Abstract

We describe a high-precision angle sensor and a new calibration system for calibrating the linearity error of an angle sensor under in situ conditions without the need for accurate reference instruments. To achieve accuracy in the nanoradian (nrad) order, we elongated the critical-angle prism to increase the sensitivity, and we added a thermal controller to the angle sensor. This served to suppress thermal drift of the light source to improve stability. The proposed angle sensor demonstrated a stability of 3 nrad over 1 min, and the maximum repeatability error of the calibration curve determined by the in situ self-calibration method was approximately 35 nrad with a measurement range of 500 μrad.

© 1998 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. S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” International Journal of the Japan Society for Precision Engineering 27, 373–378 (1993).
  3. S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” International Journal of the Japan Society for Precision Engineering 30, 349–350 (1996).
  4. 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]
  5. 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]
  6. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980) p. 38.
  7. T. Kohno, N. Ozawa, K. Miyamoto, T. Musha, “High precision optical surface sensor,” Appl. Opt. 27, 103–108 (1988).
    [CrossRef] [PubMed]
  8. S. Kiyono, W. Gao, I. Ogura, H. Seino, “In situ self-calibration of metrological sensors,” Proceedings of 14th IMEKO World Congress, Tampere, Finland, 1–6 June 1997, (Finnish Society of Automation, Tampere, Finland, 1997), Vol. 8, pp. 118–122.
  9. S. Kiyono, S. Zhang, Y. Uda, “Absolute calibration of precision angle sensors and angle standards,” Proceedings of 4th International IMEKO Symposium, Lyngby, Denmark Oct. 21–22, 1996, L. De Chiffre, ed. (Technical University of Denmark, Lyngby, Denmark, 1996), pp. 167–176.

1996 (2)

S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” International Journal of the Japan Society for Precision Engineering 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 (1)

S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” International Journal of the Japan Society for Precision Engineering 27, 373–378 (1993).

1992 (1)

1988 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980) p. 38.

Gao, W.

S. Kiyono, W. Gao, I. Ogura, H. Seino, “In situ self-calibration of metrological sensors,” Proceedings of 14th IMEKO World Congress, Tampere, Finland, 1–6 June 1997, (Finnish Society of Automation, Tampere, Finland, 1997), Vol. 8, pp. 118–122.

Huang, P. S.

Kamada, O.

Kiyono, S.

S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” International Journal of the Japan Society for Precision Engineering 30, 349–350 (1996).

S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” International Journal of the Japan Society for Precision Engineering 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]

S. Kiyono, S. Zhang, Y. Uda, “Absolute calibration of precision angle sensors and angle standards,” Proceedings of 4th International IMEKO Symposium, Lyngby, Denmark Oct. 21–22, 1996, L. De Chiffre, ed. (Technical University of Denmark, Lyngby, Denmark, 1996), pp. 167–176.

S. Kiyono, W. Gao, I. Ogura, H. Seino, “In situ self-calibration of metrological sensors,” Proceedings of 14th IMEKO World Congress, Tampere, Finland, 1–6 June 1997, (Finnish Society of Automation, Tampere, Finland, 1997), Vol. 8, pp. 118–122.

Kohno, T.

Mito, M.

S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” International Journal of the Japan Society for Precision Engineering 30, 349–350 (1996).

Miyamoto, K.

Musha, T.

Ni, J.

Ogura, I.

S. Kiyono, W. Gao, I. Ogura, H. Seino, “In situ self-calibration of metrological sensors,” Proceedings of 14th IMEKO World Congress, Tampere, Finland, 1–6 June 1997, (Finnish Society of Automation, Tampere, Finland, 1997), Vol. 8, pp. 118–122.

Ozawa, N.

Sato, H.

S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” International Journal of the Japan Society for Precision Engineering 27, 373–378 (1993).

Seino, H.

S. Kiyono, W. Gao, I. Ogura, H. Seino, “In situ self-calibration of metrological sensors,” Proceedings of 14th IMEKO World Congress, Tampere, Finland, 1–6 June 1997, (Finnish Society of Automation, Tampere, Finland, 1997), Vol. 8, pp. 118–122.

Shan, X.

S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” International Journal of the Japan Society for Precision Engineering 27, 373–378 (1993).

Uda, Y.

S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” International Journal of the Japan Society for Precision Engineering 30, 349–350 (1996).

S. Kiyono, S. Zhang, Y. Uda, “Absolute calibration of precision angle sensors and angle standards,” Proceedings of 4th International IMEKO Symposium, Lyngby, Denmark Oct. 21–22, 1996, L. De Chiffre, ed. (Technical University of Denmark, Lyngby, Denmark, 1996), pp. 167–176.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980) p. 38.

Zhang, S.

S. Zhang, S. Kiyono, Y. Uda, M. Mito, “Development of a measurement system of the angular profile of the polygon mirror surface,” International Journal of the Japan Society for Precision Engineering 30, 349–350 (1996).

S. Kiyono, S. Zhang, Y. Uda, “Absolute calibration of precision angle sensors and angle standards,” Proceedings of 4th International IMEKO Symposium, Lyngby, Denmark Oct. 21–22, 1996, L. De Chiffre, ed. (Technical University of Denmark, Lyngby, Denmark, 1996), pp. 167–176.

Appl. Opt. (4)

International Journal of the Japan Society for Precision Engineering (2)

S. Kiyono, X. Shan, H. Sato, “Development of an AFM using a critical angular sensor,” International Journal of the Japan Society for Precision Engineering 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,” International Journal of the Japan Society for Precision Engineering 30, 349–350 (1996).

Other (3)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980) p. 38.

S. Kiyono, W. Gao, I. Ogura, H. Seino, “In situ self-calibration of metrological sensors,” Proceedings of 14th IMEKO World Congress, Tampere, Finland, 1–6 June 1997, (Finnish Society of Automation, Tampere, Finland, 1997), Vol. 8, pp. 118–122.

S. Kiyono, S. Zhang, Y. Uda, “Absolute calibration of precision angle sensors and angle standards,” Proceedings of 4th International IMEKO Symposium, Lyngby, Denmark Oct. 21–22, 1996, L. De Chiffre, ed. (Technical University of Denmark, Lyngby, Denmark, 1996), pp. 167–176.

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Figures (11)

Fig. 1
Fig. 1

Reflectance of internal reflection at a glass and air interface.

Fig. 2
Fig. 2

Optical system of the differential detection method.

Fig. 3
Fig. 3

Theoretical outputs of differential detection method (p-polarized light).

Fig. 4
Fig. 4

Schematic of optical system of proposed angle sensor.

Fig. 5
Fig. 5

Calibration system of mean sensitivity of an angle sensor.

Fig. 6
Fig. 6

Calibration principle for linearity error of an angle sensor: (a) Output curve and (b) inverse function of output curve.

Fig. 7
Fig. 7

Experimental apparatus of calibration of linearity error.

Fig. 8
Fig. 8

Thermal drift of proposed angle sensor: (a) Without thermal control and (b) with thermal control.

Fig. 9
Fig. 9

Output curve of angle sensor.

Fig. 10
Fig. 10

Repeatability of calibration of linearity error of angle sensor.

Fig. 11
Fig. 11

Photograph of the proposed angle sensor.

Tables (1)

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Table 1 Mean Sensitivity of the Previously Developed Angle Sensor (with compensation)

Equations (10)

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R s = sin θ i - θ t sin θ i + θ t 2 ,
R p = tan θ i - θ t tan θ i + θ t 2 ,
n i sin   θ i = n t sin   θ t
Y = x 1 - x 2 x 1 + x 2 × 100 % ,
B 2 - A 1 + α = V 1 / S 0 , B 3 - A 2 + α = V 2 / S 0 ,   , B N - A N - 1 + α = V N - 1 / S 0 , B 1 - A N + α = V N / S 0 ,
S 0 = 1 α N i = 1 N   V i .
z = f θ = S 0 × θ + g θ .
z i = m i , f - 1 z i = z i / S 0 - h z i     i = 1 ,   2 , ,   n ,
h z i = 1 / S 0 - Δ θ / Δ m i     Δ m i = m i + - m i .
Δ θ =   Δ m i θ i + 1 - θ i / θ n - θ 1   S 0 .

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