Abstract

We show that the fractional Fourier transform is a suitable mechanism with which to analyze the diffraction patterns produced by a one-dimensional object because its intensity distribution is partially described by a linear chirp function. The three-dimensional location and the diameter of a fiber can be determined, provided that the optimal fractional order is selected. The effect of compaction of the intensity distribution in the fractional Fourier domain is discussed. A few experimental results are presented.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2001 (1)

2000 (3)

1998 (3)

1994 (5)

1993 (2)

1992 (1)

1991 (1)

C. Özkul, D. Lebrun, D. Allano, M. A. Abdelgani-Idrissi, A. Leduc, “Processing of glass cylinder diffraction patterns scanned with a photodiode array: influence of the optical transfer function of diodes on dimensional measurements,” Opt. Eng. 30, 1855–1861 (1991).
[CrossRef]

1987 (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1985 (1)

H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. 24, 462–463 (1985).
[CrossRef]

1980 (2)

1966 (1)

1948 (1)

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Abdelgani-Idrissi, M. A.

C. Özkul, D. Lebrun, D. Allano, M. A. Abdelgani-Idrissi, A. Leduc, “Processing of glass cylinder diffraction patterns scanned with a photodiode array: influence of the optical transfer function of diodes on dimensional measurements,” Opt. Eng. 30, 1855–1861 (1991).
[CrossRef]

Allano, D.

C. Özkul, D. Lebrun, D. Allano, M. A. Abdelgani-Idrissi, A. Leduc, “Processing of glass cylinder diffraction patterns scanned with a photodiode array: influence of the optical transfer function of diodes on dimensional measurements,” Opt. Eng. 30, 1855–1861 (1991).
[CrossRef]

Almeida, L.

L. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Barshan, B.

Bernardo, L. M.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

Buraga-Lefrebvre, C.

C. Buraga-Lefrebvre, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Caulfield, H. J.

H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. 24, 462–463 (1985).
[CrossRef]

Coëtmellec, S.

C. Buraga-Lefrebvre, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Erden, M. F.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2th ed. (McGraw-Hill, New York, 1996).

Haussmann, G.

Hlawatsch, F.

W. Mecklenbräuker, F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997), pp. 59–83.

Joseph, J.

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Kutay, M. A.

Lauterborn, W.

Lebrun, D.

C. Buraga-Lefrebvre, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

C. Özkul, D. Lebrun, D. Allano, M. A. Abdelgani-Idrissi, A. Leduc, “Processing of glass cylinder diffraction patterns scanned with a photodiode array: influence of the optical transfer function of diodes on dimensional measurements,” Opt. Eng. 30, 1855–1861 (1991).
[CrossRef]

Leduc, A.

C. Özkul, D. Lebrun, D. Allano, M. A. Abdelgani-Idrissi, A. Leduc, “Processing of glass cylinder diffraction patterns scanned with a photodiode array: influence of the optical transfer function of diodes on dimensional measurements,” Opt. Eng. 30, 1855–1861 (1991).
[CrossRef]

Leith, E.

Lohmann, A. W.

Marinho, F. J.

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mecklenbräuker, W.

W. Mecklenbräuker, F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997), pp. 59–83.

Mendlovic, D.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
[CrossRef]

Onural, L.

Ozaktas, H.

Ozaktas, H. M.

Ozaktas, M.

Özgen, M. T.

Özkul, C.

C. Buraga-Lefrebvre, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

C. Özkul, D. Lebrun, D. Allano, M. A. Abdelgani-Idrissi, A. Leduc, “Processing of glass cylinder diffraction patterns scanned with a photodiode array: influence of the optical transfer function of diodes on dimensional measurements,” Opt. Eng. 30, 1855–1861 (1991).
[CrossRef]

Pellat-Finet, P.

Singh, K.

Soffer, B. H.

Unnikrishnan, G.

Upatnieks, J.

VanderLugt, A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

Yetik, I. S.

Zalevsky, Z.

Appl. Opt. (4)

IEEE Trans. Signal Process. (1)

L. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Inst. Math. Its Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
[CrossRef]

J. Opt. Soc. Am. A (9)

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
[CrossRef]

M. A. Kutay, H. Ozaktas, “Optimal image restoration with the fractional Fourier transform,” J. Opt. Soc. Am. A 15, 825–833 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, “Synthesis of general linear systems with repeated filtering in consecutive fractional Fourier domains,” J. Opt. Soc. Am. A 15, 1647–1657 (1998).
[CrossRef]

F. J. Marinho, L. M. Bernardo, “Numerical calculation of fractional Fourier transforms with a single fast-Fourier-transform algorithm,” J. Opt. Soc. Am. A 15, 2111–2116 (1998).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Wigner-related phase spaces for signal processing and their optical implementation,” J. Opt. Soc. Am. A 17, 2339–2354 (2000).
[CrossRef]

L. Onural, M. T. Özgen, “Extraction of three-dimensional object-location information directly from in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A 9, 252–260 (1992).
[CrossRef]

I. S. Yetik, H. M. Ozaktas, B. Barshan, L. Onural, “Perspective projections in the space-frequency plane and fractional Fourier transforms,” J. Opt. Soc. Am. A 17, 2382–2390 (2000).
[CrossRef]

Nature (London) (1)

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Opt. Eng. (2)

H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. 24, 462–463 (1985).
[CrossRef]

C. Özkul, D. Lebrun, D. Allano, M. A. Abdelgani-Idrissi, A. Leduc, “Processing of glass cylinder diffraction patterns scanned with a photodiode array: influence of the optical transfer function of diodes on dimensional measurements,” Opt. Eng. 30, 1855–1861 (1991).
[CrossRef]

Opt. Lasers Eng. (1)

C. Buraga-Lefrebvre, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Opt. Lett. (2)

Other (3)

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

W. Mecklenbräuker, F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997), pp. 59–83.

J. W. Goodman, Introduction to Fourier Optics, 2th ed. (McGraw-Hill, New York, 1996).

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