Abstract

The fractional Fourier transform (FRT) is shown to be of potential use in analyzing the motion of a surface by use of holographic interferometry. The extra degree of freedom made available by the use of the FRT allows information regarding both translational and tilting motion to be obtained in an efficient manner.

© 2000 Optical Society of America

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References

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  1. D. Mendlovic and H. M. Ozatkas, J. Opt. Soc. Am. A 10, 1875, 2522 (1993).
    [CrossRef]
  2. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
    [CrossRef]
  3. R. G. Dorsch, Appl. Opt. 34, 6016 (1995).
    [CrossRef] [PubMed]
  4. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).
  5. Y. Katzir and I. Glaser, Appl. Opt. 21, 678 (1982).
    [CrossRef] [PubMed]
  6. S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
    [CrossRef]
  7. We have prepared a manuscript entitled “Speckle photography and the fractional Fourier transform.”
  8. M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
    [CrossRef]
  9. S. Abe and J. T. Sheridan, Opt. Lett. 19, 1801 (1994).
    [CrossRef]

1997 (2)

S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

1995 (1)

1994 (1)

1993 (2)

A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
[CrossRef]

D. Mendlovic and H. M. Ozatkas, J. Opt. Soc. Am. A 10, 1875, 2522 (1993).
[CrossRef]

1982 (1)

Abe, S.

S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
[CrossRef]

S. Abe and J. T. Sheridan, Opt. Lett. 19, 1801 (1994).
[CrossRef]

Dorsch, R. G.

Erden, M. F.

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

Glaser, I.

Katzir, Y.

Lohmann, A. W.

Mendlovic, D.

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

D. Mendlovic and H. M. Ozatkas, J. Opt. Soc. Am. A 10, 1875, 2522 (1993).
[CrossRef]

Ozaktas, H. M.

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

Ozatkas, H. M.

D. Mendlovic and H. M. Ozatkas, J. Opt. Soc. Am. A 10, 1875, 2522 (1993).
[CrossRef]

Sahin, A.

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

Sheridan, J. T.

S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
[CrossRef]

S. Abe and J. T. Sheridan, Opt. Lett. 19, 1801 (1994).
[CrossRef]

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

Appl. Opt. (2)

J. Opt. Soc. Am. A (2)

A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
[CrossRef]

D. Mendlovic and H. M. Ozatkas, J. Opt. Soc. Am. A 10, 1875, 2522 (1993).
[CrossRef]

Opt. Commun. (2)

S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

Opt. Lett. (1)

Other (2)

We have prepared a manuscript entitled “Speckle photography and the fractional Fourier transform.”

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

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Equations (10)

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ux+uxexpjκx2=ux22+exp-jκx+exp+jκx=2Ix1+cosκx,
U˜k+U˜kexpjkξ2=U˜k22+exp-jkξ+exp+jkξ=2Ik1+coskξ,
Fθux=Ux,θ=12πsin θ×exp-jπ212+Jθπ+j2x2 cot θ×-+uxexp+j2x2 cot θ-jxx csc θdx,
Fθ[ux-ξexpjκx=12πsin θ×exp-jπ212+Jθπ+j2x2 cot θ×-+ux-ξexpjκx×exp+j2x2 cot θ-jxx csc θdx.
12πsin θexp-jπ212+Jθπ+j2x2 cot θ+j2ξ2 cot θ-jxξ csc θ+jκξ-+uyexpjκy×exp+j2y2+2yξcot θ-jxy csc θdy.
12πsin θexp-jπ212+Jθπ+j2cot θx2+ξ2-2xξcos θ+2κξ tan θ×-+uyexp+j2y2 cot θ-j×x-κ sin θ-ξ cos θy csc θdy.
12πsin θexp-jπ212+Jθπ+j2cot θx-ξ2+κ22+j2κξ×-+uyexp+j2y2 cot θ-jx-ξ2+κ2y csc θ}dy.
Fθux-ξexpjκx=Fθux-ξ2+κ2expjκξ/2.
FTFθux-ξexpjκx=FθFπ/2ux-ξ2+κ2expjκξ/2=FθU˜kexpjκξ/2expjkξ2+κ2.
FθU˜k+FθU˜kexpjκξ/2expjkξ2+κ22=FθU˜k22+expjκξ/2+kξ2+κ2+exp-jκξ/2+kξ2+κ2=2FθU˜k21+coskξ2+κ2+κξ2.

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