Abstract

Based on an all-optical system, a display of a fractional Fourier transform with many fractional orders is proposed. Because digital image-processing terminology is used, this display is known as the Radon–Wigner transform. It enables new aspects for signal analysis that are related to time- and spatial-frequency analyses. The given approach for producing this display starts with a one-dimensional input signal although the output signal contains two dimensions. The optical setup for obtaining the fractional Fourier transform was adapted to include only fixed free-space propagation distances and variable lenses. With a set of two multifacet composite holograms, the Radon–Wigner display has been demonstrated experimentally.

© 1996 Optical Society of America

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References

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  1. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: Part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  2. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: Part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  3. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  4. 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]
  5. J. C. Wood, D. T. Barry, “Radon transform of the Wigner spectrum,” in Advanced Signal Processing Algorithms, Architectures, and Implementations III, F. T. Luk, ed., Proc. SPIE1770, 358–375 (1992).
    [CrossRef]
  6. J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
    [CrossRef]
  7. J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
    [CrossRef]
  8. A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
    [CrossRef]
  9. D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
    [CrossRef]
  10. A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. (to be published).
  11. H. M. Ozaktas, D. Mendlovic, “Multistage optical implementation architecture with least possible growth of system size,” Opt. Lett. 18, 296–298 (1993).
    [CrossRef] [PubMed]
  12. W. H. Lee, “Binary synthetic holograms,” Appl. Opt. 13, 1677–1682 (1974).
    [CrossRef] [PubMed]

1995 (1)

1994 (4)

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]

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
[CrossRef]

1993 (4)

1974 (1)

Barry, D. T.

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Radon transform of the Wigner spectrum,” in Advanced Signal Processing Algorithms, Architectures, and Implementations III, F. T. Luk, ed., Proc. SPIE1770, 358–375 (1992).
[CrossRef]

Bitran, Y.

Dorsch, R. G.

Lee, W. H.

Lohmann, A. W.

Mendlovic, D.

Ozaktas, H.

Ozaktas, H. M.

Soffer, B. H.

Wood, J. C.

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Radon transform of the Wigner spectrum,” in Advanced Signal Processing Algorithms, Architectures, and Implementations III, F. T. Luk, ed., Proc. SPIE1770, 358–375 (1992).
[CrossRef]

Zalevsky, Z.

Appl. Opt. (2)

IEEE Trans. Signal Process. (2)

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

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

Opt. Lett. (1)

Other (2)

J. C. Wood, D. T. Barry, “Radon transform of the Wigner spectrum,” in Advanced Signal Processing Algorithms, Architectures, and Implementations III, F. T. Luk, ed., Proc. SPIE1770, 358–375 (1992).
[CrossRef]

A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. (to be published).

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Figures (5)

Fig. 1
Fig. 1

Suggested optical setup for obtaining the (x, p) display.

Fig. 2
Fig. 2

Optical configurations: (a), and (b) Totally equivalent setups, (c) configuration equivalent to the free-space propagation of distance z, and (d) setup yielding the FRT with constant distances and varying focal lengths.

Fig. 3
Fig. 3

Experimental results for the input of a Ronchi grating of (a) 200 lines/cm, (b) 100 lines/cm, and (c) 50 lines/cm.

Fig. 4
Fig. 4

Experimental results of the input of a chirp with the constants (a) 1.5 m, and (b) 2.5 m.

Fig. 5
Fig. 5

Experimental results of the input of a plane wave.

Equations (27)

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u p ( x ) = C 1 u ( x 0 ) exp [ i π λ f 1 tan ϕ ( x 0 2 + x 2 ) ] × exp ( 2 π i λ f 1 sin ϕ x x 0 ) d x 0 , ϕ = p π 2 ,
C 1 = exp { i [ π sgn ( sin ϕ ) 4 ϕ 2 ] } | sin ϕ | 1 / 2 .
f = f 1 tan ϕ 2 , z = f 1 sin ϕ = R f 1 .
F ( x , p ) = u p ( x ) .
f = f 1 R = f 1 sin ϕ .
u ( x ) u ( x ) , W ( x , ξ ) W ( x , ξ ) ,
[ x ξ ]
[ x ξ ]
[ 1 0 0 1 ] .
[ 1 0 0 1 ] [ x ξ ] = [ x ξ ] .
[ 0 1 1 0 ] .
R f 1 [ exp ( i π x 2 R λ f 1 ) ] ,
[ 1 0 R 1 ] .
[ 1 R 0 1 ] .
[ 0 1 1 0 ] [ 1 0 R 1 ] [ 0 1 1 0 ] = [ 0 R 0 1 ] [ 1 0 0 1 ] .
f T = f 1 2 + sin ϕ .
f a = f 1 tan ϕ 2 + 1 ,
f b = f 1 sin ϕ + 2 ,
f c = f a .
t ( x , y ) = exp ( 2 π i α x ) exp ( π i x 2 λ f ) exp ( π i y 2 λ Z R ) .
exp ( π i x 2 λ f )
max | θ x | < 2 πα 2 ,
α > | x max λ f min | ,
exp ( π i y 2 λ Z R )
Z R = π w 2 2 λ ,
input = exp ( i x 2 2 f 2 )
p = 2 π tan 1 ( 2 π f 2 λ f 1 ) ,

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