Abstract

An additional degree of freedom is introduced to fractional-Fourier-transform systems by use of anamorphic optics. A different fractional Fourier order along the orthogonal principal directions is performed. A laboratory experimental system shows preliminary results that demonstrate the proposed theory. Applications such as anamorphic fractional correlation and multiplexing in fractional domains are briefly suggested.

© 1995 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theory IT-6, 386–400 (1960).
    [CrossRef]
  3. T. Szoplik, W. Kosek, C. Ferreira, “Nonsymmetric Fourier transforming with an anamorphic system,” Appl. Opt. 23, 905–909 (1984).
    [CrossRef] [PubMed]
  4. T. Szoplik, H. H. Arsenault, “Rotation-variant optical data processing using the 2-D nonsymmetrical Fourier transform,” Appl. Opt. 24, 168–174 (1985).
    [CrossRef] [PubMed]
  5. P. Andres, C. Ferreira, E. Bonet, “Fraunhofer diffraction patterns from apertures illuminated with nonparallel light in nonsymmetrical Fourier transformers,” Appl. Opt. 24, 1549–1552 (1985).
    [CrossRef] [PubMed]
  6. E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 53, 155–160 (1986).
    [CrossRef]
  7. C. Ferreira, C. Vazquez, “Anamorphic multiple matched filter for character recognition performance with signal of equal size,” J. Mod. Opt. 37, 1343–1354 (1990).
    [CrossRef]
  8. M. S. Millan, C. Ferreira, A. Pons, P. Andres, “Application of anamorphic systems to directional pseusocolor encoding,” Opt. Eng. 27, 129–134 (1988).
  9. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  10. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  11. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  12. 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]
  13. A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
    [CrossRef]
  14. D. Mendlovic, H. M. Ozaktaz, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]

1995 (1)

1994 (2)

1993 (2)

1992 (1)

1990 (1)

C. Ferreira, C. Vazquez, “Anamorphic multiple matched filter for character recognition performance with signal of equal size,” J. Mod. Opt. 37, 1343–1354 (1990).
[CrossRef]

1988 (1)

M. S. Millan, C. Ferreira, A. Pons, P. Andres, “Application of anamorphic systems to directional pseusocolor encoding,” Opt. Eng. 27, 129–134 (1988).

1986 (1)

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 53, 155–160 (1986).
[CrossRef]

1985 (2)

1984 (1)

1960 (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theory IT-6, 386–400 (1960).
[CrossRef]

Andres, P.

M. S. Millan, C. Ferreira, A. Pons, P. Andres, “Application of anamorphic systems to directional pseusocolor encoding,” Opt. Eng. 27, 129–134 (1988).

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 53, 155–160 (1986).
[CrossRef]

P. Andres, C. Ferreira, E. Bonet, “Fraunhofer diffraction patterns from apertures illuminated with nonparallel light in nonsymmetrical Fourier transformers,” Appl. Opt. 24, 1549–1552 (1985).
[CrossRef] [PubMed]

Arsenault, H. H.

Barshan, B.

Bonet, E.

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 53, 155–160 (1986).
[CrossRef]

P. Andres, C. Ferreira, E. Bonet, “Fraunhofer diffraction patterns from apertures illuminated with nonparallel light in nonsymmetrical Fourier transformers,” Appl. Opt. 24, 1549–1552 (1985).
[CrossRef] [PubMed]

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theory IT-6, 386–400 (1960).
[CrossRef]

Ferreira, C.

C. Ferreira, C. Vazquez, “Anamorphic multiple matched filter for character recognition performance with signal of equal size,” J. Mod. Opt. 37, 1343–1354 (1990).
[CrossRef]

M. S. Millan, C. Ferreira, A. Pons, P. Andres, “Application of anamorphic systems to directional pseusocolor encoding,” Opt. Eng. 27, 129–134 (1988).

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 53, 155–160 (1986).
[CrossRef]

P. Andres, C. Ferreira, E. Bonet, “Fraunhofer diffraction patterns from apertures illuminated with nonparallel light in nonsymmetrical Fourier transformers,” Appl. Opt. 24, 1549–1552 (1985).
[CrossRef] [PubMed]

T. Szoplik, W. Kosek, C. Ferreira, “Nonsymmetric Fourier transforming with an anamorphic system,” Appl. Opt. 23, 905–909 (1984).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Kosek, W.

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theory IT-6, 386–400 (1960).
[CrossRef]

Lohmann, A. W.

Mendlovic, D.

Millan, M. S.

M. S. Millan, C. Ferreira, A. Pons, P. Andres, “Application of anamorphic systems to directional pseusocolor encoding,” Opt. Eng. 27, 129–134 (1988).

Onural, L.

Ozaktas, H. M.

Ozaktaz, H. M.

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theory IT-6, 386–400 (1960).
[CrossRef]

Pons, A.

M. S. Millan, C. Ferreira, A. Pons, P. Andres, “Application of anamorphic systems to directional pseusocolor encoding,” Opt. Eng. 27, 129–134 (1988).

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 53, 155–160 (1986).
[CrossRef]

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theory IT-6, 386–400 (1960).
[CrossRef]

Szoplik, T.

Vazquez, C.

C. Ferreira, C. Vazquez, “Anamorphic multiple matched filter for character recognition performance with signal of equal size,” J. Mod. Opt. 37, 1343–1354 (1990).
[CrossRef]

Appl. Opt. (5)

IRE Trans. Inf. Theory (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theory IT-6, 386–400 (1960).
[CrossRef]

J. Mod. Opt. (1)

C. Ferreira, C. Vazquez, “Anamorphic multiple matched filter for character recognition performance with signal of equal size,” J. Mod. Opt. 37, 1343–1354 (1990).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 53, 155–160 (1986).
[CrossRef]

Opt. Eng. (1)

M. S. Millan, C. Ferreira, A. Pons, P. Andres, “Application of anamorphic systems to directional pseusocolor encoding,” Opt. Eng. 27, 129–134 (1988).

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (6)

Fig. 1
Fig. 1

Optical setup for performing a FRT.

Fig. 2
Fig. 2

Anamorphic optical setup for performing FRT’s of different orders in the x and the y axes.

Fig. 3
Fig. 3

(a) Input pattern used in experiments, (b) anamorphic FRT of orders P x = 0.667 and P y = 0 calculated digitally, and (c) experimental optical result.

Fig. 4
Fig. 4

Block diagram showing the fractional correlation procedure.

Fig. 5
Fig. 5

Space and frequency multiplexing of the signals when the main axes of their domains are aligned with the (x, ν) axes.

Fig. 6
Fig. 6

Inefficient multiplexing of signals with the domains not aligned with the (x, ν) axes.

Equations (16)

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ϕ = P π 2
f = f 1 Q ,
Z = R f 1 ,
R = tan ( ϕ 2 ) ,
Q = sin ( ϕ ) ,
u ( x 1 ) = F P { u ( x ) } = u ( x ) exp ( i π x 1 2 + x 2 T ) exp ( - i 2 π x x 1 S ) d x ,
S = λ f 1 sin ( ϕ ) ,
T = λ f 1 tan ( ϕ ) ,
P y < P x .
ϕ = ( P x - P y ) π / 2.
4 f y = R x f 1 = f 1 tan ( ϕ 2 ) .
f x = f 1 / sin ( ϕ ) ,
Z x = R x f 1 .
u ( x 2 , y 2 ) = F x P x F y P y { u ( x , y ) } = u ( x , y ) exp [ ( i π x 2 2 + x 2 T x ) + ( i π y 2 2 + y 2 T y ) ] × exp ( - i 2 π x x 2 S x - i 2 π y y 2 S y ) d x d y ,
ϕ x = P x π / 2 ,
ϕ y = P y π / 2.

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