Abstract

A novel angular probe using the Fabry-Perot etalon and angular scanning technique is proposed for absolute angular displacement determinations in this paper. The measurement theory is first derived, a setup constructed to implement the angular probe is then introduced and analyzed, and the experimental results from the uses of the setup are finally presented. The setup analyses reveal that the probe is with high measurement resolution and sensitivity. The experimental results not only confirm the validity, stability, accuracy, and repeatability, but also show an application of the angular probe.

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References

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  1. D. Malacara, Optical shop testing (John Wily & Sons, 1978), Chap. 15.
  2. F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54(12), 1648–1652 (1983).
    [CrossRef]
  3. P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect and the use of right-angle prisms,” Appl. Opt. 34(22), 4976–4981 (1995).
    [CrossRef] [PubMed]
  4. P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35(13), 2239–2241 (1996).
    [CrossRef] [PubMed]
  5. M. H. Chiu and D. C. Su, “Angle measurement using total-internal reflection heterodyne interferometry,” Opt. Eng. 36(6), 1750–1753 (1997).
    [CrossRef]
  6. M. H. Chiu and D. C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36(28), 7104–7106 (1997).
    [CrossRef]
  7. W. Zhou and L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37(25), 5957–5963 (1998).
    [CrossRef]
  8. P. S. Huang and Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37(28), 6636–6642 (1998).
    [CrossRef]
  9. W. Zhou and L. Cai, “Improved angle interferometer based on total internal reflection,” Appl. Opt. 38(7), 1179–1185 (1999).
    [CrossRef]
  10. A. Zhang and P. S. Huang, “Total internal reflection for precision small-angle measurement,” Appl. Opt. 40(10), 1617–1622 (2001).
    [CrossRef]
  11. M. H. Chiu, S. F. Wang, and R. S. Chang, “Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 43(29), 5438–5442 (2004).
    [CrossRef] [PubMed]
  12. G. Margheri, A. Mannoni, and F. Quercioli, “High-resolution angular and displacement sensing based on the excitation of surface plasma waves,” Appl. Opt. 36(19), 4521–4525 (1997).
    [CrossRef] [PubMed]
  13. F. Chen, Z. Cao, Q. Shen, and Y. Feng, “Optical approach to angular displacement measurement based on attenuated total reflection,” Appl. Opt. 44(26), 5393–5397 (2005).
    [CrossRef] [PubMed]
  14. S. T. Lin and W. J. Syu, “Heterodyne Angular Interferometer Using a Square Prism,” Opt. Lasers Eng. 47(1), 80–83 (2009).
    [CrossRef]
  15. Agilent 5529A Dynamic Calibrator: http://cp.literature.agilent.com/litweb/pdf/5968-0111E.pdf .
  16. S. T. Lin, S. L. Yeh, and C. W. Chang, “Low-coherent light-source angular interferometer using a square prism and the angular-scanning technique,” Opt. Lett. 33(20), 2344–2346 (2008).
    [CrossRef] [PubMed]
  17. S. T. Lin, S. L. Yeh, and C. W. Chang, “Absolute angular displacement determination using Mach-Zehnder interferometer,” J. Opt. A, Pure Appl. Opt. 10(9), 095304 (2008).
    [CrossRef]
  18. E. Hecht, Optics (Addison Wesley, 4th ed.), Chap. 9.

2009 (1)

S. T. Lin and W. J. Syu, “Heterodyne Angular Interferometer Using a Square Prism,” Opt. Lasers Eng. 47(1), 80–83 (2009).
[CrossRef]

2008 (2)

S. T. Lin, S. L. Yeh, and C. W. Chang, “Low-coherent light-source angular interferometer using a square prism and the angular-scanning technique,” Opt. Lett. 33(20), 2344–2346 (2008).
[CrossRef] [PubMed]

S. T. Lin, S. L. Yeh, and C. W. Chang, “Absolute angular displacement determination using Mach-Zehnder interferometer,” J. Opt. A, Pure Appl. Opt. 10(9), 095304 (2008).
[CrossRef]

2005 (1)

F. Chen, Z. Cao, Q. Shen, and Y. Feng, “Optical approach to angular displacement measurement based on attenuated total reflection,” Appl. Opt. 44(26), 5393–5397 (2005).
[CrossRef] [PubMed]

2004 (1)

M. H. Chiu, S. F. Wang, and R. S. Chang, “Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 43(29), 5438–5442 (2004).
[CrossRef] [PubMed]

2001 (1)

A. Zhang and P. S. Huang, “Total internal reflection for precision small-angle measurement,” Appl. Opt. 40(10), 1617–1622 (2001).
[CrossRef]

1999 (1)

W. Zhou and L. Cai, “Improved angle interferometer based on total internal reflection,” Appl. Opt. 38(7), 1179–1185 (1999).
[CrossRef]

1998 (2)

W. Zhou and L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37(25), 5957–5963 (1998).
[CrossRef]

P. S. Huang and Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37(28), 6636–6642 (1998).
[CrossRef]

1997 (3)

M. H. Chiu and D. C. Su, “Angle measurement using total-internal reflection heterodyne interferometry,” Opt. Eng. 36(6), 1750–1753 (1997).
[CrossRef]

M. H. Chiu and D. C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36(28), 7104–7106 (1997).
[CrossRef]

G. Margheri, A. Mannoni, and F. Quercioli, “High-resolution angular and displacement sensing based on the excitation of surface plasma waves,” Appl. Opt. 36(19), 4521–4525 (1997).
[CrossRef] [PubMed]

1996 (1)

P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35(13), 2239–2241 (1996).
[CrossRef] [PubMed]

1995 (1)

P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect and the use of right-angle prisms,” Appl. Opt. 34(22), 4976–4981 (1995).
[CrossRef] [PubMed]

1983 (1)

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54(12), 1648–1652 (1983).
[CrossRef]

Cai, L.

W. Zhou and L. Cai, “Improved angle interferometer based on total internal reflection,” Appl. Opt. 38(7), 1179–1185 (1999).
[CrossRef]

W. Zhou and L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37(25), 5957–5963 (1998).
[CrossRef]

Cao, Z.

F. Chen, Z. Cao, Q. Shen, and Y. Feng, “Optical approach to angular displacement measurement based on attenuated total reflection,” Appl. Opt. 44(26), 5393–5397 (2005).
[CrossRef] [PubMed]

Chang, C. W.

S. T. Lin, S. L. Yeh, and C. W. Chang, “Low-coherent light-source angular interferometer using a square prism and the angular-scanning technique,” Opt. Lett. 33(20), 2344–2346 (2008).
[CrossRef] [PubMed]

S. T. Lin, S. L. Yeh, and C. W. Chang, “Absolute angular displacement determination using Mach-Zehnder interferometer,” J. Opt. A, Pure Appl. Opt. 10(9), 095304 (2008).
[CrossRef]

Chang, R. S.

M. H. Chiu, S. F. Wang, and R. S. Chang, “Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 43(29), 5438–5442 (2004).
[CrossRef] [PubMed]

Chen, F.

F. Chen, Z. Cao, Q. Shen, and Y. Feng, “Optical approach to angular displacement measurement based on attenuated total reflection,” Appl. Opt. 44(26), 5393–5397 (2005).
[CrossRef] [PubMed]

Chiu, M. H.

M. H. Chiu, S. F. Wang, and R. S. Chang, “Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 43(29), 5438–5442 (2004).
[CrossRef] [PubMed]

M. H. Chiu and D. C. Su, “Angle measurement using total-internal reflection heterodyne interferometry,” Opt. Eng. 36(6), 1750–1753 (1997).
[CrossRef]

M. H. Chiu and D. C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36(28), 7104–7106 (1997).
[CrossRef]

Feng, Y.

F. Chen, Z. Cao, Q. Shen, and Y. Feng, “Optical approach to angular displacement measurement based on attenuated total reflection,” Appl. Opt. 44(26), 5393–5397 (2005).
[CrossRef] [PubMed]

Huang, P. S.

A. Zhang and P. S. Huang, “Total internal reflection for precision small-angle measurement,” Appl. Opt. 40(10), 1617–1622 (2001).
[CrossRef]

P. S. Huang and Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37(28), 6636–6642 (1998).
[CrossRef]

P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35(13), 2239–2241 (1996).
[CrossRef] [PubMed]

P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect and the use of right-angle prisms,” Appl. Opt. 34(22), 4976–4981 (1995).
[CrossRef] [PubMed]

Li, Y.

P. S. Huang and Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37(28), 6636–6642 (1998).
[CrossRef]

Lin, S. T.

S. T. Lin and W. J. Syu, “Heterodyne Angular Interferometer Using a Square Prism,” Opt. Lasers Eng. 47(1), 80–83 (2009).
[CrossRef]

S. T. Lin, S. L. Yeh, and C. W. Chang, “Absolute angular displacement determination using Mach-Zehnder interferometer,” J. Opt. A, Pure Appl. Opt. 10(9), 095304 (2008).
[CrossRef]

S. T. Lin, S. L. Yeh, and C. W. Chang, “Low-coherent light-source angular interferometer using a square prism and the angular-scanning technique,” Opt. Lett. 33(20), 2344–2346 (2008).
[CrossRef] [PubMed]

Mannoni, A.

G. Margheri, A. Mannoni, and F. Quercioli, “High-resolution angular and displacement sensing based on the excitation of surface plasma waves,” Appl. Opt. 36(19), 4521–4525 (1997).
[CrossRef] [PubMed]

Margheri, G.

G. Margheri, A. Mannoni, and F. Quercioli, “High-resolution angular and displacement sensing based on the excitation of surface plasma waves,” Appl. Opt. 36(19), 4521–4525 (1997).
[CrossRef] [PubMed]

Ni, J.

P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35(13), 2239–2241 (1996).
[CrossRef] [PubMed]

P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect and the use of right-angle prisms,” Appl. Opt. 34(22), 4976–4981 (1995).
[CrossRef] [PubMed]

Quercioli, F.

G. Margheri, A. Mannoni, and F. Quercioli, “High-resolution angular and displacement sensing based on the excitation of surface plasma waves,” Appl. Opt. 36(19), 4521–4525 (1997).
[CrossRef] [PubMed]

Schuda, F. J.

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54(12), 1648–1652 (1983).
[CrossRef]

Shen, Q.

F. Chen, Z. Cao, Q. Shen, and Y. Feng, “Optical approach to angular displacement measurement based on attenuated total reflection,” Appl. Opt. 44(26), 5393–5397 (2005).
[CrossRef] [PubMed]

Su, D. C.

M. H. Chiu and D. C. Su, “Angle measurement using total-internal reflection heterodyne interferometry,” Opt. Eng. 36(6), 1750–1753 (1997).
[CrossRef]

M. H. Chiu and D. C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36(28), 7104–7106 (1997).
[CrossRef]

Syu, W. J.

S. T. Lin and W. J. Syu, “Heterodyne Angular Interferometer Using a Square Prism,” Opt. Lasers Eng. 47(1), 80–83 (2009).
[CrossRef]

Wang, S. F.

M. H. Chiu, S. F. Wang, and R. S. Chang, “Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 43(29), 5438–5442 (2004).
[CrossRef] [PubMed]

Yeh, S. L.

S. T. Lin, S. L. Yeh, and C. W. Chang, “Low-coherent light-source angular interferometer using a square prism and the angular-scanning technique,” Opt. Lett. 33(20), 2344–2346 (2008).
[CrossRef] [PubMed]

S. T. Lin, S. L. Yeh, and C. W. Chang, “Absolute angular displacement determination using Mach-Zehnder interferometer,” J. Opt. A, Pure Appl. Opt. 10(9), 095304 (2008).
[CrossRef]

Zhang, A.

A. Zhang and P. S. Huang, “Total internal reflection for precision small-angle measurement,” Appl. Opt. 40(10), 1617–1622 (2001).
[CrossRef]

Zhou, W.

W. Zhou and L. Cai, “Improved angle interferometer based on total internal reflection,” Appl. Opt. 38(7), 1179–1185 (1999).
[CrossRef]

W. Zhou and L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37(25), 5957–5963 (1998).
[CrossRef]

Appl. Opt. (10)

M. H. Chiu and D. C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36(28), 7104–7106 (1997).
[CrossRef]

W. Zhou and L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37(25), 5957–5963 (1998).
[CrossRef]

P. S. Huang and Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37(28), 6636–6642 (1998).
[CrossRef]

W. Zhou and L. Cai, “Improved angle interferometer based on total internal reflection,” Appl. Opt. 38(7), 1179–1185 (1999).
[CrossRef]

A. Zhang and P. S. Huang, “Total internal reflection for precision small-angle measurement,” Appl. Opt. 40(10), 1617–1622 (2001).
[CrossRef]

M. H. Chiu, S. F. Wang, and R. S. Chang, “Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 43(29), 5438–5442 (2004).
[CrossRef] [PubMed]

G. Margheri, A. Mannoni, and F. Quercioli, “High-resolution angular and displacement sensing based on the excitation of surface plasma waves,” Appl. Opt. 36(19), 4521–4525 (1997).
[CrossRef] [PubMed]

F. Chen, Z. Cao, Q. Shen, and Y. Feng, “Optical approach to angular displacement measurement based on attenuated total reflection,” Appl. Opt. 44(26), 5393–5397 (2005).
[CrossRef] [PubMed]

P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect and the use of right-angle prisms,” Appl. Opt. 34(22), 4976–4981 (1995).
[CrossRef] [PubMed]

P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35(13), 2239–2241 (1996).
[CrossRef] [PubMed]

J. Opt. A, Pure Appl. Opt. (1)

S. T. Lin, S. L. Yeh, and C. W. Chang, “Absolute angular displacement determination using Mach-Zehnder interferometer,” J. Opt. A, Pure Appl. Opt. 10(9), 095304 (2008).
[CrossRef]

Opt. Eng. (1)

M. H. Chiu and D. C. Su, “Angle measurement using total-internal reflection heterodyne interferometry,” Opt. Eng. 36(6), 1750–1753 (1997).
[CrossRef]

Opt. Lasers Eng. (1)

S. T. Lin and W. J. Syu, “Heterodyne Angular Interferometer Using a Square Prism,” Opt. Lasers Eng. 47(1), 80–83 (2009).
[CrossRef]

Opt. Lett. (1)

S. T. Lin, S. L. Yeh, and C. W. Chang, “Low-coherent light-source angular interferometer using a square prism and the angular-scanning technique,” Opt. Lett. 33(20), 2344–2346 (2008).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54(12), 1648–1652 (1983).
[CrossRef]

Other (3)

D. Malacara, Optical shop testing (John Wily & Sons, 1978), Chap. 15.

E. Hecht, Optics (Addison Wesley, 4th ed.), Chap. 9.

Agilent 5529A Dynamic Calibrator: http://cp.literature.agilent.com/litweb/pdf/5968-0111E.pdf .

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Figures (8)

Fig. 1
Fig. 1

The Fabry-Perot etalon

Fig. 2
Fig. 2

Schematic diagrams of (a) the proposed angular probe and (b) the control and signal processing system

Fig. 3
Fig. 3

The etalon module

Fig. 4
Fig. 4

The calculated and measured Airy functions of beam L1

Fig. 5
Fig. 5

The least squares line of the measured S-curve

Fig. 6
Fig. 6

The result of the stability test of the proposed angular probe

Fig. 7
Fig. 7

The result of the validity test of the proposed angular probe

Fig. 8
Fig. 8

The result of the applicability test of the proposed angular probe

Equations (13)

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

I t I t max = 1 1 + F sin 2 ( ϕ / 2 ) = A ( θ ) .
ϕ = 4 π λ n d cos θ t .
sin θ = n sin θ t .
θ 1 = θ c + γ 2 + Δ θ
θ 2 = θ c γ 2 + Δ θ ,
A 1 = A ( θ c + γ / 2 + Δ θ ) = A ( θ c + γ / 2 ) + d A ( θ ) d θ | θ = θ c + γ / 2 Δ θ
A 2 = A ( θ c γ / 2 + Δ θ ) = A ( θ c γ / 2 ) + d A ( θ ) d θ | θ = θ c γ / 2 Δ θ .
d A ( θ ) d θ = 4 π λ F sin ( φ / 2 ) cos ( φ / 2 ) [ 1 + F sin 2 ( φ / 2 ) ] 2 d tan θ t cos θ .
d A ( θ ) d θ | θ = θ c ± γ / 2 = π d F θ c n λ .
A 2 A 1 = 2 π d F θ c n λ Δ θ ,
I t 2 I t 1 I t 2 + I t 1 = 2 π d F θ c n λ Δ θ ,
S = K Δ θ .
K = 2 π d F θ c n λ .

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