Abstract

A reflection-based optical implementation of two simultaneous scale-invariant fractional Fourier transforms (FRTs) is used to develop a novel compact speckle photographic system. The system allows the independent determination of both surface tilting and in-plane translational motion from two sequential mixed domain images captured using a single camera.

© 2006 Optical Society of America

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References

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  1. P. K. Rastogi, Speckle Metrology, R.S.Sirohi, ed. (Dekker, New York, 1993) pp. 50-58.
  2. H. Tiziani, Opt. Commun. 5, 271 (1972).
    [CrossRef]
  3. V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
    [CrossRef]
  4. A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
    [CrossRef]
  5. D. Mendlovic and H. M. Ozatkas, J. Opt. Soc. Am. A 10, 1875 (1993).
    [CrossRef]
  6. H. M. Ozatkas and D. Mendlovic, J. Opt. Soc. Am. A 10, 2522 (1993).
    [CrossRef]
  7. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
    [CrossRef]
  8. R. G. Dorsch, Appl. Opt. 34, 6016 (1995).
    [CrossRef] [PubMed]
  9. H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, 2001).
  10. J. T. Sheridan and R. Patten, Opt. Lett. 25, 448 (2000).
    [CrossRef]
  11. J. T. Sheridan and R. Patten, Optik (Stuttgart) 111, 329 (2000).
  12. A. W. Lohmann, Opt. Commun. 115, 427 (1995).
    [CrossRef]
  13. R. Patten, J. T. Sheridan, and A. Larkin, Opt. Eng. 40, 1438 (2001).
    [CrossRef]
  14. J. T. Sheridan, B. Hennelly, and D. Kelly, Opt. Lett. 28, 884 (2003).
    [CrossRef] [PubMed]
  15. D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, Appl. Opt. 44, 2720 (2005).
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  16. L. Z. Cai and Y. Q. Wang, Opt. Laser Technol. 34, 249 (2002).
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    [CrossRef]
  18. J. W. Goodman, Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer, 1984), Chap. 2.
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  20. H. T. Yura, B. Rose, and S. G. Hanson, J. Opt. Soc. Am. A 15, 1160 (1998).
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  21. J. M. Diazdelacruz, Appl. Opt. 44, 2250 (2005).
    [CrossRef] [PubMed]

2005 (2)

2003 (1)

2002 (2)

L. Z. Cai and Y. Q. Wang, Opt. Laser Technol. 34, 249 (2002).

K. B. Wolf and G. Krotzsch, J. Opt. Soc. Am. A 19, 1191 (2002).
[CrossRef]

2001 (1)

R. Patten, J. T. Sheridan, and A. Larkin, Opt. Eng. 40, 1438 (2001).
[CrossRef]

2000 (2)

J. T. Sheridan and R. Patten, Opt. Lett. 25, 448 (2000).
[CrossRef]

J. T. Sheridan and R. Patten, Optik (Stuttgart) 111, 329 (2000).

1998 (1)

1997 (1)

1995 (2)

1993 (3)

1987 (1)

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

1980 (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

1972 (1)

H. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

Cai, L. Z.

L. Z. Cai and Y. Q. Wang, Opt. Laser Technol. 34, 249 (2002).

Diazdelacruz, J. M.

Dorsch, R. G.

Goodman, J. W.

J. W. Goodman, Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer, 1984), Chap. 2.

Hansen, R. S.

Hanson, S. G.

Hennelly, B.

Hennelly, B. M.

Kelly, D.

Kelly, D. P.

Kerr, F. H.

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Krotzsch, G.

Kutay, M. Alper

H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, 2001).

Larkin, A.

R. Patten, J. T. Sheridan, and A. Larkin, Opt. Eng. 40, 1438 (2001).
[CrossRef]

Lohmann, A. W.

McBride, A. C.

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Mendlovic, D.

Namias, V.

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, 2001).

Ozatkas, H. M.

Patten, R.

R. Patten, J. T. Sheridan, and A. Larkin, Opt. Eng. 40, 1438 (2001).
[CrossRef]

J. T. Sheridan and R. Patten, Opt. Lett. 25, 448 (2000).
[CrossRef]

J. T. Sheridan and R. Patten, Optik (Stuttgart) 111, 329 (2000).

Rastogi, P. K.

P. K. Rastogi, Speckle Metrology, R.S.Sirohi, ed. (Dekker, New York, 1993) pp. 50-58.

Rose, B.

Sheridan, J. T.

Tiziani, H.

H. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

Wang, Y. Q.

L. Z. Cai and Y. Q. Wang, Opt. Laser Technol. 34, 249 (2002).

Wolf, K. B.

Yura, H. T.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, 2001).

Appl. Opt. (3)

IMA J. Appl. Math. (1)

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Commun. (2)

H. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

A. W. Lohmann, Opt. Commun. 115, 427 (1995).
[CrossRef]

Opt. Eng. (1)

R. Patten, J. T. Sheridan, and A. Larkin, Opt. Eng. 40, 1438 (2001).
[CrossRef]

Opt. Laser Technol. (1)

L. Z. Cai and Y. Q. Wang, Opt. Laser Technol. 34, 249 (2002).

Opt. Lett. (2)

Optik (Stuttgart) (1)

J. T. Sheridan and R. Patten, Optik (Stuttgart) 111, 329 (2000).

Other (3)

J. W. Goodman, Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer, 1984), Chap. 2.

P. K. Rastogi, Speckle Metrology, R.S.Sirohi, ed. (Dekker, New York, 1993) pp. 50-58.

H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, 2001).

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Figures (3)

Fig. 1
Fig. 1

Folded reflection optical fractional Fourier geometry.

Fig. 2
Fig. 2

Mixed domain speckle photography setup.

Fig. 3
Fig. 3

Displacement of (a) + 200 μ m and (b) + 200 μ m and rotation of 540 μ rad .

Equations (6)

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u ( x ) u ( x ξ ) exp ( + j κ x ) .
U θ ( q ) U ( q Q θ ) exp [ + j Φ θ ( q ) ] ,
U θ 1 ( q 1 ) + U θ 2 ( q 2 ) 2 = I θ 1 ( q 1 ) + I θ 2 ( q 2 ) + U θ 1 ( q 1 ) × U θ 2 * ( q 2 ) + U θ 1 * ( q 1 ) × U θ 2 ( q 2 ) ,
U θ 1 ( q 1 Q θ 1 ) exp [ + j Φ θ 1 ( q 1 ) ] + U θ 2 ( q 2 Q θ 2 ) exp [ + j Φ θ 2 ( q 2 ) ] 2 = I θ 1 ( q 1 Q θ 1 ) + I θ 2 ( q 2 Q θ 2 ) + 2 Re ( U θ 1 ( q 1 Q θ 1 ) U θ 2 * ( q 2 Q θ 2 ) exp { + j [ Φ θ 1 ( q 1 ) Φ θ 2 ( q 2 ) ] } ) ,
I θ j ( q j ) I θ i ( q i Q θ i ) I θ j ( 0 ) I θ j ( 0 ) = { 1 i = j and q i = Q θ i , 0 i j or q i Q θ i . }
[ U θ 1 ( q 1 ) × U θ 2 * ( q 2 ) ] ( U θ 1 ( q 1 Q θ 1 ) U θ 2 * ( q 2 Q θ 2 ) exp { + j [ Φ θ 1 ( q 1 ) Φ θ 2 ( q 2 ) ] } ) .

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