Abstract

Two definitions of a fractional Fourier transform have been proposed previously. One is based on the propagation of a wave field through a graded-index medium, and the other is based on rotating a function’s Wigner distribution. It is shown that both definitions are equivalent. An important result of this equivalency is that the Wigner distribution of a wave field rotates as the wave field propagates through a quadratic graded-index medium. The relation with ray-optics phase space is discussed.

© 1994 Optical Society of America

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References

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  1. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  2. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  3. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional orders and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  4. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  5. M. Minsky, Computation, Finite and Infinite Machines (Prentice-Hall, Englewood Cliffs, N.J., 1967), p. 111.
  6. E. Wigner, “On the quantum correction for thermodynamics equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  7. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979); “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  8. K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
    [CrossRef]
  9. H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  10. A. Yariv, Optical Electronics, 3rd ed. (Holt, New York, 1985).
  11. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).
  12. B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991), p. 102, Eq. (3.3.5).
  13. M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eq. (22.8.7).

1993 (4)

1982 (1)

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

1980 (1)

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

1979 (1)

1932 (1)

E. Wigner, “On the quantum correction for thermodynamics equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Abramovitz, M.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eq. (22.8.7).

Bartelt, H. O.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bastiaans, M. J.

Brenner, K. H.

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Lohmann, A. W.

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

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Mendlovic, D.

Minsky, M.

M. Minsky, Computation, Finite and Infinite Machines (Prentice-Hall, Englewood Cliffs, N.J., 1967), p. 111.

Ozaktas, H. M.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991), p. 102, Eq. (3.3.5).

Stegun, I. A.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eq. (22.8.7).

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991), p. 102, Eq. (3.3.5).

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamics equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics, 3rd ed. (Holt, New York, 1985).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (3)

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

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamics equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Other (5)

A. Yariv, Optical Electronics, 3rd ed. (Holt, New York, 1985).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991), p. 102, Eq. (3.3.5).

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eq. (22.8.7).

M. Minsky, Computation, Finite and Infinite Machines (Prentice-Hall, Englewood Cliffs, N.J., 1967), p. 111.

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Figures (1)

Fig. 1
Fig. 1

Setup for performing a two-dimensional fractional Fourier transform according to the WDF definition.

Equations (46)

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W ( x , ν ) = f ( x + x / 2 ) f * ( x x / 2 ) exp ( 2 π i ν x ) d x .
f ˜ ( ν ) = F 1 { f ( x ) } = f ( x ) exp ( 2 π i ν x ) d x ,
W ( x , ν ) = f ˜ ( ν + ν / 2 ) f ˜ * ( ν ν / 2 ) exp ( 2 π i ν x ) d ν .
n 2 ( x ) = n 1 2 [ 1 ( n 2 / n 1 ) x 2 ] ,
Ψ m ( x ) = H m ( 2 x ω ) exp ( x 2 ω 2 ) ,
H 0 ( x ) = 1 , H 1 ( x ) = 2 x , H 2 ( x ) = 4 x 2 2 .
β m = k [ 1 2 k ( n 2 n 1 ) 1 / 2 ( m + 1 2 ) ] 1 / 2 k ( n 2 n 1 ) 1 / 2 ( m + 1 2 ) ,
f ( x ) = m A m Ψ l m ( x ) ,
A m = f ( x ) Ψ m ( x ) / h m d x ,
F a { f ( x ) } = m A m Ψ m ( x ) exp ( i β m a L ) m A m Ψ m ( x ) exp { i [ k ( n 2 n 1 ) 1 / 2 ( m + 1 2 ) a L ] } .
f = f 1 / Q , z = f 1 R ,
Q = tan ( α / 2 ) , R = sin ( α ) ,
α = a π / 2 .
G a [ f ( x ) ] = C 1 exp [ i π ( Q 1 R ) x 0 2 λ f 1 ] f ( x 0 ) × exp [ i π ( Q 1 R ) x 0 2 λ f 1 ] × exp ( i 2 π x x 0 λ f 1 R ) d x 0 .
f ( x 0 ) = n A n Ψ n ( x 0 ) ,
G a f ( x ) = C 1 exp ( i π T x 2 λ f 1 ) n A n Ψ n ( x 0 ) × exp ( i π T x 0 2 λ f 1 i 2 π x x 0 λ f 1 R ) d x 0 = n A n G a Ψ n ( x ) .
G a Ψ n ( x ) = C 2 exp [ i ( n + 1 ) ϕ c ] Ψ n ( x ) .
Ψ 0 ( x ) = exp ( x 2 ω 2 ) .
G a Ψ 0 ( x ) = C 1 exp ( i π T x 2 λ f 1 ) × exp ( x 2 ω 2 i π T x 0 2 λ f 1 i 2 π x x 0 λ f 1 R ) d x 0 .
exp ( p 2 x 2 ± q x ) d x = π p exp ( q 2 4 p 2 ) ,
p 2 = 1 ω 2 + i π T λ f 1 , q = i 2 π x λ f 1 R ,
G a Ψ 0 ( x ) = C 1 exp ( i π T x 2 λ f 1 ) × exp [ ( π 2 x 2 λ 2 f 1 2 R 2 ) ( 1 1 ω 2 + π 2 T 2 λ 2 f 1 2 ) ( 1 ω 2 i π T λ f 1 ) ] × ( π 1 ω 2 + i π T λ f 1 ) 1 / 2 .
λ 2 f 1 2 R 2 π 2 ω 4 + T 2 R 2 = 1 .
G a Ψ 0 ( x ) = C 1 exp ( i π T x 2 λ f 1 + i π T x 2 λ f 1 x 2 ω 2 ) × ( π 1 ω 2 + i π T λ f 1 ) 1 / 2 = C 1 [ λ 2 f 1 2 π ( 1 ω 2 i π T λ f 1 ) ] 1 / 2 C 1 Ψ 0 ( x ) .
arg ( C 2 ) = arg [ ( π 1 ω 2 + i π T λ f 1 ) 1 / 2 ] + π 2 , ϕ c = arg ( π 1 ω 2 + i π T λ f 1 ) π 2 .
Ψ 1 ( x ) = 2 2 x ω exp ( x 2 ω ) .
G a Ψ 1 ( x ) = C 2 exp ( i 2 ϕ c ) Ψ 1 ( x ) ,
G a Ψ n + 1 ( x ) = C 2 exp [ i ( n + 2 ) ϕ c ] Ψ n + 1 ( x ) ,
Ψ n + 1 ( x ) = H n + 1 ( 2 x ω ) exp ( x 0 2 ω 2 ) .
G a Ψ n + 1 ( x ) = C 1 exp ( i π T x 2 λ f 1 ) H n + 1 ( 2 x ω ) × exp ( x 0 2 ω 2 i π T x 0 2 λ f 1 i 2 π x x 0 λ f 1 R ) d x 0 .
H n + 1 ( x ) = 2 x H n ( x ) 2 n H n 1 ( x ) .
G a Ψ n + 1 ( x ) = G 1 ( x ) + G 2 ( x ) ,
G 1 ( x ) = C 1 exp ( i π T x 2 λ f 1 ) 2 x H n ( 2 x ω ) × exp ( x 0 2 ω 2 i π T x 0 2 λ f 1 i 2 π x x 0 λ f 1 R ) d x 0 , G 2 ( x ) = C 1 exp ( i π T x 2 λ f 1 ) ( 2 n ) H n 1 ( 2 x ω ) × exp ( x 0 2 ω 2 i π T x 0 2 λ f 1 i 2 π x x 0 λ f 1 R ) d x 0 .
G 1 ( x ) = C 2 ( 2 n ) H n 1 ( 2 x ω ) exp ( x 2 ω 2 ) exp ( i n ϕ c ) .
G 2 ( x ) = C 1 exp ( i π T x 2 λ f 1 ) × F { 2 2 ω x H n ( 2 x ω ) exp ( x 0 2 ω 2 i π T x 0 2 λ f 1 ) } .
F { x g ( x ) } = 1 2 π d d ν F { g ( x ) } .
G 2 ( x ) = 2 2 ω exp ( i π T x 2 λ f 1 ) d d x { C 2 H n ( 2 x ω ) exp ( x 2 ω 2 ) × exp ( i π T x 2 λ f 1 ) exp [ i ( n + 1 ) ϕ c ] } .
d d x H n ( x ) = 2 n H n 1 ( x ) ,
G 2 ( x ) = 2 2 ω i λ f 1 2 π exp [ i ( n + 1 ) ϕ c ] C 2 × [ 2 x H 2 ( 2 x ω ) exp ( x 2 ω 2 ) ( 1 ω 2 + i π T λ f 1 ) 2 ω × 2 n H n 1 ( 2 x ω ) exp ( x 2 ω 2 ) ] .
G a Ψ n + 1 ( x ) = 2 2 ω i λ f 1 2 π exp [ i ( n + 1 ) ϕ c ] C 2 exp ( x 2 ω ) × [ π i λ f 1 2 x exp i ϕ c H n ( 2 x ω ) + 2 n H n 1 ( 2 x ω ) ω 2 ( 1 ω 2 i π T λ f 1 ) ] ,
G a Ψ n + 1 ( x ) = exp [ i ( n + 2 ) ϕ c ] C 2 exp ( x 2 ω 2 ) × [ 2 x 2 ω H n ( 2 x ω ) 2 n H n 1 ( 2 x ω ) ] .
G a Ψ n + 1 ( x ) = C 2 exp [ i ( n + 2 ) ϕ C ] Ψ n + 1 ( x ) .
F a f ( x ) = n A n exp [ i ( n + 1 ) ϕ c ] Ψ n ( x ) .
exp ( i ϕ c ) = cos ( ϕ ) i sin ( ϕ ) = exp ( i ϕ ) ,
ϕ c = ( n 2 n 1 ) 1 / 2 a L = a π 2 = ϕ ;
r ( z + Δ z ) = r ( z ) cos ( πΔ z / 2 L ) s ( z ) sin ( πΔ z / 2 L ) , s ( z + Δ z ) = r ( z ) sin ( πΔ z / 2 L ) + s ( z ) cos ( πΔ z / 2 L ) ,

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