Abstract

In this study the degree p = 1 is assigned to the ordinary Fourier transform. The fractional Fourier transform, for example with degree P = 1/2, performs an ordinary Fourier transform if applied twice in a row. Ozaktas and Mendlovic [ “ Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published)] introduced the fractional Fourier transform into optics on the basis of the fact that a piece of graded-index (GRIN) fiber of proper length will perform a Fourier transform. Cutting that piece of GRIN fiber into shorter pieces corresponds to splitting the ordinary Fourier transform into fractional transforms. I approach the subject of fractional Fourier transforms in two other ways. First, I point out the algorithmic isomorphism among image rotation, rotation of the Wigner distribution function, and fractional Fourier transforming. Second, I propose two optical setups that are able to perform a fractional Fourier transform.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published).
  2. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  3. H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  4. K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D-signals,” Opt. Commun. 42, 310–314 (1982).
    [CrossRef]
  5. A. W. Lohmann, N. Streibl, “Map transformations by optical anamorphic processing,” Appl. Opt. 22, 780–783 (1983).
    [CrossRef] [PubMed]
  6. A. W. Lohmann, D. Mendlovic, “An optical transform with odd cycles,” Opt. Commun. 93, 25–26 (1992).
    [CrossRef]
  7. A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2112 (1992).
    [CrossRef]
  8. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,”J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  9. A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  10. B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
    [CrossRef]

1992 (2)

A. W. Lohmann, D. Mendlovic, “An optical transform with odd cycles,” Opt. Commun. 93, 25–26 (1992).
[CrossRef]

A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2112 (1992).
[CrossRef]

1987 (1)

A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1983 (1)

1982 (2)

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

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

1980 (2)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,”J. Inst. Math. Appl. 25, 241–265 (1980).
[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]

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

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.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D-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]

Dickinson, B. W.

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, D. Mendlovic, “An optical transform with odd cycles,” Opt. Commun. 93, 25–26 (1992).
[CrossRef]

A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2112 (1992).
[CrossRef]

A. W. Lohmann, N. Streibl, “Map transformations by optical anamorphic processing,” Appl. Opt. 22, 780–783 (1983).
[CrossRef] [PubMed]

K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D-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]

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mendlovic, D.

A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2112 (1992).
[CrossRef]

A. W. Lohmann, D. Mendlovic, “An optical transform with odd cycles,” Opt. Commun. 93, 25–26 (1992).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published).

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,”J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published).

Steiglitz, K.

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

Streibl, N.

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (1)

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,”J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (4)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. 69, 1710–1716 (1979).
[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]

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

A. W. Lohmann, D. Mendlovic, “An optical transform with odd cycles,” Opt. Commun. 93, 25–26 (1992).
[CrossRef]

Other (1)

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Image rotation, generated by three shearing processes (top to bottom): to the left, down, to the right.

Fig. 2
Fig. 2

Setup (type I) for performing a fractional Fourier transform. Parameters R and Q determine the degree P and the angle ϕ = /2. The signals are two dimensional; the lens is spherical.

Fig. 3
Fig. 3

Setup (type II) for performing a fractional Fourier transform.

Equations (72)

Equations on this page are rendered with MathJax. Learn more.

W ( x , ν ) = u ( x + x / 2 ) u * ( x - x / 2 ) exp ( - 2 π i x ν ) d x .
u ( x ) = u ˜ ( ν ) exp ( 2 π i ν x ) d ν ,
W ( x , ν ) = u ˜ ( ν + ν / 2 ) u ˜ * ( ν - ν / 2 ) exp ( 2 π i ν x ) d ν .
W d ν = u ( x ) 2 ,
W d x = u ˜ ( ν ) 2 ,
W d ν d x = E TOTAL .
W ( x , ν ) exp ( 4 π i ν x ) d ν = u ( 2 x ) u * ( 0 ) ,
W ( x , ν ) exp ( - 4 π i ν x ) d x = u ˜ ( 2 ν ) u ˜ * ( 0 ) .
u ( 2 x - x M ) = W exp [ 4 π i ν ( x - x M ) ] d ν / u * ( x M ) .
λ f 1 ν = ξ .
u F ( ξ ) = u ˜ 0 ( ξ / λ f 1 ) = u 0 ( x ) exp ( - 2 π i x ξ / λ f 1 ) d x ,
W 0 ( x , ξ ) = u 0 ( x + x / 2 ) u 0 * ( x - x / 2 ) × exp ( - 2 π i x ξ / λ f 1 ) d x ,
W 0 = u F ( ξ + ξ / 2 ) u F * ( ξ - ξ / 2 ) exp ( 2 π i x ξ / λ f 1 ) d ξ .
u 0 ( x ) u F ( x ) ,
W 0 ( x , ξ ) W F ( x , ξ ) = W 0 ( - ξ , x ) .
u 0 ( x ) u 0 ( x ) exp ( - i π x 2 / λ f ) = u L ( x ) .
f = f 1 Q ,
W 0 ( x , ξ ) W 0 ( x , ξ + Q x ) = W L ( x , ξ ) .
u ˜ 0 ( ν ) u ˜ 0 ( ν ) exp ( - i π λ z ν 2 ) = u ˜ Z ( ν ) .
Z = R f 1 ,
u F ( ξ ) u F ( ξ ) exp ( - i π R ξ 2 / λ f 1 ) = u Z ( ξ ) ,
W 0 ( x , ξ ) W Z ( x , ξ ) = W 0 ( x - R ξ , ξ ) .
[ A B C D ] [ x ξ ] = [ x ξ ] ,
[ 0 - 1 1 0 ] ;             [ 1 0 Q 1 ] ;             [ 1 - R 0 1 ] .
F ( 1 ) [ u 0 ( x ) ] = u ˜ 0 ( ξ / λ f 1 ) = u F ( ξ ) .
F ( P ) [ u 0 ( x ) ] = u P .
W 0 ( x , ξ ) W 0 ( - ξ , x ) = W 1 ( x , ξ ) .
[ cos ( ϕ ) - sin ( ϕ ) sin ( ϕ ) cos ( ϕ ) ] [ x ξ ] = [ x cos ( ϕ ) - ξ sin ( ϕ ) ξ cos ( ϕ ) + x sin ( ϕ ) ] .
ϕ = P π / 2.
W 0 ( x , ξ ) W P ( x , ξ ) = W 0 ( x cos ϕ - ξ sin ϕ , ξ cos ϕ + x sin ϕ ) .
u 0 ( x ) W 0 ( x , ξ ) ,
W 0 ( x , ξ ) W P ( x , ξ ) ,
W P ( x , ξ ) u P ( x ) = F ( P ) [ u 0 ( x ) ] .
u P ( 2 x ) = u 0 ( x cos ϕ - ξ sin ϕ + x / 2 ) × u 0 * ( ξ cos ϕ + x sin ϕ - x / 2 ) × exp ( .. ) d x d ξ , exp ( .. ) = exp { ( 2 π i / λ f 1 ) [ 2 ξ x - x ( ξ cos ϕ + x sin ϕ ) ] } .
I ( r , θ ) I ( r , θ + ϕ ) .
θ 0 θ 0 + 2 ( α - θ 0 ) = - θ 0 + 2 α ,
θ 1 θ 1 + 2 ( β - θ 1 ) = θ 0 + 2 ( β - α ) .
( 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 ) .
x 3 = x 0 ( 1 - B C ) - y 0 ( A + C - A B C ) ,
y 3 = y 0 ( 1 - A B ) + x 0 B .
B = sin ( ϕ ) ,             A = C = tan ( ϕ / 2 ) .
( x 0 , y 0 ) ( x 0 , y 0 + a x 0 ) = ( x 1 , y 1 ) ,
( x 1 , y 1 ) ( x 1 - b y 1 , y 1 ) = ( x 2 , y 2 ) ,
( x 2 , y 2 ) ( x 2 , y 2 + c x 2 ) = ( x 3 , y 3 ) .
x 3 = x 0 ( 1 - a b ) - b y 0 ,
y 3 = y 0 ( 1 - b c ) + x 0 ( a + c - a b c ) .
b = sin ( ϕ ) ,             a = c = tan ( ϕ / 2 ) .
W 0 ( x , ξ ) W P ( x , ξ ) = W 0 ( x cos ϕ - ξ sin ϕ , ξ cos ϕ + x sin ϕ ) ,
ϕ = P π / 2.
W Z ( x , ξ ) = W 0 ( x - R ξ , ξ ) ,
W L ( x , ξ ) = W 0 ( x , ξ + Q x ) ,
Z = R f 1 ;             f = f 1 / Q .
I : R = tan ( ϕ / 2 ) , Q = sin ( ϕ ) ,
II : Q = tan ( ϕ / 2 ) , R = sin ( ϕ ) .
W ( x , y , ξ , η ) = u ( x + x / 2 , y + y / 2 ) × u * ( - , - ) exp [ .. ] d x d y , [ .. ] = - ( 2 π i / λ f 1 ) ( x ξ + y η ) .
exp [ - ( i π / λ f 1 ) R ( ξ 2 + η 2 ) ] .
exp [ - ( i π / λ f 1 ) Q ( x 2 + y 2 ) ] .
W Z = W 0 ( x - R ξ , y - R η , ξ , η ) ,
W L = W 0 ( x , y , ξ + Q x , η + Q y ) .
[ cos ( ϕ ) - sin ( ϕ ) sin ( ϕ ) cos ( ϕ ) 0 0 cos ( ϕ ) - sin ( ϕ ) sin ( ϕ ) cos ( ϕ ) ] .
P x = P y .
W exp [ ( 4 π i / λ f 1 ) ( x ξ + y η ) ] d ξ d η = u ( 2 x , 2 y ) u * ( 0 , 0 ) .
Z = 0 - : u 0 ( x 0 ) ,
Z = 0 + : u 0 ( x 0 ) exp [ - ( i π / λ f 1 ) Q x 0 2 ] = u 1 ( x 0 ) ,
Z = R f 1 - 0 : u ˜ 1 ( ξ ) exp [ - ( i π / λ f 1 ) R ξ 2 ] = u ˜ 2 ( ξ ) ,
Z = R f 1 + 0 : u 2 exp [ - ( i π / λ f 1 ) Q x 2 ] = u P ( x ) .
u ˜ 1 ( ξ ) = u 1 ( x 0 ) exp [ - ( 2 π i / λ f 1 ) x 0 ξ ] d x 0 .
u 2 ( x ) = u ˜ 2 ( ξ ) exp [ ( 2 π i / λ f 1 ) x ξ ] d ξ .
u P ( x ) = F ( P ) [ u 0 ( x 0 ) ] = u 0 ( 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.

Metrics