Abstract

A detailed analysis of a carousel interferometer is presented for the measurement of an unknown angle and axis of rotation. The technique exploits a set of compensator glass plates and a right-angle prism that is placed in each of the two arms of the interferometer. The two sets are placed at the same rotational stage, while the end mirrors of the interferometer are static. When rotation takes place, individual and relative optical path differences are generated in the two beams of the interferometer. The generated phase differences contribute toward finding the angle and axis of rotation. The analysis is presented for any initial position of the interferometer, i.e., the radial vector from the axis of rotation to the apex of one of the prisms used. The results show the slight variations in the error and nonlinearity when different parameters are manipulated. Moreover, the trade-off between the maximum size of the prisms and the radial distances are also presented.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Ikram and G. Hussain, “Michelson interferometer for precision angle measurement,” Appl. Opt. 38, 113-120 (1999).
    [CrossRef]
  2. G. Hussain and M. Ikram, “Optimization of linearity by use of a glass plate in carousel interferometers,” Opt. Lett. 29, 1930-1932 (2004).
    [CrossRef] [PubMed]
  3. G. Hussain and M. Ikram, “Optical measurements of angle and axis of rotation,” Opt. Lett. 33, 2419-2421 (2008).
    [CrossRef] [PubMed]
  4. D. Malacara and O. Harris, “Interferometric measurement of angles,” Appl. Opt. 9, 1630-1633 (1970).
    [CrossRef] [PubMed]
  5. J. D. Chapman, “Interferometric angular measurement,” Appl. Opt. 13, 1646-1651 (1974).
    [CrossRef] [PubMed]
  6. P. S. Huang, S. Kiyano, and O. Kamada, “Angle measurement based on the internal-reflection effect: a new method,” Appl. Opt. 31, 6047-6055 (1992).
    [CrossRef] [PubMed]
  7. H. K. Chiang, R. P. Kenan, N. F. Hartman, and C. J. Summers, “Optical alignment and tilt angle measurement technique based on Lloyd's mirror arrangement,” Opt. Lett. 17, 1024-1025 (1992).
    [CrossRef] [PubMed]
  8. X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, “Measurement of small rotation angles by using a parallel interference pattern,” Appl. Opt. 34, 6380-6388 (1995).
    [CrossRef] [PubMed]
  9. L. Zeng, H. Mataumoto, and K. Kawachi, “Scanning beam collimation method for measuring dynamic angle variation using an acousto-optic deflector,” Opt. Eng. 35, 1662-1667(1996).
    [CrossRef]
  10. J. K. Kauppinen, I. K. Salomaa, and J. O. Partanen, “Carousel interferometer,” Appl. Opt. 34, 6081-6085 (1995).
    [CrossRef] [PubMed]
  11. P. Shi and E. Stijns, “New optical method for measuring small angle rotations,” Appl. Opt. 27, 4342-4344 (1988).
    [CrossRef] [PubMed]
  12. P. Shi and E. Stijns, “Improving the linearity of the Michelson interferometric angular measurement by a parametric compensation method,” Appl. Opt. 32, 44-51 (1993).
    [CrossRef] [PubMed]
  13. L.-c. Li, O.-f. Yu, Z. Lei, and J. Li, “High-accuracy measurement of rotation angle based on image,” Acta Opt. Sinica 25, 491-496 (2005).
    [CrossRef]
  14. T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
    [CrossRef]
  15. P. J. Giblin, F. E. Pollick, and J. E. Rycroft, “Recovery of an unknown axis of rotation from the profiles of a rotating surface,” J. Opt. Soc. Am. A 11, 1976-1984 (1994).
    [CrossRef]
  16. T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotation-angle measurement using an imaging method,” Opt. Eng. 40, 426-432 (2001).
    [CrossRef]

2008 (1)

2006 (1)

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

2005 (1)

L.-c. Li, O.-f. Yu, Z. Lei, and J. Li, “High-accuracy measurement of rotation angle based on image,” Acta Opt. Sinica 25, 491-496 (2005).
[CrossRef]

2004 (1)

2001 (1)

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotation-angle measurement using an imaging method,” Opt. Eng. 40, 426-432 (2001).
[CrossRef]

1999 (1)

1996 (1)

L. Zeng, H. Mataumoto, and K. Kawachi, “Scanning beam collimation method for measuring dynamic angle variation using an acousto-optic deflector,” Opt. Eng. 35, 1662-1667(1996).
[CrossRef]

1995 (2)

1994 (1)

1993 (1)

1992 (2)

1988 (1)

1974 (1)

1970 (1)

Chapman, J. D.

Chiang, H. K.

Dai, X.

Endo, T.

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

Giblin, P. J.

Greivenkamp, J. E.

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotation-angle measurement using an imaging method,” Opt. Eng. 40, 426-432 (2001).
[CrossRef]

X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, “Measurement of small rotation angles by using a parallel interference pattern,” Appl. Opt. 34, 6380-6388 (1995).
[CrossRef] [PubMed]

Harris, O.

Hartman, N. F.

Huang, P. S.

Hussain, G.

Ikram, M.

Kamada, O.

Kauppinen, J. K.

Kawachi, K.

L. Zeng, H. Mataumoto, and K. Kawachi, “Scanning beam collimation method for measuring dynamic angle variation using an acousto-optic deflector,” Opt. Eng. 35, 1662-1667(1996).
[CrossRef]

Kenan, R. P.

Kiyano, S.

Lei, Z.

L.-c. Li, O.-f. Yu, Z. Lei, and J. Li, “High-accuracy measurement of rotation angle based on image,” Acta Opt. Sinica 25, 491-496 (2005).
[CrossRef]

Li, J.

L.-c. Li, O.-f. Yu, Z. Lei, and J. Li, “High-accuracy measurement of rotation angle based on image,” Acta Opt. Sinica 25, 491-496 (2005).
[CrossRef]

Li, L.-c.

L.-c. Li, O.-f. Yu, Z. Lei, and J. Li, “High-accuracy measurement of rotation angle based on image,” Acta Opt. Sinica 25, 491-496 (2005).
[CrossRef]

Malacara, D.

Mataumoto, H.

L. Zeng, H. Mataumoto, and K. Kawachi, “Scanning beam collimation method for measuring dynamic angle variation using an acousto-optic deflector,” Opt. Eng. 35, 1662-1667(1996).
[CrossRef]

Nakamura, H.

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotation-angle measurement using an imaging method,” Opt. Eng. 40, 426-432 (2001).
[CrossRef]

Partanen, J. O.

Pollick, F. E.

Rycroft, J. E.

Salomaa, I. K.

Sasaki, O.

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotation-angle measurement using an imaging method,” Opt. Eng. 40, 426-432 (2001).
[CrossRef]

X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, “Measurement of small rotation angles by using a parallel interference pattern,” Appl. Opt. 34, 6380-6388 (1995).
[CrossRef] [PubMed]

Shi, P.

Stijns, E.

Summers, C. J.

Suzuki, T.

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotation-angle measurement using an imaging method,” Opt. Eng. 40, 426-432 (2001).
[CrossRef]

X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, “Measurement of small rotation angles by using a parallel interference pattern,” Appl. Opt. 34, 6380-6388 (1995).
[CrossRef] [PubMed]

Yu, O.-f.

L.-c. Li, O.-f. Yu, Z. Lei, and J. Li, “High-accuracy measurement of rotation angle based on image,” Acta Opt. Sinica 25, 491-496 (2005).
[CrossRef]

Zeng, L.

L. Zeng, H. Mataumoto, and K. Kawachi, “Scanning beam collimation method for measuring dynamic angle variation using an acousto-optic deflector,” Opt. Eng. 35, 1662-1667(1996).
[CrossRef]

Acta Opt. Sinica (1)

L.-c. Li, O.-f. Yu, Z. Lei, and J. Li, “High-accuracy measurement of rotation angle based on image,” Acta Opt. Sinica 25, 491-496 (2005).
[CrossRef]

Appl. Opt. (8)

J. Opt. Soc. Am. A (1)

Opt. Eng. (3)

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotation-angle measurement using an imaging method,” Opt. Eng. 40, 426-432 (2001).
[CrossRef]

L. Zeng, H. Mataumoto, and K. Kawachi, “Scanning beam collimation method for measuring dynamic angle variation using an acousto-optic deflector,” Opt. Eng. 35, 1662-1667(1996).
[CrossRef]

Opt. Lett. (3)

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Experimental setup for angle and axis of rotation measurement. FI, Faraday isolator; MI 1 MI 3 , Michelson inter ferometers; B 1 B 4 , beam splitters; PR 1 , PR 2 , prisms; S1, S2, glass strips; M 1 M 4 , plane reflecting mirrors; MR 1 , MR 2 , are retroreflecting mirrors; R 1 , R 2 , distances of the apexes of the two prisms from the axis of rotation; F 1 , F 2 , fringes from the individual prisms; F r , fringes from the two prisms.

Fig. 2
Fig. 2

OPDs P 1 ( θ ) , P 2 ( θ ) , and Δ P 2 ( θ ) generated in beams I 1 , I 2 , and between I 1 and I 2 , respectively, due to rotation of the disk for α = 0 ° and ψ = 15.55 ° .

Fig. 3
Fig. 3

OPDs P 1 ( θ ) , P 2 ( θ ) , and Δ P 2 ( θ ) generated due to the rotation of the disk at α = 90 ° and ψ = 15.55 ° .

Fig. 4
Fig. 4

Optimized OPD Δ P 12 between beams I 1 and I 2 .

Fig. 5
Fig. 5

Nonlinearity that exists in OPD Δ P 12 ( θ ) .

Fig. 6
Fig. 6

Error in OPD P 12 ( θ ) due to nonlinearity present in the system.

Fig. 7
Fig. 7

Error in the retrieved angle of rotation: solid curve, ψ = 1.550 ; dashed curve, ψ = 1.543 ; dotted curve, ψ = 1.560 .

Fig. 8
Fig. 8

Variation in the inverse of interferometer sensitivity at different values of strip angles.

Fig. 9
Fig. 9

Path intercepts by beams at the two reflecting faces of prism P R 1 .

Tables (4)

Tables Icon

Table 1 Optimized and Given Parameters

Tables Icon

Table 2 Values of Optical Path Differences P 1 and P 2 for different values of α for strip angle ψ s = 15.55 °

Tables Icon

Table 3 Values of Δ P , Nonlinearity, Error in Δ P 12 ( θ ) , and Error in θ r

Tables Icon

Table 4 Error in Retrieved Radial Distance R r 1 for Different Values of Strip Angle ψ

Equations (5)

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

P 1 ( θ ) = 2 x 1 sin θ + 2 y 1 ( cos θ 1 ) + Φ 1 ( θ ) ,
P 2 ( θ ) = 2 x 2 sin θ + 2 y 2 ( cos θ 1 ) + Φ 2 ( θ ) ,
Δ P 12 ( θ ) = 2 Δ x sin θ + 2 Δ y ( cos θ 1 ) + d [ n s cos ( ψ + θ γ ) cos γ n s cos ( ψ θ β ) cos β ] ,
L n ( θ ) = ( 2 / λ ) Δ P 12 ( θ ) m θ m θ ,
θ r = Δ P 12 ( θ ) ( λ / 2 ) m ,

Metrics