng. 30, 349–350 (1996).

Ulbrich, G.-J.

W. Duis, J. Trede, G.-J. Ulbrich, M. Mross, “Design and performance of a high resolution, high accuracy automatic autocollimator,” in Precision Engineering and Optomechanics, D. Vukobratovich, ed., Proc. SPIE1167, 297–304 (1989).
[CrossRef]

Virdee, M. S.

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Prec. Eng. 4, 5–8 (1982).
[CrossRef]

Wuerz, L. J.

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. 34, 3–13 (1983).

Xu, X.

P. S. Huang, X. Xu, “Optical probe for surface profiling of aspherical mirrors,” in Proceedings of the Tenth Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1995), pp. 112–115.

Yoder, P. R.

Yonehara, S.

T. Takano, S. Yonehara, “Basic investigations on an angle measurement system using a laser,” IEEE Trans. Aerosp. Electron. Syst. 26, 657–662 (1990).
[CrossRef]

Zhang, S.

S. Zhang, S. Kiyono, Y. Uda, “Nanoradian angle sensor and in situ self-calibration,” Appl. Opt. 37, 4154–4159 (1998).
[CrossRef]

S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” Int. J. Jpn. Soc. Prec. Eng. 30, 349–350 (1996).

Zhou, W.

W. Zhou, L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37, 5957–5963 (1998).
[CrossRef]

W. Zhou, L. Cai, “An angular displacement interferometer based on total internal reflection,” Meas. Sci. Technol. 9, 1647–1652 (1998).
[CrossRef]

Appl. Opt. (13)

D. Malacara, O. Harris, “Interferometric measurement of angles,” Appl. Opt. 9, 1630–1633 (1970).
[CrossRef] [PubMed]

G. D. Chapman, “Interferometric angular measurement,” Appl. Opt. 13, 1646–1651 (1974).
[CrossRef] [PubMed]

P. R. Yoder, E. R. Schlesinger, J. L. Chickvary, “Active annular-beam laser autocollimator system,” Appl. Opt. 14, 1890–1895 (1975).
[CrossRef] [PubMed]

P. Shi, E. Stijns, “New optical methods for measuring small-angle rotations,” Appl. Opt. 27, 4342–4344 (1988).
[CrossRef] [PubMed]

P. Shi, E. Stijns, “Improving the linearity of the Michelson interferometric angular measurement by a parameter compensation method,” Appl. Opt. 32, 44–51 (1993).
[CrossRef] [PubMed]

M.-H. Chiu, D.-C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36, 7104–7106 (1997).
[CrossRef]

S. Zhang, S. Kiyono, Y. Uda, “Nanoradian angle sensor and in situ self-calibration,” Appl. Opt. 37, 4154–4159 (1998).
[CrossRef]

W. Zhou, L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37, 5957–5963 (1998).
[CrossRef]

P. S. Huang, Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37, 6636–6642 (1998).
[CrossRef]

P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect and the use of right-angle prisms,” Appl. Opt. 34, 4976–4981 (1995).
[CrossRef] [PubMed]

P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35, 2239–2241 (1996).
[CrossRef] [PubMed]

P. S. Huang, S. Kiyono, O. Kamada, “Angle measurement based on the internal-reflection effect: a new method,” Appl. Opt. 31, 6047–6055 (1992).
[CrossRef] [PubMed]

J. Rohlin, “An interferometer for precision angle measurement,” Appl. Opt. 2, 762–763 (1963).
[CrossRef]

Hewlett-Packard J. (1)

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. 34, 3–13 (1983).

IEEE Trans. Aerosp. Electron. Syst. (1)

T. Takano, S. Yonehara, “Basic investigations on an angle measurement system using a laser,” IEEE Trans. Aerosp. Electron. Syst. 26, 657–662 (1990).
[CrossRef]

Int. J. Jpn. Soc. Prec. Eng. (2)

S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” Int. J. Jpn. Soc. Prec. Eng. 27, 373–378 (1993).

S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” Int. J. Jpn. Soc. Prec. Eng. 30, 349–350 (1996).

Meas. Sci. Technol. (1)

W. Zhou, L. Cai, “An angular displacement interferometer based on total internal reflection,” Meas. Sci. Technol. 9, 1647–1652 (1998).
[CrossRef]

Opt. Eng. (2)

M.-H. Chiu, D.-C. Su, “Angle measurement using total-internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750–1753 (1997).
[CrossRef]

L. D. Hutcheson, “Practical electro-optic deflection measurements system,” Opt. Eng. 15, 61–63 (1976).
[CrossRef]

Prec. Eng. (1)

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Prec. Eng. 4, 5–8 (1982).
[CrossRef]

Rev. Sci. Instrum. (2)

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

G. G. Luther, R. D. Deslattes, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747–750 (1984).
[CrossRef]

Other (6)

W. Duis, J. Trede, G.-J. Ulbrich, M. Mross, “Design and performance of a high resolution, high accuracy automatic autocollimator,” in Precision Engineering and Optomechanics, D. Vukobratovich, ed., Proc. SPIE1167, 297–304 (1989).
[CrossRef]

P. S. Huang, X. Xu, “Optical probe for surface profiling of aspherical mirrors,” in Proceedings of the Tenth Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1995), pp. 112–115.

P. S. Huang, Y. Li, “Laser measurement instrument for fast calibration of machine tools,” in Proceedings of the Eleventh Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1996), pp. 644–647.

P. S. Huang, J. Ni, “Angle measurement based on the internal-reflection effect and its application in measurement of geometric errors of machine tools,” in Proceedings of the Eighth Annual Meeting of the American Society for Precision Engineering (American Society of Precision Engineering, Raleigh, N.C., 1993), pp. 350–353.

K. Kato, T. Musha, K. Ito, “Method and apparatus for detecting focussing error signal of objective lens,” U.S. Patent No.4,505,584, (19March1985).

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991), Chap. 4, p. 46.

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Figures (9)

Fig. 1
Fig. 1

Light reflection at a prism coated with multiple layers of thin films.

Fig. 2
Fig. 2

Reflectance curves of s-polarized light at a prism coated with as many as three layers of thin films. Here k is the number of thin-film layers

Fig. 3
Fig. 3

Reflectance curves of p-polarized light at a prism coated with as many as three layers of thin films. Here k is the number of thin-film layers.

Fig. 4
Fig. 4

Optical layout of the differential method with thin films.

Fig. 5
Fig. 5

Nonlinearity error versus the angle of incidence for p-polarized light. Here k is the number of thin-film layers. The substrate material is SF 11 glass (n 0 = 1.78), the high-refractive-index thin-film material is TiO2 (n h = 2.59), and the low-refractive-index thin-film material is MgF2 (n l = 1.38).

Fig. 6
Fig. 6

Linearized reflectance curves of p-polarized light at a prism with different numbers of thin-film layers.

Fig. 7
Fig. 7

Optical layout of the single-prism method with thin films.

Fig. 8
Fig. 8

Nonlinearity error versus the angle of incidence for p-polarized light. Here k is the number of thin-film layers. The substrate material is SF 11 glass (n 0 = 1.78), the high-refractive-index thin-film material is TiO2 (n h = 2.59), and the low-refractive-index thin-film material is MgF2 (n l = 1.38).

Fig. 9
Fig. 9

Linearized reflectance curves of p-polarized light at a prism with different numbers of thin-film layers. Here k is the number of thin-film layers.

Tables (2)

Tables Icon

Table 1 Optimized Parameters for the Differential Method

Tables Icon

Table 2 Optimized Parameters for the Single-Prism Method

Equations (10)

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

ρk expiΔk=rk+rk+1 exp-2iδk1+rkrk+1 exp-2iδk.
δk=2πλ nkdk cos θk,
ρk=rk2+rk+12+2rkrk+1 cos 2δk1+rk2rk+12+2rkrk+1 cos 2δk1/2,
Δk=ηk-ξk,
ξk=tan-1rk+1 sin 2δkrk+rk+1 cos 2δk,
ξk=tan-1rkrk+1 sin 2δk1+rkrk+1 cos 2δk.
ρk-1 expiΔk-1=rk-1+ρk expiΔkexp-2iδk-11+rk-1ρk expiΔkexp-2iδk-1,
R=ρ12.
Rl=RΔθ-R-ΔθRΔθ+R-Δθ=I1-I2I1+I2,
S=1Rθ0+Δθ1Rθ0 Δθ.

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