Abstract

The fractional Fourier transform is a new topic in optics. To make use of the fractional Fourier transform as an experimental tool, I design a fractional Fourier transformer of variable order: I introduce a lens system that is able to perform equidistant fractional Fourier transforms that cover the whole range of orders and that consist of a minimum number of modules. By module, I mean an elementary fractional Fourier transform of certain order that consists of a lens between two free-space lengths. Because of the commutative additivity of the transform, various fractional orders can be achieved by means of different constellations of the modules. It is possible to perform a large variety of fractional Fourier transforms with a small number of modules.

© 1995 Optical Society of America

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References

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  1. E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
    [CrossRef] [PubMed]
  2. V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform, Part I,” Commun. Pure Appl. Math. 14, 187–214 (1961).
    [CrossRef]
  3. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  4. A. C. McBride, F. H. Kerr, “On Namias's fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  5. H. M. Ozaktas, D. Mendlovic, “Fourier transform of fractional order and their optical interpretation,” Opt. Commun. 110, 163–169 (1993).
    [CrossRef]
  6. A. W. Lohmann, “Image rotation, Wigner rotation and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  7. Y. Bitran, D. Mendlovic, R. G. Dorsch, A. W. Lohmann, H. M. Ozaktas, “Fractional Fourier transforms: simulations and experimental results,” Appl. Opt. 34, 1329–1332 (1995).
    [CrossRef] [PubMed]
  8. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
    [CrossRef] [PubMed]
  9. D. Mendlovic, Y. Bitran, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A (to be published).
  10. R. G. Dorsch, A. W. Lohmann, “Fractional Fourier transform, used for a lens design problem,” Appl. Opt. (to be published).
  11. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]

1995 (1)

1994 (2)

1993 (2)

H. M. Ozaktas, D. Mendlovic, “Fourier transform of fractional order and their optical interpretation,” Opt. Commun. 110, 163–169 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

1987 (1)

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

1980 (1)

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

1961 (1)

V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform, Part I,” Commun. Pure Appl. Math. 14, 187–214 (1961).
[CrossRef]

1937 (1)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Bargmann, V.

V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform, Part I,” Commun. Pure Appl. Math. 14, 187–214 (1961).
[CrossRef]

Bitran, Y.

Condon, E. U.

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Dorsch, R. G.

Y. Bitran, D. Mendlovic, R. G. Dorsch, A. W. Lohmann, H. M. Ozaktas, “Fractional Fourier transforms: simulations and experimental results,” Appl. Opt. 34, 1329–1332 (1995).
[CrossRef] [PubMed]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

R. G. Dorsch, A. W. Lohmann, “Fractional Fourier transform, used for a lens design problem,” Appl. Opt. (to be published).

D. Mendlovic, Y. Bitran, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A (to be published).

Kerr, F. H.

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

Lohmann, A. W.

McBride, A. C.

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

Mendlovic, D.

Namias, V.

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

Ozaktas, H. M.

Appl. Opt. (3)

Commun. Pure Appl. Math. (1)

V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform, Part I,” Commun. Pure Appl. Math. 14, 187–214 (1961).
[CrossRef]

IMA J. Appl. Math. (1)

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

J. Inst. Math. Appl. (1)

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

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

H. M. Ozaktas, D. Mendlovic, “Fourier transform of fractional order and their optical interpretation,” Opt. Commun. 110, 163–169 (1993).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Other (2)

D. Mendlovic, Y. Bitran, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A (to be published).

R. G. Dorsch, A. W. Lohmann, “Fractional Fourier transform, used for a lens design problem,” Appl. Opt. (to be published).

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Figures (1)

Fig. 1
Fig. 1

Type-I module for performing a FRT.

Tables (1)

Tables Icon

Table 1 Specific Design Example of a Fractional Fourier Transformer

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

u P ( x ) = C u 0 ( x 0 ) exp [ ( x 2 + x 0 2 ) / λ F 1 tan ϕ ] × exp ( 2 π ix x 0 / λ F 1 sin ϕ ) d x 0 ,
ϕ = P π / 2 .
P total = k 0 P 0 + k 1 P 1 + + k N P N = n = 0 N k n P n ,
P total = 2 .
u 0 ( x 0 ) u P ( x ) u 2 ( x )
F 1 / f = sin ( P π / 2 ) ,
z / F 1 = tan ( P π / 4 ) = [ 1 cos ( P π / 2 ) ] / sin ( P π / 2 ) ;
z = f ( 1 cos ϕ ) ,
F 1 f n = sin ϕ n ,
F 1 f n + 1 = sin ϕ n + 1 ,
F 1 / f n F 1 / f n + 1 = sin ϕ n sin ϕ n + 1 = f n + 1 f n .
z = f ( 1 + cos ϕ ) .
P m = m δ P ,
P 0 = 1 / 2 0 , P 1 = 1 / 2 1 , P 2 = 1 / 2 2 , P N = 1 / 2 N = δ P ,
P = 2 n = 0 N 1 / 2 n = 1 / 2 N .
P 0 = 2 0 δ P , P 1 = 2 1 δ P , P 2 = 2 2 δ P , , P N = 2 N δ P .
sin [ π / 2 ( 0.1 ) ] sin [ π / 2 ( 0.05 ) ] 2
sin [ π / 2 ( 0.2 ) ] sin [ π / 2 ( 0.1 ) ] 2 ,
sin [ π / 2 ( 0.4 ) ] sin [ π / 2 ( 0.2 ) ] 2 ,
sin [ π / 2 ( 0.8 ) ] sin [ π / 2 ( 0.4 ) ] 3 / 2 .

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