Abstract

The fractional Fourier transform is a mathematical operation that generalizes the well-known Fourier transform. This operation has been shown to have physical and optical fundamental meanings, and it has been experimentally implemented by relatively simple optical setups. Based on the fractional Fourier-transform operation, a new space-frequency chart definition is introduced. By the application of various geometric operations on this new chart, such as radial and angular shearing and rotation, optical systems may be designed or analyzed. The field distribution, as well as full information about the spectrum and the space–bandwidth product, can be easily obtained in all the stages of the optical system.

© 1995 Optical Society of America

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References

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  1. E. Wigner, “On the quantum correction for thermodynamics equilibrium,” Phys. Rev. 40, 749 (1932).
    [CrossRef]
  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. Soc. Photo-Opt. Instrum. Eng.1770, 358–375 (1992).
    [CrossRef]
  3. 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]
  4. J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Acoust. Speech Signal Process. 42, 2105–2111 (1994).
    [CrossRef]
  5. 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. Acoust. Speech Signal Process. 42, 2094–2104 (1994).
    [CrossRef]
  6. 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]
  7. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  8. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  9. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional orders and their optical interpretation,” Opt. Commun. 101, 163–165 (1993).
    [CrossRef]
  10. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  11. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,”Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  12. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domain and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]

1994 (5)

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]

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Acoust. Speech 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. Acoust. Speech Signal Process. 42, 2094–2104 (1994).
[CrossRef]

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]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domain and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

1993 (4)

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,”Opt. Commun. 25, 26–30 (1978).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamics equilibrium,” Phys. Rev. 40, 749 (1932).
[CrossRef]

Barry, D. T.

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Acoust. Speech 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. Acoust. Speech 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. Soc. Photo-Opt. Instrum. Eng.1770, 358–375 (1992).
[CrossRef]

Barshan, B.

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,”Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Lohmann, A. W.

Mendlovic, D.

Onural, L.

Ozaktas, H. M.

Soffer, B. H.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamics equilibrium,” Phys. Rev. 40, 749 (1932).
[CrossRef]

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. Acoust. Speech Signal Process. 42, 2094–2104 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Acoust. Speech 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. Soc. Photo-Opt. Instrum. Eng.1770, 358–375 (1992).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (2)

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Acoust. Speech 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. Acoust. Speech Signal Process. 42, 2094–2104 (1994).
[CrossRef]

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

Opt. Commun. (2)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,”Opt. Commun. 25, 26–30 (1978).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional orders and their optical interpretation,” Opt. Commun. 101, 163–165 (1993).
[CrossRef]

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamics equilibrium,” Phys. Rev. 40, 749 (1932).
[CrossRef]

Other (1)

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. Soc. Photo-Opt. Instrum. Eng.1770, 358–375 (1992).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic illustration of the Yω representation.

Fig. 2
Fig. 2

Schematic illustration of X- and Y-shearing.

Fig. 3
Fig. 3

Illustration of the (x, p) chart.

Fig. 4
Fig. 4

Illustration of the (r, p) chart.

Fig. 5
Fig. 5

Sketch of the interesting area in the (r, p) chart.

Fig. 6
Fig. 6

Schematic illustration of the shear operation in polar coordinates.

Fig. 7
Fig. 7

(a) (r, p) chart of a square, (b) its X-shearing transformation, (c) its radial-shearing transformation.

Fig. 8
Fig. 8

Same as Fig. 7 but with an (r, p) chart of a circle.

Equations (45)

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W [ f ( x ) ] = W ( x , ν ) = f ( x + x 2 ) f * ( x x 2 ) × exp ( 2 π i ν x ) d x .
f ( x ) = 1 f * ( 0 ) W ( x 2 , ν ) exp ( 2 π i ν x ) d ν .
{ p [ u ( x ) ] } ( x ) = B P ( x , x ) u ( x ) d x ,
B P ( x , x ) = 2 exp [ π ( x 2 + x 2 ) ] n = 0 i P n 2 n n ! × H n ( 2 π x ) H n ( 2 π x ) ,
B p ( x , x ) = exp { i [ π sgn ( sin ϕ ) 4 ϕ 2 ] } | sin ϕ | 1 / 2 × exp ( i π x 2 + x 2 tan ϕ 2 i π x x sin ϕ ) .
u P ( x ) = C 1 u ( x 0 ) exp [ i π tan ϕ ( x 0 2 + x 2 ) ] × exp ( 2 π i 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 ( x , p ) = u P ( x ) .
F ( r , p ) = u P ( r ) .
u p + 2 ( r ) = u p ( r ) .
u P ( r ) = F ( r , p ) ,
u 0 ( r ) = F ( r , 0 ) .
u ( x 0 ) W [ u ( x 0 ) ] = W ( x , ν ) Rot [ W ( x , ν ) ] Inverse Winger = u P ( x ) .
u ( x 0 ) W [ ( x , ν ) ] X shear [ W ( x , ν ) ] Y shear [ W ( x , ν ) ] X shear W [ ( x , ν ) ] Inverse Wigner = u P ( x ) ,
X shear [ f ( x , y ) ] = f ( x + α y , y ) , Y shear [ f ( x , y ) ] = f ( x , y + α x ) .
F ( r , p ) = u p ( r ) = C 1 u ( x 0 ) exp ( π i r 2 + x 0 2 tan ϕ ) × exp ( 2 π i r x 0 sin ϕ ) d x 0 .
F q ( r , p ) = ( u q ) p ( r ) = u q + p ( r ) = F ( r , p + q ) .
u ( x 0 ) F ( r , p ) Rot [ F ( r , p ) ] Inverse ( r , p ) chart u P ( x ) .
α = 1 λ f ,
F ( lens ) ( r , p ) = C 1 u 0 ( x 0 ) exp ( i α π x 0 2 ) × exp ( i π x 0 2 + r 2 tan ϕ ) exp ( i 2 π r x 0 sin ϕ ) d x 0 = C 1 exp ( i π r 2 tan ϕ ) exp [ i π x 0 2 ( 1 tan ϕ + α ) ] × exp ( i 2 π r x 0 sin ϕ ) d x 0 .
β = 1 tan ϕ + α = 1 tan θ .
1 sin θ = β 2 + 1 .
s = sin θ sin ϕ ,
F ( lens ) ( r , p ) = C 1 exp ( i π r 2 tan ϕ ) u 0 ( x 0 ) × exp ( i π x 0 2 tan θ ) exp ( i 2 π x 0 r s sin θ ) d x 0 = ψ u θ ( r s ) ,
tan θ = tan ϕ 1 + α tan ϕ , s = 1 sin ϕ [ ( 1 tan ϕ + α ) 2 + 1 ] 1 / 2 .
θ = tan 1 ( r sin ϕ r cos ϕ + α r sin ϕ ) , s = r [ ( r sin ϕ ) 2 + ( r cos ϕ + α r sin ϕ ) 2 ] 1 / 2 .
θ = tan 1 ( tan ϕ 1 + α tan ϕ ) , s = 1 sin ϕ [ 1 + ( 1 tan ϕ + α ) 2 ] 1 / 2 .
u i ( x , z ) = exp ( i 2 π λ z ) i λ z u 0 ( x 0 ) exp [ i π λ z ( x 0 x ) 2 ] d x 0 .
SW = ( Δ F 0 ) ( Δ F 1 ) ,
Δ F p = r 2 | F ( r , p ) | 2 d r | F ( r , p ) | 2 d r .
SW ( p ) = ( Δ F p ) ( Δ F p + 1 ) .
F total ( r , p ) = α F u ( r , p ) + β F υ ( r , p ) ,
θ = tan 1 [ tan ϕ 1 + ( α 1 + α 2 ) tan ϕ ] , s = 1 sin ϕ { [ 1 tan ϕ + ( α 1 + α 2 ) ] 2 + 1 } 1 / 2 .
( r , ϕ ) ( s 1 r , θ 1 ) , ( s 1 r , θ 1 ) ( s 2 s 1 r , θ ) ,
θ 1 = tan 1 ( tan ϕ 1 + α 1 tan ϕ ) , s 1 = 1 sin ϕ [ ( 1 tan ϕ + α 1 ) 2 + 1 ] 1 / 2 , θ = tan 1 ( tan θ 1 1 + α 2 tan θ 2 ) , s 2 = 1 sin θ 1 [ ( 1 tan θ 1 + α 2 ) 2 + 1 ] 1 / 2 .
sin 2 β = tan 2 β 1 + tan 2 β ,
( x 0 , y 0 ) ( x 0 A y 0 , y 0 ) = ( x 1 , y 1 ) , ( x 1 , y 1 ) ( x 1 , y 1 + B x 1 ) = ( x 2 , y 2 ) , ( x 2 , y 2 ) ( x 2 C y 2 , y 2 ) = ( x 3 , y 3 ) .
A = C = tan γ 2 , B = sin γ .
( r , ϕ ) ( s 1 r , θ 1 ) , ( s 1 r , θ 1 ) ( s 1 r , θ 2 ) , ( s 1 r , θ 2 ) ( s 2 s 1 r , θ 3 ) , ( s 2 s 1 r , θ 3 ) ( s 2 s 1 r , θ 4 ) , ( s 2 s 1 r , θ 4 ) ( s 3 s 2 s 1 r , θ 5 ) ,
θ 1 = tan 1 ( tan ϕ 1 + α tan ϕ ) , θ 2 = θ 1 + ( π / 2 ) , θ 3 = tan 1 ( tan θ 2 1 + β tan θ 2 ) , θ 4 = θ 3 ( π / 2 ) , θ 5 = tan 1 ( tan θ 4 1 + α tan θ 4 ) , s 1 = 1 sin ϕ [ ( 1 tan ϕ + α ) 2 + 1 ] 1 / 2 , s 2 = 1 sin θ 2 [ ( 1 tan θ 2 + β ) 2 + 1 ] 1 / 2 , s 3 = 1 sin θ 4 [ ( 1 tan θ 4 + α ) 2 + 1 ] 1 / 2 .
tan γ 2 = sin γ 1 + ( 1 sin 2 γ ) 1 / 2 ,
β = 2 α α 2 + 1 .
1 tan θ 4 = tan θ 3 , sin θ 4 = cos θ 3 , 1 tan θ 2 = tan θ 1 , sin θ 2 = cos θ 1 ,
tan ( γ 1 + γ 2 ) = tan γ 1 + tan γ 2 1 tan γ 1 tan γ 2 ,
θ 5 = ϕ tan 1 2 α 1 α 2 , s 3 s 2 s 1 r = r ;

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