Abstract

The fractional Fourier transform (FRT) is becoming important in optics and can be used as a new tool to analyze many optical problems. However, we point out that the FRT might be much more sensitive to parameters than the conventional Fourier transform. This sensitivity leads to higher requirements on the optical implementation. On the other hand, high parametric sensitivity can be used in optical diffraction measurements. We give the first proposal, to our knowledge, of the FRT’s applications in optical measurement.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
    [CrossRef]
  2. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  3. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  4. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  5. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  6. 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]
  7. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded index fibers, Wigner-distribution functions, and the fractional Fourier transforms,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  8. Z. Jiang, “Scaling law and simultaneous optical implementation of various-order fractional Fourier transforms,” Opt. Lett. 20, 2408–2410 (1995).
    [CrossRef]
  9. Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769–770 (1994).
    [CrossRef] [PubMed]
  10. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  11. L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
    [CrossRef]
  12. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
    [CrossRef] [PubMed]
  13. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
    [CrossRef]
  14. P. Pallat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef]
  15. P. Pallat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
    [CrossRef]
  16. A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. 115, 437–443 (1995).
    [CrossRef]
  17. H. M. Ozaktas, D. Mendolovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  18. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
    [CrossRef] [PubMed]
  19. R. G. Dorsch, A. W. Lohmann, “Fractional Fourier transform used for a lens-design problem,” Appl. Opt. 34, 4111–4112 (1995).
    [CrossRef] [PubMed]
  20. Y. Bitran, Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Fractional correlation operation: performance analysis,” Appl. Opt. 35, 297–303 (1996).
    [CrossRef] [PubMed]
  21. X. Deng, Y. Li, Y. Qiu, D. Fan, “Diffraction interpreted through fractional Fourier transforms,” Opt. Commun. 131, 241–245 (1996).
    [CrossRef]
  22. Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef] [PubMed]
  23. Z. Zalevsky, D. Mendlovic, “Fractional Radon transform: definition,” Appl. Opt. 35, 4628–4631 (1996).
    [CrossRef] [PubMed]
  24. D. Dragoman, “Fractional Wigner distribution function,” J. Opt. Soc. Am. A 13, 474–478 (1996).
    [CrossRef]
  25. Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
    [CrossRef] [PubMed]
  26. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]

1996 (6)

1995 (5)

1994 (9)

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
[CrossRef]

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]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded index fibers, Wigner-distribution functions, and the fractional Fourier transforms,” Appl. Opt. 33, 6188–6193 (1994).
[CrossRef] [PubMed]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769–770 (1994).
[CrossRef] [PubMed]

P. Pallat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
[CrossRef] [PubMed]

P. Pallat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

1993 (4)

1980 (1)

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

1970 (1)

Barshan, B.

Bernardo, L. M.

Bitran, Y.

Bonnet, G.

P. Pallat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Collins, S. A.

Deng, X.

X. Deng, Y. Li, Y. Qiu, D. Fan, “Diffraction interpreted through fractional Fourier transforms,” Opt. Commun. 131, 241–245 (1996).
[CrossRef]

Dorsch, R. G.

Dragoman, D.

Fan, D.

X. Deng, Y. Li, Y. Qiu, D. Fan, “Diffraction interpreted through fractional Fourier transforms,” Opt. Commun. 131, 241–245 (1996).
[CrossRef]

Jiang, Z.

Karasik, Y. B.

Li, Y.

X. Deng, Y. Li, Y. Qiu, D. Fan, “Diffraction interpreted through fractional Fourier transforms,” Opt. Commun. 131, 241–245 (1996).
[CrossRef]

Lohmann, A. W.

Mendlovic, D.

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

Z. Zalevsky, D. Mendlovic, “Fractional Radon transform: definition,” Appl. Opt. 35, 4628–4631 (1996).
[CrossRef] [PubMed]

Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
[CrossRef] [PubMed]

Y. Bitran, Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Fractional correlation operation: performance analysis,” Appl. Opt. 35, 297–303 (1996).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
[CrossRef] [PubMed]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

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]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded index fibers, Wigner-distribution functions, and the fractional Fourier transforms,” Appl. Opt. 33, 6188–6193 (1994).
[CrossRef] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

Mendolovic, D.

Namias, V.

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

Onural, L.

Ozaktas, H. M.

H. M. Ozaktas, D. Mendolovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
[CrossRef] [PubMed]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

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]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded index fibers, Wigner-distribution functions, and the fractional Fourier transforms,” Appl. Opt. 33, 6188–6193 (1994).
[CrossRef] [PubMed]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 11, 7599–7602 (1994).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

Pallat-Finet, P.

P. Pallat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

P. Pallat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef]

Qiu, Y.

X. Deng, Y. Li, Y. Qiu, D. Fan, “Diffraction interpreted through fractional Fourier transforms,” Opt. Commun. 131, 241–245 (1996).
[CrossRef]

Soares, O. D. D.

Zalevsky, Z.

Appl. Opt. (8)

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

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Commun. (4)

X. Deng, Y. Li, Y. Qiu, D. Fan, “Diffraction interpreted through fractional Fourier transforms,” Opt. Commun. 131, 241–245 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

P. Pallat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. 115, 437–443 (1995).
[CrossRef]

Opt. Lett. (5)

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Optical setup for a (1/Q)th-order two-dimensional FRT. For the one-dimensional FRT, the lens should be replaced by a cylindrical lens. Parameters: l = f tan (π/4Q) and f1/Q = f/sin (π/2Q).

Fig. 2
Fig. 2

Dependence of the FRT on the parametric errors: w′ = 1.0, Q = 2.0. Curve a: Δ = Δ1 = Δ2 = 0; curve b: Δ = 0.05, Δ1 = Δ2 = 0; curve c: Δ1 = 0.05, Δ = Δ2 = 0; curve d: Δ2 = 0.05, Δ = Δ1 = 0. The transversal coordinate is x 2′; the vertical axis is the normalized intensity.

Fig. 3
Fig. 3

Same as for Fig. 2 but with w′ = 1.6. According to the scaling law, the change of w′ can also be considered to be the change of the FRT order.8

Fig. 4
Fig. 4

Same as for Fig. 2 but with Q = 1.0 (i.e., the conventional Fourier transform). Curves a, c, and d are practically indistinguishable.

Fig. 5
Fig. 5

Differences between curves a and b in Figs. 24.

Equations (8)

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

l=f tanπ/4Q, f1/Q=f/sinπ/2Q,
ABCD=cosπ/2Qf sinπ/2Q-sinπ/2Q/fcosπ/2Q.
E2x2=1λB-E1x1×expiπλBAx12-2x1x2+Dx22dx1=1λf sin ϕ1/2- E1x1expiπλfx12 cot ϕ-2x1x2 csc ϕ+x22 cot ϕdx1,
x1=x1/s, x2=x2/s,
E2x2/s=1sin1/2 ϕ-E1x1/sexpiπx12 cot ϕ-2x1 x2 csc ϕ+x22 cot ϕdx1.
ABCD=1l1+Δ20110-1/f1/Q11l1+Δ101=cos ϕ-2Δ2 sin2ϕ/2f sin ϕ+Δ1+Δ2f cos ϕ-sin ϕ/fcos ϕ-2Δ1 sin2ϕ/2.
I2x2/s=E2x2/s2=1sin ϕ+Δ1+Δ2cos ϕ×- E1x1/sexpiπsin ϕ+Δ1+Δ2cos ϕ×x12cos ϕ-2Δ1 sin2ϕ/2-2x1 x2dx12.
I2x2/s=E2x2/s2=1sin ϕ+Δ1+Δ2cos ϕ×-1+Δw1+Δw E1x1/s×expiπsin ϕ+Δ1+Δ2cos ϕ×x12cos ϕ-2Δ1 sin2ϕ/2-2x1 x2dx12.

Metrics