Abstract

Fourier transforms of fractional order a are defined in a manner such that the common Fourier transform is a special case with order a = 1. An optical interpretation is provided in terms of quadratic graded index media and discussed from both wave and ray viewpoints. Several mathematical properties are derived.

© 1993 Optical Society of America

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References

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  1. R. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).
  2. A. Zygmund, Trigonometric Series, 1 (Cambridge U. Press, Cambridge, 1979).
  3. P. I. Lizorkin, “Fractional integration and differentiation,” in Encyclopedia of Mathematics (Kluwer, Dordrecht, The Netherlands), Vol. 4.
  4. A. Vander Lugt, Optical Signal Processing (Wiley, New York, 1992).
  5. A. Lohmann, D. Mendlovic, “Image formation and self-Fourier object,” submitted to Appl. Opt.
  6. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 28, p. 3–110.
  7. A. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
    [CrossRef]
  8. A. Lohmann, D. Mendlovic, “Optical self-transform with odd order,” Opt. Commun. 93, 25–26 (1992).
    [CrossRef]
  9. H. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. (to be published).
  10. H. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A (to be published).
  11. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,”J. Inst. Math. Its Appl. 25, 241–265 (1980).
    [CrossRef]
  12. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  13. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  14. A. Lohmann, “Ein neues Dualitatsprinzip in der Optik,” Optik (Stuttgart) 11, 478–488 (1954). An English version appeared in Optik (Stuttgart) 89, 93–97 (1992).
  15. A. Yariv, Optical Electronics, 3rd ed. (Holt Reinhart, New York, 1985).
  16. A. Yariv, Quantum Electronics (Wiley, London, 1975).
  17. A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 369.
  18. B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991).
    [CrossRef]
  19. M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

1992

A. Lohmann, D. Mendlovic, “Optical self-transform with odd order,” Opt. Commun. 93, 25–26 (1992).
[CrossRef]

A. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
[CrossRef]

1987

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

1980

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

1954

A. Lohmann, “Ein neues Dualitatsprinzip in der Optik,” Optik (Stuttgart) 11, 478–488 (1954). An English version appeared in Optik (Stuttgart) 89, 93–97 (1992).

Abramovitz, M.

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

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

Erdelyi, A.

A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 369.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Kerr, F. H.

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

Lizorkin, P. I.

P. I. Lizorkin, “Fractional integration and differentiation,” in Encyclopedia of Mathematics (Kluwer, Dordrecht, The Netherlands), Vol. 4.

Lohmann, A.

A. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
[CrossRef]

A. Lohmann, D. Mendlovic, “Optical self-transform with odd order,” Opt. Commun. 93, 25–26 (1992).
[CrossRef]

A. Lohmann, “Ein neues Dualitatsprinzip in der Optik,” Optik (Stuttgart) 11, 478–488 (1954). An English version appeared in Optik (Stuttgart) 89, 93–97 (1992).

A. Lohmann, D. Mendlovic, “Image formation and self-Fourier object,” submitted to Appl. Opt.

McBride, A. C.

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

Mendlovic, D.

A. Lohmann, D. Mendlovic, “Optical self-transform with odd order,” Opt. Commun. 93, 25–26 (1992).
[CrossRef]

A. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
[CrossRef]

A. Lohmann, D. Mendlovic, “Image formation and self-Fourier object,” submitted to Appl. Opt.

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

H. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A (to be published).

Namias, V.

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

Ozaktas, H.

H. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A (to be published).

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

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 28, p. 3–110.

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991).
[CrossRef]

Stegun, I. A.

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

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991).
[CrossRef]

Vander Lugt, A.

A. Vander Lugt, Optical Signal Processing (Wiley, New York, 1992).

Yariv, A.

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

A. Yariv, Quantum Electronics (Wiley, London, 1975).

Zygmund, A.

A. Zygmund, Trigonometric Series, 1 (Cambridge U. Press, Cambridge, 1979).

IMA J. Appl. Math.

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

J. Inst. Math. Its Appl.

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

J. Opt. Soc. Am. A

Opt. Commun.

A. Lohmann, D. Mendlovic, “Optical self-transform with odd order,” Opt. Commun. 93, 25–26 (1992).
[CrossRef]

Optik (Stuttgart)

A. Lohmann, “Ein neues Dualitatsprinzip in der Optik,” Optik (Stuttgart) 11, 478–488 (1954). An English version appeared in Optik (Stuttgart) 89, 93–97 (1992).

Other

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

A. Yariv, Quantum Electronics (Wiley, London, 1975).

A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, p. 369.

B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991).
[CrossRef]

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

R. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

A. Zygmund, Trigonometric Series, 1 (Cambridge U. Press, Cambridge, 1979).

P. I. Lizorkin, “Fractional integration and differentiation,” in Encyclopedia of Mathematics (Kluwer, Dordrecht, The Netherlands), Vol. 4.

A. Vander Lugt, Optical Signal Processing (Wiley, New York, 1992).

A. Lohmann, D. Mendlovic, “Image formation and self-Fourier object,” submitted to Appl. Opt.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 28, p. 3–110.

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

H. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A (to be published).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (4)

Fig. 1
Fig. 1

Optical setup for defining the 1/2 operator.

Fig. 2
Fig. 2

Typical refractive index profile of a quadratic GRIN medium.

Fig. 3
Fig. 3

Effect of (a) propagation and (b) focusing by a lens on a given ray in (r, s) space.

Fig. 4
Fig. 4

Phase-space representation of the original bundle of rays (a), after free-space propagation through a distance f (b), after passage through a lens of focal length f (c), and after another free-space propagation through a distance f (d).

Equations (47)

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F [ f ( x ) ] = - + f ( x ) exp ( - i 2 π f x d ) d x ,
( - i 2 π f x ) a F [ f ( x ) ] = F [ d a f ( x ) d x a ]
( F 1 f ) ( x , y ) = - + - + f ( x , y ) exp [ - i 2 π ( x x + y y ) s 2 ] d x d y ,
F a [ F b f ] = F a F b f = F b F a f = F a + b f ;
exp [ - i π ( x 2 + y 2 ) / C ] ,
f 1 / 2 = f 1 / sin ( π / 4 ) ,             l = f 1 tan ( π / 8 ) .
f 1 / Q = f 1 sin ( π / 2 Q ) ,             l = f 1 tan ( π 4 Q ) ,
n 2 ( r ) = n 1 2 [ 1 - ( n 2 / n 1 ) r 2 ] ,
Ψ l m ( x , y ) = H l ( 2 x ω ) H m ( 2 y ω ) exp ( - x 2 + y 2 ω 2 ) ,
ω = ( 2 / k ) 1 / 2 ( n 1 / n 2 ) 1 / 4 ,
β l m = k [ 1 - 2 k ( n 2 n 1 ) 1 / 2 ( l + m + 1 ) ] 1 / 2 k - ( n 2 n 1 ) 1 / 2 ( l + m + 1 ) .
E l m ( x , y , z ) = Ψ l m ( x , y ) exp ( i β l m z ) .
f ( x , y ) = l m A l m Ψ l m ( x , y ) ,
A l m = - - f ( x , y ) Ψ l m ( x , y ) h l m d x d y ,
h l m = 2 l + m l ! m ! π ω 2 / 2.
F a [ f ( x , y ) ] = l m A l m Ψ l m ( x , y ) exp ( i β l m a L ) .
F b [ F a f ] = F b [ l m A l m Ψ l m ( x , y ) exp ( i β l m a L ) ] = l m A l m Ψ l m ( x , y ) exp [ i β l m ( a + b ) L ] = F a + b f .
s = ω π .
H n ( x ) = ( - 1 ) n exp ( x 2 ) d n d x n exp ( - x 2 ) ,
H 0 ( x ) = 1 ,             H 1 ( x ) = 2 x ,             H 2 ( x ) = 4 x 2 - 2.
H n + 1 ( x ) = 2 x H n ( x ) - 2 n H n - 1 ( x ) ,
d d x H n ( x ) = 2 n H n - 1 ( x ) .
F 1 [ Ψ n ( x ) ] = π ω Ψ n ( x ) ( - i ) n .
F 1 [ Ψ 0 ( x ) ] = - exp ( - x 2 ω 2 - i 2 π x x s 2 ) d x .
- exp ( - p 2 x 2 ± q x ) d x = exp ( q 2 4 p 2 ) ( π p ) ,
- x exp ( - p x 2 + 2 q x ) d x = ( q p ) ( π p ) 1 / 2 exp ( q 2 p ) .
F 1 [ Ψ n + 1 ( x ) ] = ( - i ) n + 1 π ω Ψ n + 1 ( x ) ,
F 1 [ Ψ n ( x ) ] = Ψ n ( x ) π ω exp ( - i n π / 2 ) .
L = ( 2 π / 4 ) ( n 1 / n 2 ) 1 / 2 ,
F a [ c 1 f ( x ) + c 2 g ( x ) ] = c 1 F a f ( x ) + c 2 F a g ( x ) ,
f ( x ) = n A n f Ψ n ( x ) ,             g ( x ) = n n A n g Ψ n ( x ) .
c 1 f ( x ) + c 2 g ( x ) = n ( c 1 A n f + c 2 A n g ) Ψ n ( x ) ,
F c 1 a 1 + c 2 a 2 f ( x ) = F c 1 a 1 F c 2 a 2 f ( x ) = F c 2 a 2 F c 1 a 1 f ( x ) .
F c 1 a 1 + c 2 a 2 f ( x ) = n A n f Ψ n ( x ) exp [ - i β n L ( c 1 a 1 + c 2 a 2 ) ] = n A n f exp ( - i β n L c 1 a 1 ) exp ( - i β n L c 2 a 2 ) = F c 1 a 1 [ n A n f exp ( - i β n L c 2 a 2 ) Ψ n ( x ) ] = F c 1 a 1 F c 2 a 2 f ( x ) .
F c 1 a 1 + 4 f ( x ) = F c 1 a 1 F 4 f ( x ) = F c 1 a 1 f ( x ) .
CONV ( f , g ) = F - 1 [ F 1 f ( x ) × F 1 g ( x ) ] .
CONV a ( f , g ) = F - a [ F a f ( x ) × F a g ( x ) ] .
CORR a [ f ( x ) , g ( x ) ] = CONV a [ f ( x ) , g * ( - x ) ] .
F a [ y m f ( y ) ] ( x ) = [ x cos ( 2 π a ) - i sin ( 2 π a ) D ] m F a [ f ( y ) ] ( x ) ,
F a [ D m f ( y ) ] ( x ) = [ - i x sin ( 2 π a ) + cos ( 2 π a ) D ] m F a [ f ( y ) ] ( x ) .
F a [ ( y D ) m f ( y ) ] ( x ) = { - [ sin ( 2 π a ) + i x 2 cos ( 2 π a ) ] sin ( 2 π a ) + x cos ( 2 π a ) D - i sin ( 2 π a ) × cos ( 2 π a ) D 2 } m F a [ f ( y ) ] ( x ) .
F a T b [ f ( y ) ] ( x ) = exp ( - i b sin { 2 π a [ x + 0.5 b cos ( 2 π a ) ] } ) × F a { f ( y ) } [ x + b cos ( 2 π a ) ] ,
T b [ f ] ( x ) = f ( x + b ) .
F a { exp ( i b y ) f ( y ) } ( x ) = exp ( i b cos { 2 π a [ x + 0.5 b sin ( 2 π a ) ] } ) × F a { f ( y ) } [ x + b sin ( 2 π a ) ] .
Δ r = r ( z + Δ z ) - r ( z ) = s ( z ) Δ z .
Δ s = s ( z + Δ z ) - s ( z ) = - r ( z ) / f .
r ( z + Δ z ) = r ( z ) cos ( π Δ z / 2 L ) - s ( z ) sin ( π Δ z / 2 L ) , s ( z + Δ z ) = r ( z ) sin ( π Δ z / 2 L ) + s ( z ) cos ( π Δ z / 2 L ) ,

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