Abstract

The problem of expressing the kernel of a fractional Fourier transform in elementary functions is posed and solved.

© 1994 Optical Society of America

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References

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  1. H. M. Ozaktas, D. Mendlovic, Opt. Commun. 101, 163 (1993).
    [CrossRef]
  2. D. Mendlovic, H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
    [CrossRef]
  3. V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
    [CrossRef]
  4. L. B. Almeida, “The angular Fourier transform,” IEEE Trans. Signal Process. (to be published).
  5. A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
    [CrossRef]
  6. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
    [CrossRef]

1993

1987

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

1980

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Almeida, L. B.

L. B. Almeida, “The angular Fourier transform,” IEEE Trans. Signal Process. (to be published).

Kerr, F. H.

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Lohmann, A. W.

McBride, A. C.

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Mendlovic, D.

H. M. Ozaktas, D. Mendlovic, Opt. Commun. 101, 163 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
[CrossRef]

Namias, V.

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, D. Mendlovic, Opt. Commun. 101, 163 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
[CrossRef]

IMA J. Appl. Math.

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

J. Inst. Math. Appl.

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

H. M. Ozaktas, D. Mendlovic, Opt. Commun. 101, 163 (1993).
[CrossRef]

Other

L. B. Almeida, “The angular Fourier transform,” IEEE Trans. Signal Process. (to be published).

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Equations (16)

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F a [ f ( x , y ) ] = l m A l m Ψ l m ( x , y ) exp ( i β l m a L ) ,
A l m = - - f ( x , y ) Ψ l m ( x , y ) / ( 2 l + m - 1 l ! m ! π ω 2 ) d x d y ,
L = ( π / 2 ) ( n 1 / n 2 ) 1 / 2 ,
F a [ f ( x , y ) ] = - - f ( x , y ) K a ( x , y , u , v ) d x d y ,
K a ( x , y , u , v ) = 2 ω 2 1 π exp [ - ( x 2 + u 2 ) / ω 2 ] × exp [ - ( y r + v 2 ) / ω 2 ] exp [ i k a L - i ( π / 2 ) a ] × l m exp [ - i a l ( π / 2 ) ] 2 l l ! exp [ - i a m ( π / 2 ) ] 2 m m ! × H l ( 2 x / ω ) H l ( 2 u / ω ) H m ( 2 y / ω ) H m ( 2 v / ω ) .
F a [ f ( x ) ] = - f ( x ) K a ( x , u ) d x ,
K a ( x , u ) = 2 ω 1 π exp [ i k a L - i ( π / 4 ) a ] × exp [ - ( x 2 + u 2 ) / ω 2 ] × l exp [ - i a l ( π / 2 ) ] 2 l l ! H l ( 2 x / ω ) H l ( 2 u / ω ) .
K a ( x , u ) = exp [ i π 4 ( 1 - a ) ] [ 2 π sin ( a π / 2 ] 1 / 2 exp [ i x u sin ( a π / 2 ) ] × exp [ - 1 2 i ( x 2 + u 2 ) cot ( a π / 2 ) ] .
K a ( x 1 , u 1 , , x N , u N ) = i = 1 N K a ( x i , u i ) .
Q a [ f ( x 1 , , x N ) ] = - - f ( x 1 , , x N ) × i = l N K a ( x i , u i ) d x i .
K a ( x 1 , u 1 , , x N , u N ) = i = 1 N K a ( x i , u i ) .
Q a Q b = - - [ i = 1 N K a ( x i , u i ) - - f ( p 1 , , p N ) × i = 1 N K b ( p i , x i ) d p i ] d x i = - - f ( p 1 , , p N ) × [ i = 1 N - K a ( p i , x i ) K b ( x i , u i ) d x i ] d p 1 d p N .
- K a ( u , x ) K b ( x , v ) d x = K a + b ( u , v ) ,
- - f ( p 1 , , p N ) × [ i = 1 N - K a ( p i , x i ) K b ( x i , u i ) d x i ] d p 1 d p N = - - f ( p 1 , , p N ) i - 1 N K a + b ( p i , u i ) d p 1 = Q a + b .
K a ( x , y , u , v ) = K a ( x , u ) K a ( y , v ) = exp [ i π 2 ( 1 - a ) ] 2 π sin ( a π / 2 ) exp [ i ( x u + u v ) sin ( a π / 2 ) ] × exp [ - 1 2 i ( x 2 + y 2 + u 2 + v 2 ) cot ( a π / 2 ) ] ,
K a ( x , y , z , u , v , w ) = K a ( x , u ) K a ( y , v ) K a ( z , w ) = exp [ 3 i π 4 ( 1 - a ) ] [ 2 π sin ( a π / 2 ) ] 3 / 2 exp [ i ( x u + y v + z w ) sin ( a π / 2 ) ] × exp [ - 1 2 i ( x 2 + y 2 + z 2 + u 2 + v 2 + w 2 ) cot ( a π / 2 ) ] .

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