Abstract

The scaled fractional Fourier transform is suggested and is implemented optically by one lens for different values of ϕ and output scale. In addition, physically it relates the FRT with the general lens transform—the optical diffraction between two asymmetrically positioned planes before and after a lens.

© 1997 Optical Society of America

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References

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  1. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
    [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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  14. A. W. Lohmann, D. Mendlovic, “Fractional Fourier transform: photonic implementation,” Appl. Opt. 33, 7661–7664 (1994).
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  15. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
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  16. L. Bernardo, O. D. D. Soares, “Fractional Fourier transform and optical system,” Opt. Commun. 110, 517–522 (1994).
    [Crossref]
  17. C.-C. Shih, “Optical interpretation of a complex-order Fourier transform,” Opt. Lett. 20, 1178–1180 (1995).
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  18. L. Bernardo, O. D. D. Soares, “Optical fractional Fourier transforms with complex orders,” Appl. Opt. 35, 3163–3166 (1996).
    [Crossref] [PubMed]
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1997 (1)

1996 (3)

1995 (5)

1994 (7)

1993 (3)

1987 (1)

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

1980 (1)

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

Alieva, T.

Barshan, B.

H. M. Ozaktas, B. Barshan, D. Mendlovic, H. Urey, “Space-variant filtering in fractional Fourier domains,” Inst. Phys. Conf. Ser. 139, 285–288 (1994).

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

Bernardo, L.

L. Bernardo, O. D. D. Soares, “Optical fractional Fourier transforms with complex orders,” Appl. Opt. 35, 3163–3166 (1996).
[Crossref] [PubMed]

L. Bernardo, O. D. D. Soares, “Fractional Fourier transform and optical system,” Opt. Commun. 110, 517–522 (1994).
[Crossref]

Bernardo, L. M.

Bitran, Y.

Chen, L.

Dorsch, G.

Dorsch, R. G.

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 22, 842–844 (1996).
[Crossref]

Granieri, S.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Frenel domain,” Opt. Commun. 119, 275–278 (1995).
[Crossref]

Hua, J.

Kerr, F. H.

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

Li, C.

Li, G.

Liu, L.

Liu, S.

Lohmann, A. W.

McBride, A. C.

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

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]

Ozaktas, H. M.

Pellat-Finet, P.

Shih, C.-C.

Sicre, E. E.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Frenel domain,” Opt. Commun. 119, 275–278 (1995).
[Crossref]

Soares, O. D.

Soares, O. D. D.

L. Bernardo, O. D. D. Soares, “Optical fractional Fourier transforms with complex orders,” Appl. Opt. 35, 3163–3166 (1996).
[Crossref] [PubMed]

L. Bernardo, O. D. D. Soares, “Fractional Fourier transform and optical system,” Opt. Commun. 110, 517–522 (1994).
[Crossref]

Trabocchi, O.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Frenel domain,” Opt. Commun. 119, 275–278 (1995).
[Crossref]

Urey, H.

H. M. Ozaktas, B. Barshan, D. Mendlovic, H. Urey, “Space-variant filtering in fractional Fourier domains,” Inst. Phys. Conf. Ser. 139, 285–288 (1994).

Xu, J.

Zalevsky, Z.

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 22, 842–844 (1996).
[Crossref]

Zhang, Y.

Appl. Opt. (6)

IMA J. Appl. Math. (1)

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

Inst. Phys. Conf. Ser. (1)

H. M. Ozaktas, B. Barshan, D. Mendlovic, H. Urey, “Space-variant filtering in fractional Fourier domains,” Inst. Phys. Conf. Ser. 139, 285–288 (1994).

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 (5)

Opt. Commun. (2)

L. Bernardo, O. D. D. Soares, “Fractional Fourier transform and optical system,” Opt. Commun. 110, 517–522 (1994).
[Crossref]

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Frenel domain,” Opt. Commun. 119, 275–278 (1995).
[Crossref]

Opt. Lett. (5)

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Figures (2)

Fig. 1
Fig. 1

Geometry of general lens transform.

Fig. 2
Fig. 2

Function and its scaled FRT: a, function to be transformed; b, optical experimental result of its scaled FRT. In the experiment, ϕ = 38°, M = 0.79, q = 1.2, f = 3900, z 1 = 978, and z 2 = 667.

Equations (11)

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Fpux=-+ uxexpiπλf1x2+y2tan ϕ-2xysin ϕdx,
Fϕux=-+uxexpiπx2+My2tan ϕ-2xMysin ϕdx,
vy=-+expiπλXL-y2z2-+ux/q×expiπλx-xL2z1dxexp-iπλxL2fdxL =-+ ux/qexpiπλx2z1+y2z2×exp-iπλx/z1+y/z221/z1+1/z2-1/fIdx
I=-+expiπλxL1z1+1z2-1f1/2-x/z1+y/z21/z1+1/z2-1/f1/22dxL.
vy=-+ ux/qexpiπλx2z1+y2z2×exp-iπλx/z1+y/z221/z1+1/z2-1/fdx=-+ ux/q×expiπλf-z2x2+f-z1y2-2fxyz1f+z2f-z1z2dx.
x/q=t, cos ϕ=f-z11/2f-z21/2f,
M=f-z11/2λf-z21/2z1f+z2f-z1z21/21/2,
q=λf-z11/2z1f+z2f-z1z21/2f-z21/21/2=1Mf-z11/2f-z21/2,
vy=-+ utexpiπt2+My2tan ϕ-2tMysin ϕdt,
z1=f-λM2f2 sin ϕcos ϕ,
z2=f-1λM2 tan ϕ.

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