Abstract

We describe a novel method of angle measurement by borrowing the concept of the circular Dammann grating (CDG). A three-order CDG is employed in this experiment. The displacement of the tilted angle can be determined accurately by measuring the projection from the distorted CDG image. This method is controlled only by the initial radius of the image and the converging ratio of the lens. Compared with conventional techniques, this technique has the advantages of a simple design with superior resolution to within 1°, low cost, and compactness. A theoretical analysis together with experimental results is presented.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
  8. X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, “High accuracy, wide range, rotation angle measurement by the use of two parallel interference patterns,” Appl. Opt. 36, 6190-6195 (1997).
    [CrossRef]
  9. S. Singh, S. Rana, and S. Prakash, “Design, development and performance evaluation of Lau based array illuminators,” presented at the International Conference on Photonics, International School of Photonics, Cochin, India, December 9-11, 2004.
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  13. E. N. Hogert and N. G. Gaggioli, “Tilting of a diffusing surface determined by the direct observation of its photographically recorded speckle,” J. Opt. 17, 203-207 (1986).
    [CrossRef]
  14. R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photorefractive speckle correlator: theoretical and experimental analysis,” Opt. Eng 37, 2988-2997 (1998).
    [CrossRef]
  15. F. J. Wen and P. S. Chung, “Design of circular Dammann gratings by employing the circular spot rotation method,” Appl. Opt. 46, 5452-5455 (2007).
    [CrossRef] [PubMed]
  16. F. J. Wen and P. S. Chung, “A new circular Dammann grating using Hankel transform,” J. Opt. A 10, 075306 (2008).
    [CrossRef]

2008 (1)

F. J. Wen and P. S. Chung, “A new circular Dammann grating using Hankel transform,” J. Opt. A 10, 075306 (2008).
[CrossRef]

2007 (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)

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]

1998 (1)

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photorefractive speckle correlator: theoretical and experimental analysis,” Opt. Eng 37, 2988-2997 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

1992 (1)

1988 (1)

1986 (1)

E. N. Hogert and N. G. Gaggioli, “Tilting of a diffusing surface determined by the direct observation of its photographically recorded speckle,” J. Opt. 17, 203-207 (1986).
[CrossRef]

1984 (1)

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

1982 (1)

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

1963 (1)

Chung, P. S.

Dai, X.

Deslattes, R. D.

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

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]

Ennos, A. E.

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

Gaggioli, N. G.

E. N. Hogert and N. G. Gaggioli, “Tilting of a diffusing surface determined by the direct observation of its photographically recorded speckle,” J. Opt. 17, 203-207 (1986).
[CrossRef]

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, “High accuracy, wide range, rotation angle measurement by the use of two parallel interference patterns,” Appl. Opt. 36, 6190-6195 (1997).
[CrossRef]

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

Hogert, E. N.

E. N. Hogert and N. G. Gaggioli, “Tilting of a diffusing surface determined by the direct observation of its photographically recorded speckle,” J. Opt. 17, 203-207 (1986).
[CrossRef]

Huang, P. S.

Kamada, O.

Kiyono, S.

Kumar, A.

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photorefractive speckle correlator: theoretical and experimental analysis,” Opt. Eng 37, 2988-2997 (1998).
[CrossRef]

Luther, G. G.

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

Ni, J.

Pati, G. S.

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photorefractive speckle correlator: theoretical and experimental analysis,” Opt. Eng 37, 2988-2997 (1998).
[CrossRef]

Prakash, S.

S. Prakash, S. Singh, and S. Rana, “Automated small tilt-angle measurement using Lau interferometry,” Appl. Opt. 44, 5905-5909 (2005).
[CrossRef] [PubMed]

S. Singh, S. Rana, and S. Prakash, “Design, development and performance evaluation of Lau based array illuminators,” presented at the International Conference on Photonics, International School of Photonics, Cochin, India, December 9-11, 2004.

Rana, S.

S. Prakash, S. Singh, and S. Rana, “Automated small tilt-angle measurement using Lau interferometry,” Appl. Opt. 44, 5905-5909 (2005).
[CrossRef] [PubMed]

S. Singh, S. Rana, and S. Prakash, “Design, development and performance evaluation of Lau based array illuminators,” presented at the International Conference on Photonics, International School of Photonics, Cochin, India, December 9-11, 2004.

Rohlin, J.

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, “High accuracy, wide range, rotation angle measurement by the use of two parallel interference patterns,” Appl. Opt. 36, 6190-6195 (1997).
[CrossRef]

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

Shi, P.

Singh, K.

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photorefractive speckle correlator: theoretical and experimental analysis,” Opt. Eng 37, 2988-2997 (1998).
[CrossRef]

Singh, S.

S. Prakash, S. Singh, and S. Rana, “Automated small tilt-angle measurement using Lau interferometry,” Appl. Opt. 44, 5905-5909 (2005).
[CrossRef] [PubMed]

S. Singh, S. Rana, and S. Prakash, “Design, development and performance evaluation of Lau based array illuminators,” presented at the International Conference on Photonics, International School of Photonics, Cochin, India, December 9-11, 2004.

Stijns, E.

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, “High accuracy, wide range, rotation angle measurement by the use of two parallel interference patterns,” Appl. Opt. 36, 6190-6195 (1997).
[CrossRef]

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

Tripathi, R.

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photorefractive speckle correlator: theoretical and experimental analysis,” Opt. Eng 37, 2988-2997 (1998).
[CrossRef]

Virdee, M. S.

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

Wen, F. J.

Appl. Opt. (8)

J. Opt. (1)

E. N. Hogert and N. G. Gaggioli, “Tilting of a diffusing surface determined by the direct observation of its photographically recorded speckle,” J. Opt. 17, 203-207 (1986).
[CrossRef]

J. Opt. A (1)

F. J. Wen and P. S. Chung, “A new circular Dammann grating using Hankel transform,” J. Opt. A 10, 075306 (2008).
[CrossRef]

Opt. Eng (1)

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photorefractive speckle correlator: theoretical and experimental analysis,” Opt. Eng 37, 2988-2997 (1998).
[CrossRef]

Opt. Eng. (2)

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]

Precision engineering (1)

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

Rev. Sci. Instrum. (1)

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

Other (1)

S. Singh, S. Rana, and S. Prakash, “Design, development and performance evaluation of Lau based array illuminators,” presented at the International Conference on Photonics, International School of Photonics, Cochin, India, December 9-11, 2004.

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Figures (6)

Fig. 1
Fig. 1

(a) Transition points in the cross section of the CDG. (b) Output image of the three-order CDG.

Fig. 2
Fig. 2

Optical setup for angle measurement.

Fig. 3
Fig. 3

(a) Principles of the rotation angle from the distorted grating that is projected onto the CCD image plane. (b) Optical schematic of the distorted grating.

Fig. 4
Fig. 4

Distorted images with different tilted level: (a)  θ = 0 ° , (b)  θ = 40 ° , (c)  θ = 55 ° .

Fig. 5
Fig. 5

Comparison between calculated and measured values in the relationship between the tilted angle θ i and the extra radius Δ r i [lighter curve (red online), measurement; darker curve (blue online), simulation].

Fig. 6
Fig. 6

Simulation of how different parameters can affect the results: (a) radius, r 0 ; (b) converging ratio of the lens, tan φ .

Equations (3)

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l i sin ( π / 2 + φ ) = c i sin θ i & sin φ = Δ r i sin φ Δ r i sin φ = l i sin θ i cos φ sin θ i = Δ r i l i tan φ sin θ i = Δ r i l i tan φ & l i = r 0 + Δ r i cos θ i tan θ i = Δ r i ( r 0 + Δ r i ) tan φ ,
r 0 = N λ D f ,
sin φ = Δ r i c i & c i sin θ i = r 0 sin [ 90 φ θ i ] sin φ = Δ r i cos [ φ + θ i ] r 0 sin θ i r 0 = Δ r i cos [ φ + θ i ] sin θ i sin φ Δ r i = r 0 cot θ i cot φ 1 .

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