Abstract

A method for the calculation of the fractional Fourier transform (FRT) by means of the fast Fourier transform (FFT) algorithm is presented. The process involves mainly two FFT’s in cascade; thus the process has the same complexity as this algorithm. The method is valid for fractional orders varying from −1 to 1. Scaling factors for the FRT and Fresnel diffraction when calculated through the FFT are discussed.

© 1996 Optical Society of America

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References

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  1. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1880 (1993).
    [CrossRef]
  2. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  3. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” J. Appl. Math. 39, 159–175 (1987).
  4. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
    [CrossRef]
  5. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  6. D. Mendlovic, H. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  7. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  8. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  9. D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. García, H. M. Ozaktas, “Anamorphic fractional Fourier transforming optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995).
    [CrossRef] [PubMed]
  10. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  11. P. Pellat-Finet, “Fresnel diffraction and the fractional Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
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    [CrossRef]
  13. J. Shamir, N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415–2423 (1995).
    [CrossRef]
  14. H. M. Ozaktas, O. Arikan, A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. (to be published).
  15. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of a complex Fourier series,” Math. Computat. 19, 297–301 (1965).
    [CrossRef]

1995

1994

1993

1987

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

1980

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

1965

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of a complex Fourier series,” Math. Computat. 19, 297–301 (1965).
[CrossRef]

Arikan, O.

H. M. Ozaktas, O. Arikan, A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. (to be published).

Barshan, B.

Bernardo, L. M.

Bitran, Y.

Bozdagi, G.

H. M. Ozaktas, O. Arikan, A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. (to be published).

Cohen, N.

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of a complex Fourier series,” Math. Computat. 19, 297–301 (1965).
[CrossRef]

Dorsch, R. G.

Ferreira, C.

García, J.

Kerr, F. H.

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

Kutay, A.

H. M. Ozaktas, O. Arikan, A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. (to be published).

Lohmann, A. W.

McBride, A. C.

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

Mendlovic, D.

Namias, V.

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

Onural, L.

Ozaktas, H.

Ozaktas, H. M.

Pellat-Finet, P.

Shamir, J.

Soares, O. D. D.

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of a complex Fourier series,” Math. Computat. 19, 297–301 (1965).
[CrossRef]

Appl. Opt.

J. Appl. Math.

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

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

Math. Computat.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of a complex Fourier series,” Math. Computat. 19, 297–301 (1965).
[CrossRef]

Opt. Lett.

Other

H. M. Ozaktas, O. Arikan, A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. (to be published).

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Figures (2)

Fig. 1
Fig. 1

Schematic diagrams of the optical setups for obtaining an FRT: (a) a type I module and (b) a type II module.

Fig. 2
Fig. 2

(a) Input vector for the calculation of the FRT. (b) Modulus of the FRT of the order 0.5 of the vector in (a), calculated through direct integration. (c) Same as (b) but calculated through Hermite–Gaussian functions. (d) Same as (b) but calculated through the algorithm introduced in this paper.

Equations (23)

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u ˜ ( x ˜ ) = exp ( j k f ) j λ f - + u ( x ) exp ( - j 2 π λ f x x ˜ ) d x ,
u ˜ m ˜ = m = - N / 2 N / 2 - 1 u m exp ( - j 2 π N m m ˜ ) ,
N Δ x Δ x ˜ = λ f ,             L x L x ˜ = λ f N ,
u ( x ) = exp ( j k z ) j λ z [ u ( x ) * exp ( j π λ z x 2 ) ] ,
u ( x ) = exp ( j k z ) j λ z exp ( j π λ z x 2 ) × - + [ u ( x ) exp ( j π λ z x 2 ) ] exp ( - j 2 π λ z x x ) d x .
u m = exp ( j k z ) j λ z exp ( j π λ z L x 2 m 2 ) × m = - N / 2 N / 2 - 1 { [ u m exp ( j π λ z L x 2 N 2 m 2 ) ] × exp ( - j 2 π N m m ) } .
z z 1 L x 2 λ N .
u ˜ ( ν ) = exp ( j k z ) j λ z u ˜ ( ν ) exp ( - j π λ z ν 2 ) .
exp [ - j π λ z ( Δ ν ) 2 m ˜ 2 ] = exp [ - j π λ z ( Δ x ˜ λ z ) 2 m ˜ 2 ] = exp ( - j π λ z L x 2 m ˜ 2 ) .
u m = exp [ j ( 2 π / λ ) z ] j λ z × m ˜ = - N / 2 N / 2 - 1 { [ m = - N / 2 N / 2 - 1 u m exp ( - j 2 π N m m ˜ ) ] × exp ( - j π λ z L x 2 m ˜ 2 ) exp ( j 2 π N m ˜ m ) } .
z z 2 L x 2 λ N .
u p ( a ) = C p exp { j π tan [ p ( π / 2 ) ] a 2 } × - + u ( a ) exp { - j π tan [ p ( π / 2 ) ] a 2 } × exp { - j 2 π sin [ p ( π / 2 ) ] a a } d a ,
C p = exp ( - j π sgn { sin [ p ( π / 2 ) ] } / 4 + j p ( π / 4 ) sin [ p ( π / 2 ) ] 1 / 2 .
u 0 ( a ) = - + u ( a ) δ ( a - a ) d a , u ± 2 ( a ) = - + u ( a ) δ ( a + a ) d a ,
( F 1 u ) ( a ) = F [ u ( a ) ] ( a ) = - + - + u ( a ) exp ( - j 2 π a a ) d x .
F a [ F b f ] = F a + b f .
u p ( x ) = exp ( j k z ) j λ z [ u ( x ) exp ( - j π λ f x 2 ) ] exp [ j π λ z × ( x 2 + x 2 - 2 x x ) ] exp ( - j π λ f x 2 ) d x .
exp [ - j π N tan ( p π 4 ) m 2 ] .
exp [ - j π N sin ( p π 2 ) m ˜ 2 ] ,
exp [ - j π N tan ( p π 4 ) m 2 ] .
| sin ( p π 2 ) | < 1 ,
| tan ( p π 4 ) | < 1.
M p = exp { - i π sgn [ sin ( p π 2 ) ] / 4 + i p π 4 + π / 4 } ,

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