Abstract

A new set of optical implementations of the fractional Fourier transform (FRT) is developed by use of Wigner matrix algebra. The reinterpretation of some elementary operations that synthesize a rotation in the phase-space domain allows us to propose a lensless setup for obtaining the FRT. This compact configuration is also very flexible, because the fractional degree of the transformation can be varied continuously by shifting the input and the output planes along the optical axis by proper amounts. The above results permit one to build an optical FRT processor formed by two FRT systems in cascade, with a spatial filter between them. We present the design of such a variable FRT processor, which contains only one lens.

© 1997 Optical Society of America

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Corrections

Pedro Andrés, Walter D. Furlan, Genaro Saavedra, and Adolf W. Lohmann, "Variable fractional Fourier processor: a simple implementation: erratum," J. Opt. Soc. Am. A 14, 3432-3432 (1997)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-14-12-3432

References

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  1. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  2. 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]
  3. D. Mendlovic, Y. Bitran, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1677–1681 (1995).
    [CrossRef]
  4. S. Liu, J. Wu, C. Li, “Cascading the multiple stages of optical fractional Fourier transforms under different variable scales,” Opt. Lett. 20, 1415–1417 (1995).
    [CrossRef] [PubMed]
  5. A. W. Lohmann, “Image rotation, Wigner rotation and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  6. L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transform and optical systems,” Opt. Commun. 110, 517–522 (1994).
    [CrossRef]
  7. R. G. Dorsch, “Fractional Fourier transformer of variable order based on a modular lens system,” Appl. Opt. 34, 6016–6020 (1995).
    [CrossRef] [PubMed]
  8. S. Liu, J. Xu, Z. Zhang, L. Chen, C. Li, “General optical implementations of fractional Fourier transforms,” Opt. Lett. 20, 1053–1055 (1995).
    [CrossRef] [PubMed]
  9. A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. 115, 437–443 (1995).
    [CrossRef]
  10. S. Granieri, O. Trabochi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–279 (1995).
    [CrossRef]
  11. P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. 59, 1314–1321 (1969).
    [CrossRef]
  12. 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]

1995 (6)

1994 (3)

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]

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

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]

1993 (2)

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

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

1969 (1)

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]

Bernardo, L. M.

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

Bitran, Y.

Chen, L.

Dorsch, R. G.

Granieri, S.

S. Granieri, O. Trabochi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–279 (1995).
[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]

Mendlovic, D.

Ozaktas, H. M.

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

Sicre, E. E.

S. Granieri, O. Trabochi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–279 (1995).
[CrossRef]

Soares, O. D. D.

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

Stokseth, P. A.

Trabochi, O.

S. Granieri, O. Trabochi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–279 (1995).
[CrossRef]

Wu, J.

Xu, J.

Zhang, Z.

Appl. Opt. (2)

J. Mod. Opt. (1)

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. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (4)

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

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

S. Granieri, O. Trabochi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–279 (1995).
[CrossRef]

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

Opt. Lett. (2)

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic Wigner representation of the input signal, (b) after a magnification M, (c) after the action of a lens of focal power Q/f1, and (d) after free propagation over a distance Rf1.

Fig. 2
Fig. 2

Fractional Fourier transform systems: (a) Lohmann’s setup of type II, (b) lensless configuration.

Fig. 3
Fig. 3

Two configurations for a lensless FRT processor: (a) proper scale at the output plane, wrong scale at the filtering plane; (b) proper scale at the filtering plane, wrong scale at the output plane.

Fig. 4
Fig. 4

Single-lens FRT processor.

Tables (1)

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Table 1 All Complete Triplets Involving the Magnification Matrix M

Equations (41)

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uP(x)=FP[u0(x0)]=Cu0(x0)exp{iπ/[λf1 tan(Pπ/2)](x2+x02)}×exp{-2πi/[λf1 sin(Pπ/2)]xx0}dx0,
1/C2=λf1|sin(Pπ/2)|.
F1-R01,L10Q1.
M1/M00M,
1-R01 10Q1=1-RQ-RQ1,
ROTcos ϕ-sin ϕsin ϕcos ϕ.
1-RQ/M-R/MQMM,
M=cos ϕ,
Q=tan ϕ,
R=sin ϕ cos ϕ.
f=f1/tan ϕ,
d=Rf1=f1 sin ϕ cos ϕ,
d=f1 sin ϕ,
f=f1tan(ϕ/2).
zP=-f=-f1tan(Pπ/4),
R(P; zP)=d=f1 sin(Pπ/2).
U[x; zP, R(P; zP)]=exp[-ik/(2zP)x2]FP[t(x)].
FP[t(x)]=exp [ik/(2zP)x2]U[x; zP, R(P; zP)].
1z+1R(P, z)=1zP+1R(P; zP),
M(P; z)=R(P; z)R(P; zP).
R(P; z)=f1 tan(Pπ/2)1-f1/z tan(Pπ/2),
M(P; z)=1+tan(Pπ/2)tan(Pπ/4)1-f1/z tan(Pπ/2).
R(P+Q; zP)
=f1 tan[(P+Q)π/2]1+tan[(P+Q)π/2]tan(Pπ/4),
M(P+Q; zP)
=1+tan[(P+Q)π/2]tan[(P+Q)π/4]1+tan[(P+Q)π/2]tan(Pπ/4).
R(P+Q; zP)=R(P; zP)+R(Q;-zP),
M(P+Q; zP)=M(Q;-zP).
ML=±1M(P+Q; zP).
ML=-fa-f,
a=[1±M(P+Q; zP)]f,
a=[M(P+Q; zP)±1]M(P+Q; zP)f,
a+R(P+Q; zP)-R(P; zP)>0,
a>0.
[1±M(Q;-zP)]f+R(Q;-zP)>0,
M(Q;-zP)±1M(Q;-zP)f>0.
R(1-P;-zP)=f1 cos(Pπ/2)tan (Pπ/4),
M(1-P;-zP)=1tan(Pπ/4).
R(2-P;-zP)=-2f1cotan(Pπ/2)+tan(Pπ/2),
M(2-P;-zP)=-1.
ff1sin(Pπ/2)2.

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