Abstract

An optical implementation of the fractional Fourier transform (FRT) with broadband illumination is proposed by use of a single imaging element, namely, a blazed diffractive lens. The setup displays an achromatized version of the FRT of order P of any two-dimensional input function. This fractional order can be tuned continuously by shifting of the input along the optical axis. Our compact and flexible configuration is tested with a chirplike input signal, and the good experimental results obtained support the theory.

© 2000 Optical Society of America

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  1. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 265–342 (1998).
  2. T. Alieva, V. López, F. Agulló-López, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
    [CrossRef]
  3. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  4. B. Lü, F. Kong, B. Zhang, “Optical systems expressed in terms of fractional Fourier transforms,” Opt. Commun. 137, 13–16 (1997).
    [CrossRef]
  5. D. Dragoman, “Fractional Wigner distribution function,” J. Opt. Soc. Am. A 13, 474–478 (1996).
    [CrossRef]
  6. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  7. J. García, D. Mendlovic, Z. Zalevsky, A. W. Lohmann, “Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier-transform filters,” Appl. Opt. 35, 3945–3952 (1996).
    [CrossRef] [PubMed]
  8. S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
    [CrossRef]
  9. Z. Zalevsky, D. Mendlovic, J. H. Caulfield, “Localized, partially space-invariant filtering,” Appl. Opt. 36, 1086–1092 (1997).
    [CrossRef] [PubMed]
  10. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  11. S. Liu, J. Xu, Y. Zhang, L. Chen, C. Li, “General optical implementations of fractional Fourier transforms,” Opt. Lett. 20, 1053–1055 (1995).
    [CrossRef] [PubMed]
  12. A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. 115, 437–443 (1995).
    [CrossRef]
  13. R. G. Dorsch, “Fractional Fourier transformer of variable order based on a modular lens system,” Appl. Opt. 34, 6016–6020 (1995).
    [CrossRef] [PubMed]
  14. P. Andrés, W. D. Furlan, G. Saavedra, A. W. Lohmann, “Variable fractional Fourier processor: a simple implementation,” J. Opt. Soc. Am. A 14, 853–858 (1997).
    [CrossRef]
  15. J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
    [CrossRef]
  16. S. Granieri, W. D. Furlan, G. Saavedra, P. Andrés, “Radon–Wigner display: a compact optical implementation with a single varifocal lens,” Appl. Opt. 36, 8363–8369 (1997).
    [CrossRef]
  17. G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, San Diego, Calif., 1987), pp. 23–71.
  18. J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
    [CrossRef]
  19. E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
    [CrossRef] [PubMed]
  20. V. Namias, “The fractional Fourier transform and its applications to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
    [CrossRef]
  21. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]

1998

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 265–342 (1998).

1997

B. Lü, F. Kong, B. Zhang, “Optical systems expressed in terms of fractional Fourier transforms,” Opt. Commun. 137, 13–16 (1997).
[CrossRef]

Z. Zalevsky, D. Mendlovic, J. H. Caulfield, “Localized, partially space-invariant filtering,” Appl. Opt. 36, 1086–1092 (1997).
[CrossRef] [PubMed]

P. Andrés, W. D. Furlan, G. Saavedra, A. W. Lohmann, “Variable fractional Fourier processor: a simple implementation,” J. Opt. Soc. Am. A 14, 853–858 (1997).
[CrossRef]

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

S. Granieri, W. D. Furlan, G. Saavedra, P. Andrés, “Radon–Wigner display: a compact optical implementation with a single varifocal lens,” Appl. Opt. 36, 8363–8369 (1997).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

1996

1995

1994

T. Alieva, V. López, F. Agulló-López, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

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

1993

1980

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

1937

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Agulló-López, F.

T. Alieva, V. López, F. Agulló-López, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Alieva, T.

T. Alieva, V. López, F. Agulló-López, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Almeida, L. B.

T. Alieva, V. López, F. Agulló-López, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Andrés, P.

Caulfield, J. H.

Chen, L.

Climent, V.

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

Condon, E. U.

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Dorsch, R. G.

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

R. G. Dorsch, “Fractional Fourier transformer of variable order based on a modular lens system,” Appl. Opt. 34, 6016–6020 (1995).
[CrossRef] [PubMed]

Dragoman, D.

Ferreira, C.

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

Furlan, W. D.

García, J.

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

J. García, D. Mendlovic, Z. Zalevsky, A. W. Lohmann, “Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier-transform filters,” Appl. Opt. 35, 3945–3952 (1996).
[CrossRef] [PubMed]

Granieri, S.

S. Granieri, W. D. Furlan, G. Saavedra, P. Andrés, “Radon–Wigner display: a compact optical implementation with a single varifocal lens,” Appl. Opt. 36, 8363–8369 (1997).
[CrossRef]

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

Kong, F.

B. Lü, F. Kong, B. Zhang, “Optical systems expressed in terms of fractional Fourier transforms,” Opt. Commun. 137, 13–16 (1997).
[CrossRef]

Lancis, J.

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

Li, C.

Liu, S.

Lohmann, A. W.

López, V.

T. Alieva, V. López, F. Agulló-López, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Lü, B.

B. Lü, F. Kong, B. Zhang, “Optical systems expressed in terms of fractional Fourier transforms,” Opt. Commun. 137, 13–16 (1997).
[CrossRef]

Mendlovic, D.

Morris, G. M.

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, San Diego, Calif., 1987), pp. 23–71.

Namias, V.

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

Ozaktas, H. M.

Pellat-Finet, P.

Saavedra, G.

Sicre, E. E.

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

Tajahuerce, E.

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

Tepichin, E.

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

Trabocchi, O.

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

Xu, J.

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 265–342 (1998).

Z. Zalevsky, D. Mendlovic, J. H. Caulfield, “Localized, partially space-invariant filtering,” Appl. Opt. 36, 1086–1092 (1997).
[CrossRef] [PubMed]

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

J. García, D. Mendlovic, Z. Zalevsky, A. W. Lohmann, “Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier-transform filters,” Appl. Opt. 35, 3945–3952 (1996).
[CrossRef] [PubMed]

Zhang, B.

B. Lü, F. Kong, B. Zhang, “Optical systems expressed in terms of fractional Fourier transforms,” Opt. Commun. 137, 13–16 (1997).
[CrossRef]

Zhang, Y.

Zweig, D. A.

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, San Diego, Calif., 1987), pp. 23–71.

Appl. Opt.

J. Inst. Math. Its Appl.

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

J. Mod. Opt.

T. Alieva, V. López, F. Agulló-López, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, E. Tepichin, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 (1997).
[CrossRef]

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

B. Lü, F. Kong, B. Zhang, “Optical systems expressed in terms of fractional Fourier transforms,” Opt. Commun. 137, 13–16 (1997).
[CrossRef]

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

Opt. Lett.

Proc. Natl. Acad. Sci. USA

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Prog. Opt.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 265–342 (1998).

Other

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, San Diego, Calif., 1987), pp. 23–71.

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Figures (8)

Fig. 1
Fig. 1

Free-space FRT setup.

Fig. 2
Fig. 2

Relative chromatic errors obtained when broadband illumination is used in the setup shown in Fig. 1: (a) fractional-order error, (b) magnification error. Note that only values P ∈ (0, 1) are represented in (b), since the same curves are obtained by exchange of P by 2 - P. In both plots we choose λ0 = 600 nm.

Fig. 3
Fig. 3

White-light fractional Fourier transformer.

Fig. 4
Fig. 4

Fractional errors affecting each chromatic channel in the optimal design of the setup presented in Fig. 3: (a) fractional-order error, (b) magnification error. As in Fig. 2, only values P ∈ (0, 1) are considered for the magnification error because of the invariance of this function under the change of P by 2 - P. We also choose here λ0 = 600 nm.

Fig. 5
Fig. 5

Input function consisting of a binary grating with linearly increasing spatial frequency (chirplike function).

Fig. 6
Fig. 6

Achromatic FRT obtained by use of the setup in Fig. 3 under white-light illumination: (a) gray-scale display of the irradiance, (b) profile of the irradiance along the horizontal direction for each RGB component of the registered image.

Fig. 7
Fig. 7

Monochromatic FRT obtained by the free-space propagation setup in Fig. 1: (a) gray-scale register of the irradiance, (b) horizontal profile of the image in (a).

Fig. 8
Fig. 8

Same as in Fig. 7 but obtained with polychromatic illumination: (a) gray-scale display of the irradiance, (b) horizontal profile of each RGB component of the registered image.

Equations (33)

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GPw=i expiϕsin ϕexpiπ|w|2tan ϕ -+ gρ×expiπ|ρ|2tan ϕexp-i2πρ·wsin ϕd2ρ,
URr; λ, z=iλRexpiπλR|r|2 -+ tr0×expiπλz+RzR|r0|2×exp-i2πr0·rλRd2r0,
ρ=r0/s,  w=r/s,
UˆRw; λ, z=is2λRexpiπs2λR|w|2 -+ gρ×expiπλz+RzR s2|ρ|2×exp-i2πs2ρ·wλRd2ρ,
UˆRw; λ, z  GPwMP; λ, z.
RP; λ, z=fλtanPπ/21-fλ/ztanPπ/2
MP; λ, z=11-fλ/ztanPπ/2cosPπ/2.
RQ; λ, z=RP; λ0, z.
ΔQλ, P=Qλ, P-PP,
ΔMλ, P=MQλ, P; λ, z-MP; λ0, zMP; λ0, z.
ΔQλ, P=2Pπarctanλλ0tanPπ/2-1,
ΔMλ, P=1+λ/λ02 tan2Pπ/21+tan2Pπ/21/2-1.
Dr=exp-iπ|r|2λ0Z0.
Udr; λ, z, Z0=-+ tr0exp-iπ|r0|2λαz×expi2πr0·rλαdd2r0,
α=z1d-1z-λλ0Z0.
Uˆdw; λ, z, Z0=-+ gρexp-iπs2|ρ|2λαz×expi2πs2ρ·wλαdd2ρ,
Uˆdw; λ, z, Z0  GPwMP; λ, z, Z0,
dP; λ, z, Z0=zZ0Z0+z λλ0-Z0fλztanPπ/2.
MP; λ, z, Z0=dP; λ, z, Z0z cosPπ/2.
dQ; λ, z, Z0=dP; λ0, z, Z0;
tanQπ/2=λλ0-1λλ0z2fλ0Z0+λλ0tanPπ/2.
MQ; λ, z, Z0=dP; λ0, z, Z0z cosQπ/2=MP; λ0, z, Z0×1+tan2Qπ/21+tan2Pπ/21/2.
ddλ Qλ, P, z, Z0λ0=0,
ddλ MQ; λ, z, Z0λ0=0.
z2=-Z0fλ0tanPπ/2.
tanQπ/2=1-λ/λ0-12tanPπ/2,
MQ; λ, z, Z0=MP; λ0, z, Z0×1+1-λ/λ0-122 tan2Pπ/21+tan2Pπ/21/2.
MP; λ0, z, Z0=dP; λ0, z, Z0z1+tan2Pπ/21/2
dP; λ0, z, Z0=z2z-2fλ0tanPπ/2
ΔQλ, P=Qλ, P, z, Z0-PP=2Pπarctan(1-λ/λ0-12×tanPπ/2)-1.
ΔMλ, P=MQλ, P, z, Z0; λ, z, Z0-MP; λ0, z, Z0MP; λ0, z, Z0=1+1-λ/λ0-122 tan2Pπ/21+tan2Pπ/21/2-1.
gρ=n=0 gn cosπnαρx2,if -L<ρx00,otherwise  ,
tanP±nπ/2=±1/nα for all n0/gn0.

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