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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  1. J. Upatnieks, A. VanderLugt, E. Leith, “Correction of lens aberrations by means of holograms,” Appl. Opt. 5, 589–593 (1966).
    [CrossRef] [PubMed]
  2. D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
    [CrossRef]
  3. G. Haussmann, W. Lauterborn, “Determination of size and position of fast moving glass bubbles in liquids by digital 3-D image processing of hologram reconstructions,” Appl. Opt. 19, 3529–3535 (1980).
    [CrossRef] [PubMed]
  4. H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. 24, 462–463 (1985).
    [CrossRef]
  5. 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]
  6. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
    [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
    [CrossRef]
  10. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  11. G. Unnikrishnan, J. Joseph, K. Singh, “Fractional Fourier domain encrypted holographic memory by use of an anamorphic optical system,” Appl. Opt. 40, 299–306 (2001).
    [CrossRef]
  12. 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]
  13. 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]
  14. M. A. Kutay, H. Ozaktas, “Optimal image restoration with the fractional Fourier transform,” J. Opt. Soc. Am. A 15, 825–833 (1998).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).
  16. 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]
  17. D. Mendlovic, M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  18. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  19. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  20. 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]
  21. W. Mecklenbräuker, F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997), pp. 59–83.
  22. L. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  23. 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]
  24. J. W. Goodman, Introduction to Fourier Optics, 2th ed. (McGraw-Hill, New York, 1996).
  25. 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]

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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Figures (7)

Fig. 1
Fig. 1

Experimental setup for in-line hologram recording.

Fig. 2
Fig. 2

Setup for performing the FRFT according to the WDF definition.

Fig. 3
Fig. 3

(a) Corrected intensity distribution I c (x) with z=129 mm and d=90 µm. (b) WDF of I c (x). (c) WDF of I ca (x a ) at a opt = 0.580. (d) I ca (x a /d)2 at a opt = 0.580.

Fig. 4
Fig. 4

FRFT of I c (x) with reduced compaction. z = 129 mm, d = 90 µm.

Fig. 5
Fig. 5

(a) WDF of I ca (x a ) at a opt = 0.295. z = 50 mm and d = 90 µm. (b) I ca (x a /d)2 at a opt = 0.295.

Fig. 6
Fig. 6

Diffraction pattern of a fiber with d = 20 µm and z = 119 mm digitized on 256 × 256 pixels.

Fig. 7
Fig. 7

(a) Intensity distribution profile obtained from the diffraction pattern of Fig. 6. (b) WDF of the diffraction pattern. (c) WDF at a opt = 0.555. (d) |I ca (x a /d| 2 at a opt = 0.555.

Equations (43)

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

Ax, y=1iλzexpi 2πλ z-+-+1-Ox, y×expiπλzx-x2+y-y2dxdy.
Ax=1λzexpi 2πλ z-+1-Ox×expiπλzx-x2-π4dx.
Ox=rectxd=1|x|<d/2½|x|=d/20otherwise.
Ax=1-exp-iπ/42Cμ1-Cμ2+iSμ1-Sμ2,
μ1=2/λzx+d/2,  μ2=2/λzx-d/2,
Ix=1-2λzcosπx2λz-π4Fx+1λz F2x,
cosπx2λz-π4.
Γαuxxa=exp-i π4λf1|R|-+ ux×exp-i πλf1Q-1Rxa2+x2×exp-i 2πλf1xaxRdx,
α=a π2,  R=sin α,Q=tanα2,  Q-1R=-1tan α.
αuxxa=C1 expiπxa2λf1 tan α-+ ux×expiπx2λf1 tan αexp- i2πxaxλf1 sin αdx,
C1=exp-iπ/4signsin α-α/2|λf1 sin α|1/2.
αux-χrxa=exp-i2π χr sin αxa-1/2χr cos αλf1×αuxxa-χr cos α.
Wu,ux, ν=Wux, ν=-+ux+x2u*x-x2×exp-i2πνxdx.
Wau+bvx, ν=|a|2Wux, ν+|b|2Wvx, ν+2×Reab*Wu, vx, ν,
Uν=TFuxν,
WUx, ν=Wu-ν, x.
cx=uxexp-πβx2,
Wcx, ν=Wux, ν+βx.
Wuax, ν=R-αWux, ν,
R-α=cos αsin α-sin αcos α.
cosπx2λz-π4
Sx=exp-iπ/42 S+x+exp+iπ/42 S-x,
S+x=expi πx2λzFx,S-x=exp-i πx2λzFx.
Wsx, ν=¼Ws+x, ν+¼Ws-x, ν+NLx, ν,
NLx, ν=12-+ sinπλz2x2+x2/2Fx+x/2×F*x-x/2exp-i2πνxdx.
Ws+x, ν+Ws-x, ν=WOxλz-ν, x+WO-xλz -ν, x,
WOx, ν=d-|x|sincνd-|x|rectx/d.
WOxλz-ν, x,  WO- xλz-ν, x
αSxxa=C1 expiπxa2λf1 tan α-+cosπx2λz-π4×Fxexpiπx2λf1 tan α×exp- i2πxaxλf1 sin αdx.
αSxxa=Sa+xa+Sa-xa.
Sa+xa=1λzexp-i π4exp-i πxa2λz×θ+Fxsxa,
Sa-xa=1λzexp+i π4exp+i πxa2λz×θ-Fxsxa,
tan θ±=tan α1±f1/ztan α,  s=sin θsin α.
α=αopt=arctanz/f1,
χr=χacos αopt=-27 μm.
R=sinα,  Q=tanα/2,
Z=Rf1,  f=f1/Q.
u1x=uxexp-i πλf1 Qx2,
ũ1ξ=-+u1xexp-i 2πλf1 ξxdx.
ũ2ξ=exp-i πλf1 Rξ2-+uxexp-i πλf1 Qx2×exp-i 2πλf1 ξxdx.
-+exp-α2ξ+β2dξ=π/α,
u2xa=exp-i π4λf1|R|-+uxexp-i πQλf1 x2×expi πλf1x-xa2λf1 Rdx.
Γαuxxa=exp-i π4λf1|R|-+ux×exp-i πλf1Q-1Rxa2+x2×exp-i 2πλf1xaxRdx.

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