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

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References

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  1. H. M. Ozaktas and 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, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [Crossref]
  4. K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1D-signals,” Opt. Commun. 42, 310–314 (1982).
    [Crossref]
  5. A. W. Lohmann and N. Streibl, “Map transformations by optical anamorphic processing,” Appl. Opt. 22, 780–783 (1983).
    [Crossref] [PubMed]
  6. A. W. Lohmann and D. Mendlovic, “An optical transform with odd cycles,” Opt. Commun. 93, 25–26 (1992).
    [Crossref]
  7. A. W. Lohmann and 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 and F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [Crossref]
  10. B. W. Dickinson and 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 and D. Mendlovic, “An optical transform with odd cycles,” Opt. Commun. 93, 25–26 (1992).
[Crossref]

A. W. Lohmann and 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 and F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

1983 (1)

1982 (2)

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

B. W. Dickinson and 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, and 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, and 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 and A. W. Lohmann, “Wigner distribution function display of complex 1D-signals,” Opt. Commun. 42, 310–314 (1982).
[Crossref]

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

Dickinson, B. W.

B. W. Dickinson and 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 and F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

Lohmann, A. W.

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

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

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

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

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

McBride, A. C.

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

Mendlovic, D.

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

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

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

Steiglitz, K.

B. W. Dickinson and 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 and 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 and 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, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

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

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

Other (1)

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

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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)

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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.

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