Abstract

It is demonstrated that the definition of a fractional-order Fourier transform can be extended into the complex-order regime. A complex-order Fourier transform deals with the imaginary part as well as the real part of the exponential function in the integral. As a result, while the optical implementation of fractional-order Fourier transform involves gradient-index media or spherical lenses, the optical interpretation of complex-order Fourier transform is practically based on the convolution operation and Gaussian apertures. The beam width of a Gaussian beam subjected to the complex-order Fourier transform is obtained analytically with the ABCD matrix approach.

© 1995 Optical Society of America

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References

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1994 (3)

1993 (4)

1987 (1)

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

1980 (1)

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

1975 (1)

A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
[CrossRef]

1970 (1)

Barshan, B.

Bernardo, L.

Collins, S. A.

Kerr, F. H.

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

Lee, S.

S. Lee, H. H. Szu, Opt. Eng. 33, 2326 (1994).
[CrossRef]

Lohmann, A. W.

Mcbride, A. C.

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

Mendlovic, D.

Namias, V.

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

Onural, L.

Ozaktas, H. M.

Soares, O. D. D.

Szu, H. H.

S. Lee, H. H. Szu, Opt. Eng. 33, 2326 (1994).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
[CrossRef]

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6, pp. 106–135.

Yeh, P.

A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
[CrossRef]

Inst. Math. J. Appl. Math. (1)

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

J. Inst. Math. Appl. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (2)

A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
[CrossRef]

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

Opt. Eng. (1)

S. Lee, H. H. Szu, Opt. Eng. 33, 2326 (1994).
[CrossRef]

Other (1)

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6, pp. 106–135.

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Figures (2)

Fig. 1
Fig. 1

(a) Two-lens and (b) one-lens optical systems for achieving a fractional-order Fourier transform with order a.

Fig. 2
Fig. 2

(a) Imaginary-order Fourier transform with a self-imaging optical system and Gaussian apertures placed at both ends and at the focus. (b) Complex-order Fourier transform in a double-pass optical configuration with combination of real and imaginary Fourier transforms.

Equations (18)

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g ( y ) = F [ f ( x ) ] = 1 2 π f ( x ) exp ( i y x ) d x .
M = [ A B C D ] = [ cos ( h z ) ( 1 / h ) sin ( h z ) - h sin ( h z ) cos ( h z ) ] ,
E ( x ) = i k / 2 π B × exp [ i k 2 B ( A x 2 - 2 x x + D x 2 ) ] E ( x ) d x ,
M a = [ A B C D ] a = [ cos ( π a / 2 ) ( 1 / h ) sin ( π a / 2 ) - h sin ( π a / 2 ) cos ( π a / 2 ) ] .
M a = [ 1 0 - h tan ( π a / 4 ) 1 ] [ 1 ( 1 / h ) sin ( π a / 2 ) 0 1 ] × [ 1 0 - h tan ( π a / 4 ) 1 ] ,
M a = [ 1 ( 1 / h ) tan ( π a / 4 ) 0 1 ] [ 1 0 - h sin ( π a / 2 ) 1 ] × [ 1 ( 1 / h ) tan ( π a / 4 ) 0 1 ] ,
F c = F a + i b = F a F i b ,
M i b = [ A B C D ] i b = [ cosh ( π b / 2 ) ( i / h ) sinh ( π b / 2 ) - i h sinh ( π b / 2 ) cosh ( π b / 2 ) ] .
M i b = [ 1 0 - i h tanh ( π b / 4 ) 1 ] [ 1 ( i / h ) sinh ( π b / 2 ) 0 1 ] × [ 1 0 - i h tanh ( π b / 4 ) 1 ] .
E ( x ) = [ h k 2 π sinh ( π b / 2 ) ] 1 / 2 × exp [ h k ( x - x ) 2 2 sinh ( π b / 2 ) ] E ( x ) d x ,
1 q = 1 R - i π w ,
q = A q + B C q + D .
R 1 = ,             w 1 = 1 / π 2 h 2 w .
R a = tan ( π a / 2 ) + ( π h w ) 2 cot ( π a / 2 ) h [ 1 - ( π h w ) 2 ] w a = w cos 2 ( π a / 2 ) + ( 1 / π 2 h 2 w ) sin 2 ( π a / 2 ) .
R i b = , W i b = w 1 + ( 1 / π h w ) tanh ( π b / 2 ) 1 + π h w tanh ( π b / 2 ) = ( 1 / π h ) tanh [ π b / 2 ] + tanh - 1 ( π h w ) ] .
w a + i b = w i b cos 2 ( π a / 2 ) + ( 1 / π 2 h 2 w i b ) sin 2 ( π a / 2 ) .
w 1 + i b = ( 1 / π h ) tanh [ π b / 2 + coth - 1 ( π h w ) ] ,
w a + i b = cosh [ π b + 2 tanh - 1 ( π h w ) ] - cos ( π a ) π h sinh [ π b + 2 tanh - 1 ( π h w ) ] .

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