Abstract

A method for precision small-angle measurement is proposed. This method is based on the total-internal-reflection effect of a light beam at a pair of glass prisms. Angular displacement of the light beam is measured when the intensity change of the reflected beam is detected as a result of the relative phase shift between the s- and the p-polarized beams. An initial phase shift between the s- and the p-polarized components is introduced to increase measurement sensitivity. For increased measurement linearity and reduced effect of laser power fluctuation on the output, a differential method is used in which the light beam is split equally into two beams, each reflected at a prism and detected by a photodiode. The output is obtained as the difference of the two detected intensities divided by their sum. A prototype device was built, which demonstrated a nonlinearity error of 1.3% in a measurement range of ±0.6° or 0.4% in ±0.3°. The peak-to-peak noise level was found to be at approximately 0.5 arc sec. This noise level can be reduced further and resolution increased by a reduction of the measurement range.

© 2001 Optical Society of America

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References

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  1. 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]
  2. 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]
  3. 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]
  4. P. S. Huang, Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37, 6636–6642 (1998).
    [CrossRef]
  5. P. S. Huang, “Use of thin films for high-sensitivity angle measurement,” Appl. Opt. 38, 4831–4836 (1999).
    [CrossRef]
  6. 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 American Society for Precision Engineering Annual Meeting (American Society for Precision Engineering, Raleigh, N.C., 1993), pp. 350–353.
  7. P. S. Huang, Y. Li, “Laser measurement instrument for fast calibration of machine tools,” in Proceedings: American Society for Precision Engineering Annual Meeting (American Society for Precision Engineering, Raleigh, N.C., 1996), pp. 112–115.
  8. P. S. Huang, X. Xu, “Design of an optical probe for surface profile measurement,” Opt. Eng. 38, 1223–1228 (1999).
    [CrossRef]
  9. 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. Precis. Eng. 30, 349–350 (1996).
  10. S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” Int. J. Jpn. Soc. Precis. Eng. 27, 373–378 (1993).
    [CrossRef]
  11. S. Zhang, S. Kiyono, Y. Uda, “Nanoradian angle sensor and in situ self-calibration,” Appl. Opt. 37, 4154–4159 (1998).
    [CrossRef]
  12. M.-H. Chiu, D.-C. Su, “Angle measurement using total-internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750–1753 (1997).
    [CrossRef]
  13. M.-H. Chiu, D.-C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36, 7104–7106 (1997).
    [CrossRef]
  14. W. Zhou, L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37, 5957–5963 (1998).
    [CrossRef]
  15. W. Zhou, L. Cai, “An angular displacement interferometer based on total internal reflection,” Meas. Sci. Technol. 9, 1647–1652 (1998).
    [CrossRef]
  16. W. Zhou, L. Cai, “Improved angle interferometer based on total internal reflection,” Appl. Opt. 38, 1179–1185 (1999).
    [CrossRef]
  17. M. P. Kothiyal, C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–4442 (1985).
    [CrossRef] [PubMed]

1999 (3)

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)

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]

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. Precis. Eng. 30, 349–350 (1996).

1995 (1)

1993 (1)

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

1992 (1)

1985 (1)

Cai, 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]

Delisle, C.

Huang, P. S.

P. S. Huang, “Use of thin films for high-sensitivity angle measurement,” Appl. Opt. 38, 4831–4836 (1999).
[CrossRef]

P. S. Huang, X. Xu, “Design of an optical probe for surface profile measurement,” Opt. Eng. 38, 1223–1228 (1999).
[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 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 American Society for Precision Engineering Annual Meeting (American Society for 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: American Society for Precision Engineering Annual Meeting (American Society for Precision Engineering, Raleigh, N.C., 1996), pp. 112–115.

Kamada, O.

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. Precis. Eng. 30, 349–350 (1996).

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

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]

Kothiyal, M. P.

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: American Society for Precision Engineering Annual Meeting (American Society for Precision Engineering, Raleigh, N.C., 1996), pp. 112–115.

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. Precis. Eng. 30, 349–350 (1996).

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 American Society for Precision Engineering Annual Meeting (American Society for Precision Engineering, Raleigh, N.C., 1993), pp. 350–353.

Sato, H.

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

Shan, X.

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

Su, D.-C.

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]

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. Precis. Eng. 30, 349–350 (1996).

Xu, X.

P. S. Huang, X. Xu, “Design of an optical probe for surface profile measurement,” Opt. Eng. 38, 1223–1228 (1999).
[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. Precis. Eng. 30, 349–350 (1996).

Zhou, W.

Appl. Opt. (10)

Int. J. Jpn. Soc. Precis. Eng. (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. Precis. Eng. 30, 349–350 (1996).

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

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)

P. S. Huang, X. Xu, “Design of an optical probe for surface profile measurement,” Opt. Eng. 38, 1223–1228 (1999).
[CrossRef]

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

Other (2)

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 American Society for Precision Engineering Annual Meeting (American Society for 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: American Society for Precision Engineering Annual Meeting (American Society for Precision Engineering, Raleigh, N.C., 1996), pp. 112–115.

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Figures (10)

Fig. 1
Fig. 1

Light propagation through a right-angle prism.

Fig. 2
Fig. 2

Relative phase shift between the s- and the p-polarized components.

Fig. 3
Fig. 3

Principle of the total-internal-reflection method for small-angle measurement.

Fig. 4
Fig. 4

Output versus angular displacement at differential initial angles of incidence.

Fig. 5
Fig. 5

Measurement sensitivity and nonlinearity error versus initial phase shift when θ0 = -4.2°.

Fig. 6
Fig. 6

Optimized initial phase shift and maximum nonlinearity error versus measurement range.

Fig. 7
Fig. 7

Comparison of measurement sensitivity with and without the initial phase shift.

Fig. 8
Fig. 8

Experimental setup for the calibration of the prototype device.

Fig. 9
Fig. 9

Experimental result of calibration.

Fig. 10
Fig. 10

Change of the output curves that is due to the difference in the initial angles.

Tables (1)

Tables Icon

Table 1 Optimized Initial Phase Shift, Measurement Range, Sensitivity, and Nonlinearity Error for Different Values of the Initial Angle

Equations (9)

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

ϕ=2 tan-1sin2 θi-1/n21/2tan θi sin θi,
θi=π4+sin-1sin θ1n,
I1=I0/21+cos ϕ1,
I2=I0/21+cos ϕ2,
ϕ1=4 tan-1sin2 θi1-1/n21/2tan θi1 sin θi1,
ϕ2=4 tan-1sin2 θi2-1/n21/2tan θi2 sin θi2,
θ11=θ0+Δθ,
θ12=θ0-Δθ.
S=I1-I2I1+I2=cos ϕ1-cos ϕ22+cos ϕ1+cos ϕ2.

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