Abstract

We present experimental results for a fractional Fourier transform (FRFT) system implemented with programmable lenses written onto a liquid-crystal spatial light modulator (LCSLM). Because the focal length can be changed, different orders of the FRFT can be obtained without changing the optical setup. The LCSLM can very easily implement more complicated operations, including the realization of simultaneous orders of the FRFT and anamorphic transforms.

© 2003 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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2000 (1)

1999 (1)

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

1998 (1)

1997 (3)

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. 103, 393–400 (1997).
[CrossRef]

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

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]

1996 (2)

1995 (8)

1994 (1)

D. Mendlovic, H. M. Ozatkas, A. W. Lohmann, “Self-Fourier functions and fractional Fourier transforms,” Opt. Commun. 105, 36–38 (1994).
[CrossRef]

1993 (3)

1992 (1)

1989 (1)

Amako, J.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Andrés, P.

Bernardo, L. M.

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
[CrossRef]

Bitran, Y.

Britan, Y.

Campos, J.

E. Carcolé, M. S. Millán, J. Campos, “Derivation of weighting coefficients for multiplexed phase-diffractive elements,” Opt. Lett. 20, 2360–2362 (1995).
[CrossRef] [PubMed]

I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. Part 1 34, 6423–6432 (1995).
[CrossRef]

Carcolé, E.

Chen, L.

Connely, S. W.

Cottrell, D. M.

Davis, J. A.

Dorsch, R. G.

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. 103, 393–400 (1997).
[CrossRef]

D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. García, H. M. Ozaktas, “Anamorphic fractional Fourier transform: optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995).
[CrossRef] [PubMed]

Furlan, W. D.

García, J.

Gorecki, C.

I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. Part 1 34, 6423–6432 (1995).
[CrossRef]

Guralnik, I. R.

Highnote, S. M.

Kong, F.

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

Li, C.

Lilly, R. A.

Liu, S.

Lohmann, A. W.

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. 103, 393–400 (1997).
[CrossRef]

A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. 119, 275–279 (1995).

D. Mendlovic, Y. Britan, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
[CrossRef]

Y. Bitran, D. Mendlovic, R. G. Dorsch, A. W. Lohmann, H. M. Ozaktas, “Fractional Fourier transform: simulations and experimental results,” Appl. Opt. 34, 1329–1332 (1995).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozatkas, A. W. Lohmann, “Self-Fourier functions and fractional Fourier transforms,” Opt. Commun. 105, 36–38 (1994).
[CrossRef]

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

Loktev, M. Y.

Lü, B.

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

Mas, D.

Mendlovic, D.

Millán, M. S.

Moreno, I.

I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. Part 1 34, 6423–6432 (1995).
[CrossRef]

Naumov, A. F.

Nestorovic, N.

Ozaktas, H. M.

Ozatkas, H. M.

Saavedra, G.

Sicre, E. E.

Sonehara, T.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Tajahuerce, E.

Tsai, P.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Vdovin, G.

Xu, J.

Yzuel, M. J.

I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. Part 1 34, 6423–6432 (1995).
[CrossRef]

Zalevsky, Z.

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. 103, 393–400 (1997).
[CrossRef]

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.

Appl. Opt. (6)

J. Opt. Soc. Am. A (5)

Jpn. J. Appl. Phys. Part 1 (1)

I. Moreno, J. Campos, C. Gorecki, M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. Part 1 34, 6423–6432 (1995).
[CrossRef]

Opt. Commun. (4)

D. Mendlovic, H. M. Ozatkas, A. W. Lohmann, “Self-Fourier functions and fractional Fourier transforms,” Opt. Commun. 105, 36–38 (1994).
[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. 119, 275–279 (1995).

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. 103, 393–400 (1997).
[CrossRef]

Opt. Eng. (2)

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
[CrossRef]

Opt. Lett. (4)

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Figures (7)

Fig. 1
Fig. 1

Lens system to produce the FRFT.

Fig. 2
Fig. 2

Relation between the order n of the FRFT versus the quotient d/ f.

Fig. 3
Fig. 3

Computer simulation results for a FRFT of an amplitude slit. The order of the FRFT is (a) n = 2 (imaging), (b) n = 3/2, (c) n = 5/4, (d) n = 1 (Fourier transform), (e) n = 3/4, (f) n = 1/2.

Fig. 4
Fig. 4

Focusing of the different harmonic lenses for different orders of the FRFT: (a) n = 2 (imaging), (b) n = 1 (Fourier transform), (c) n = 1/2 (d) formation of harmonic lenses for the case n = 1/2.

Fig. 5
Fig. 5

Experimental results with a diffractive lens implemented with a LCSLM. The order of the FRFT is (a) n = 2 (imaging), (b) n = 3/2, (c) n = 5/4, (d) n = 1 (Fourier transform), (e) n = 3/4, (f) n = 1/2.

Fig. 6
Fig. 6

Experimental results with multiplexed FRFTs. (a) n = 2 (imaging) and n = 1 (Fourier transform); (b) n = 3/2 and n = 1/2.

Fig. 7
Fig. 7

Experimental results with anamorphic FRFTs. (a) n x = 1 (Fourier transform) and n y = 2 (imaging); (b) n y = 5/4 and n x = 3/4; (c) n y = 3/4 and n x = 5/4.

Equations (7)

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M=1-dfd2-df-1f1-df.
ux= uxexpi πλx2+x2f1 tan ϕ×exp-i 2πλxxf1 sin ϕdx,
f1=fd1-df1/2, cos ϕ=1-df.
expjψβ=m=-+ cm expjmβ,
cm=12π02πexpjψβ-mβdβ.
βx, y=-πr2/λf,
expψr=m=-+ cm expjmβr=m=-+ cm exp-j πr2λf/m.

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