Abstract

Continuum extensions of common dual pairs of operators are presented and consolidated, based on the fractional Fourier transform. In particular, the fractional chirp multiplication, fractional chirp convolution, and fractional scaling operators are defined and expressed in terms of their common nonfractional special cases, revealing precisely how they are interpolations of their conventional counterparts. Optical realizations of these operators are possible with use of common physical components. These three operators can be interpreted as fractional lenses, fractional free space, and fractional imaging systems, respectively. Any optical system consisting of an arbitrary concatenation of sections of free space and thin lenses can be interpreted as a fractional imaging system with spherical reference surfaces. As a special case, a system departing from the classical single-lens imaging condition can be interpreted as a fractional imaging system.

© 2003 Optical Society of America

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    [CrossRef]
  13. L. Barker, C̣. Candan, T. Hakioğlu, M. A. Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
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    [CrossRef]
  25. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  26. H. M. Ozaktas, M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
    [CrossRef]
  27. H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
    [CrossRef]
  28. O. Aytür, H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995).
    [CrossRef]
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  31. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
  32. R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962–982 (1967).
    [CrossRef]
  33. M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
    [CrossRef]
  34. K. B. Wolf, “Canonical transforms. I. Complex linear transforms,” J. Math. Phys. 15, 1295–1301 (1974).
    [CrossRef]
  35. R. Gilmore, “Baker–Campbell–Hausdorff formulas,” J. Math. Phys. 15, 2090–2092 (1974).
    [CrossRef]
  36. A. J. Dragt, J. M. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math. Phys. 17, 2215–2227 (1976).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  45. M. Kauderer, “Fourier-optics approach to the symplectic group,” J. Opt. Soc. Am. A 7, 231–239 (1990).
    [CrossRef]
  46. M. Kauderer, Symplectic Matrices: First Order Systems and Special Relativity (World Scientific, Singapore, 1994).
  47. R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
    [CrossRef]
  48. 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]
  49. It should be noted, however, that the fractional power of an operator is not unique. For instance, see Ref. 21 or Ref. 1, pp. 137–143.
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  51. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

2001 (3)

S. T. Liu, L. Yu, B. H. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187, 57–63 (2001).
[CrossRef]

Z. Zalevsky, D. Mendlovic, M. A. Kutay, H. M. Ozaktas, J. Solomon, “Improved acoustic signals discrimination using fractional Fourier transform based phase-space representations,” Opt. Commun. 190, 95–101 (2001).
[CrossRef]

D. M. Zhao, S. M. Wang, “Effect of misalignment on optical fractional Fourier transforming systems,” Opt. Commun. 198, 281–286 (2001).
[CrossRef]

2000 (8)

1999 (3)

1998 (3)

1997 (2)

H. M. Ozaktas, M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

J. Hua, L. Liu, G. Li, “Performing the fractional Fourier transform by one Fresnel diffraction and one lens,” Opt. Commun. 137, 11–12 (1997).
[CrossRef]

1996 (1)

X. Deng, Y. Li, Y. Qiu, D. Fan, “Diffraction interpreted through fractional Fourier transforms,” Opt. Commun. 131, 241–245 (1996).
[CrossRef]

1995 (3)

H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

O. Aytür, H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

1994 (1)

1993 (3)

1990 (1)

1982 (3)

1981 (1)

1980 (1)

1979 (2)

1977 (1)

1976 (1)

A. J. Dragt, J. M. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math. Phys. 17, 2215–2227 (1976).
[CrossRef]

1974 (2)

K. B. Wolf, “Canonical transforms. I. Complex linear transforms,” J. Math. Phys. 15, 1295–1301 (1974).
[CrossRef]

R. Gilmore, “Baker–Campbell–Hausdorff formulas,” J. Math. Phys. 15, 2090–2092 (1974).
[CrossRef]

1971 (1)

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

1967 (1)

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962–982 (1967).
[CrossRef]

1954 (1)

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

Abe, S.

Akay, O.

O. Akay, G. F. Boudreaux-Bartels, “Unitary and Hermitian fractional operators and their relation to the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 312–314 (1998).
[CrossRef]

Alieva, T.

Arikan, O.

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arıkan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

Aytür, O.

H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

O. Aytür, H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995).
[CrossRef]

Barker, L.

L. Barker, C̣. Candan, T. Hakioğlu, M. A. Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Barshan, B.

Bastiaans, M. J.

Basu, D.

D. Basu, K. B. Wolf, “The unitary irreducible representations of SL(2, R) in all subgroup reductions,” J. Math. Phys. 23, 189–205 (1982).
[CrossRef]

Bernardo, L. M.

L. M. Bernardo, “Independent adjustment of the scale and the order of polychromatic fractional Fourier transforms,” Opt. Commun. 176, 61–64 (2000).
[CrossRef]

Boudreaux-Bartels, G. F.

O. Akay, G. F. Boudreaux-Bartels, “Unitary and Hermitian fractional operators and their relation to the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 312–314 (1998).
[CrossRef]

Brenner, K. H.

Butterweck, H. J.

Candan, C?.

L. Barker, C̣. Candan, T. Hakioğlu, M. A. Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977).

Deng, X.

X. Deng, Y. Li, Y. Qiu, D. Fan, “Diffraction interpreted through fractional Fourier transforms,” Opt. Commun. 131, 241–245 (1996).
[CrossRef]

Ding, J. J.

Diu, B.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977).

Dong, B.-Z.

Dragoman, D.

Dragoman, M.

Dragt, A. J.

A. J. Dragt, J. M. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math. Phys. 17, 2215–2227 (1976).
[CrossRef]

Erden, M. F.

H. M. Ozaktas, M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

Fan, D.

X. Deng, Y. Li, Y. Qiu, D. Fan, “Diffraction interpreted through fractional Fourier transforms,” Opt. Commun. 131, 241–245 (1996).
[CrossRef]

Finn, J. M.

A. J. Dragt, J. M. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math. Phys. 17, 2215–2227 (1976).
[CrossRef]

Gilmore, R.

R. Gilmore, “Baker–Campbell–Hausdorff formulas,” J. Math. Phys. 15, 2090–2092 (1974).
[CrossRef]

Gu, B.-Y.

Hakioglu, T.

L. Barker, C̣. Candan, T. Hakioğlu, M. A. Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Hua, J.

J. Hua, L. Liu, G. Li, “Performing the fractional Fourier transform by one Fresnel diffraction and one lens,” Opt. Commun. 137, 11–12 (1997).
[CrossRef]

Kauderer, M.

M. Kauderer, “Fourier-optics approach to the symplectic group,” J. Opt. Soc. Am. A 7, 231–239 (1990).
[CrossRef]

M. Kauderer, Symplectic Matrices: First Order Systems and Special Relativity (World Scientific, Singapore, 1994).

Kutay, M. A.

Z. Zalevsky, D. Mendlovic, M. A. Kutay, H. M. Ozaktas, J. Solomon, “Improved acoustic signals discrimination using fractional Fourier transform based phase-space representations,” Opt. Commun. 190, 95–101 (2001).
[CrossRef]

L. Barker, C̣. Candan, T. Hakioğlu, M. A. Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arıkan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, “Optimal image restoration with the fractional Fourier transform,” J. Opt. Soc. Am. A 15, 825–833 (1998).
[CrossRef]

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 2000), Vol. I, pp. 93–96.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Laloë, F.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977).

Li, G.

J. Hua, L. Liu, G. Li, “Performing the fractional Fourier transform by one Fresnel diffraction and one lens,” Opt. Commun. 137, 11–12 (1997).
[CrossRef]

Li, Y.

X. Deng, Y. Li, Y. Qiu, D. Fan, “Diffraction interpreted through fractional Fourier transforms,” Opt. Commun. 131, 241–245 (1996).
[CrossRef]

Liu, L.

J. Hua, L. Liu, G. Li, “Performing the fractional Fourier transform by one Fresnel diffraction and one lens,” Opt. Commun. 137, 11–12 (1997).
[CrossRef]

Liu, S. T.

S. T. Liu, L. Yu, B. H. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187, 57–63 (2001).
[CrossRef]

Lohmann, A.

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

Mendlovic, D.

Z. Zalevsky, D. Mendlovic, M. A. Kutay, H. M. Ozaktas, J. Solomon, “Improved acoustic signals discrimination using fractional Fourier transform based phase-space representations,” Opt. Commun. 190, 95–101 (2001).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

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]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, H. Urey, “Space-variant filtering in fractional Fourier domains,” in Optical Computing, Institute of Physics Conference Series, B. S. Wherrett, P. Chavel, eds. (Institute of Physics, Bristol, UK, 1995), pp. 285–288.

Moshinsky, M.

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

Nazarathy, M.

Onural, L.

Ozaktas, H. M.

Z. Zalevsky, D. Mendlovic, M. A. Kutay, H. M. Ozaktas, J. Solomon, “Improved acoustic signals discrimination using fractional Fourier transform based phase-space representations,” Opt. Commun. 190, 95–101 (2001).
[CrossRef]

İ. Ş. Yetik, H. M. Ozaktas, B. Barshan, L. Onural, “Perspective projections in the space–frequency plane and fractional Fourier transforms,” J. Opt. Soc. Am. A 17, 2382–2390 (2000).
[CrossRef]

L. Barker, C̣. Candan, T. Hakioğlu, M. A. Kutay, H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arıkan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, “Optimal image restoration with the fractional Fourier transform,” J. Opt. Soc. Am. A 15, 825–833 (1998).
[CrossRef]

H. M. Ozaktas, M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

O. Aytür, H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

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]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

H. M. Ozaktas, B. Barshan, D. Mendlovic, H. Urey, “Space-variant filtering in fractional Fourier domains,” in Optical Computing, Institute of Physics Conference Series, B. S. Wherrett, P. Chavel, eds. (Institute of Physics, Bristol, UK, 1995), pp. 285–288.

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 2000), Vol. I, pp. 93–96.

Özaktas, H.

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arıkan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 2000), Vol. I, pp. 93–96.

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, New York, 1968).

Patten, R.

Pei, S. C.

Qiu, Y.

X. Deng, Y. Li, Y. Qiu, D. Fan, “Diffraction interpreted through fractional Fourier transforms,” Opt. Commun. 131, 241–245 (1996).
[CrossRef]

Quesne, C.

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

Shamir, J.

Sheridan, J. T.

Simon, R.

Solomon, J.

Z. Zalevsky, D. Mendlovic, M. A. Kutay, H. M. Ozaktas, J. Solomon, “Improved acoustic signals discrimination using fractional Fourier transform based phase-space representations,” Opt. Commun. 190, 95–101 (2001).
[CrossRef]

Stoler, D.

Urey, H.

H. M. Ozaktas, B. Barshan, D. Mendlovic, H. Urey, “Space-variant filtering in fractional Fourier domains,” in Optical Computing, Institute of Physics Conference Series, B. S. Wherrett, P. Chavel, eds. (Institute of Physics, Bristol, UK, 1995), pp. 285–288.

Wang, S. M.

D. M. Zhao, S. M. Wang, “Effect of misalignment on optical fractional Fourier transforming systems,” Opt. Commun. 198, 281–286 (2001).
[CrossRef]

Wilcox, R. M.

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962–982 (1967).
[CrossRef]

Wolf, K. B.

R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

R. Simon, K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000).
[CrossRef]

D. Basu, K. B. Wolf, “The unitary irreducible representations of SL(2, R) in all subgroup reductions,” J. Math. Phys. 23, 189–205 (1982).
[CrossRef]

K. B. Wolf, “Canonical transforms. I. Complex linear transforms,” J. Math. Phys. 15, 1295–1301 (1974).
[CrossRef]

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

Yang, G.-Z.

Yetik, I. S.

İ. Ş. Yetik, H. M. Ozaktas, B. Barshan, L. Onural, “Perspective projections in the space–frequency plane and fractional Fourier transforms,” J. Opt. Soc. Am. A 17, 2382–2390 (2000).
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S. T. Liu, L. Yu, B. H. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187, 57–63 (2001).
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Zalevsky, Z.

Z. Zalevsky, D. Mendlovic, M. A. Kutay, H. M. Ozaktas, J. Solomon, “Improved acoustic signals discrimination using fractional Fourier transform based phase-space representations,” Opt. Commun. 190, 95–101 (2001).
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S. T. Liu, L. Yu, B. H. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187, 57–63 (2001).
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M. A. Kutay, H. M. Ozaktas, “Optimal image restoration with the fractional Fourier transform,” J. Opt. Soc. Am. A 15, 825–833 (1998).
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[CrossRef]

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S. T. Liu, L. Yu, B. H. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187, 57–63 (2001).
[CrossRef]

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[CrossRef]

Z. Zalevsky, D. Mendlovic, M. A. Kutay, H. M. Ozaktas, J. Solomon, “Improved acoustic signals discrimination using fractional Fourier transform based phase-space representations,” Opt. Commun. 190, 95–101 (2001).
[CrossRef]

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Other (9)

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 2000), Vol. I, pp. 93–96.

H. M. Ozaktas, B. Barshan, D. Mendlovic, H. Urey, “Space-variant filtering in fractional Fourier domains,” in Optical Computing, Institute of Physics Conference Series, B. S. Wherrett, P. Chavel, eds. (Institute of Physics, Bristol, UK, 1995), pp. 285–288.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

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A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, New York, 1968).

M. Kauderer, Symplectic Matrices: First Order Systems and Special Relativity (World Scientific, Singapore, 1994).

It should be noted, however, that the fractional power of an operator is not unique. For instance, see Ref. 21 or Ref. 1, pp. 137–143.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977).

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

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Figures (1)

Fig. 1
Fig. 1

The ath fractional Fourier domain.

Tables (1)

Tables Icon

Table 1 Summary of Fractional Operators

Equations (89)

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Fa[f(u)]{Faf}(u)-Ka(u, u)f(u)du,
Ka(u, u)exp{-i[π sgn(α)/4-α/2]}|sin α|1/2×exp[iπ(cot αu2-2 csc αuu+cot αu2)],
αaπ/2.
{CM f}(u)=CM(u, u)f(u)du,
CM(u, u)=AM exp[iπ(αu2-2βuu+γu2)],
Mγ/β1/β-β+αγ/βα/βABCD.
Fa=exp(iaπ/4)CM,
M=cos αsin α-sin αcos α.
Wfa(u, μ)=Wf (u cos α-μ sin α, u sin α+μ cos α).
{RDNα[Wf (u, μ)]}(ua)=|fa(ua)|2.
Aa=F-aAFa.
{U f}(u)=uf(u),
{Df}(u)=(i2π)-1df(u)du.
D=F-1UF.
{Uafa}(ua)=uafa(ua),
{Dafa}(ua)=(i2π)-1dfa(ua)dua.
Ua=F-aUFa,
Da=F-aDFa,
U0=U,U1=D,U-1=-D,
D0=D,D1=-U,D-1=U.
Ua=cos αU+sin αD,
Da=-sin αU+cos αD.
PH(ξ)=exp(i2πξU),
SH(ξ)=exp(i2πξD).
{PH(ξ)f}(u)=exp(i2πξu)f(u),
{SH(ξ)f}(u)=f(u+ξ).
SH(ξ)=F-1PH(ξ)F,
PHa(ξ)=exp(i2πξUa),
SHa(ξ)=exp(i2πξDa),
PH0(ξ)=PH(ξ),PH1(ξ)=SH(ξ),
PH-1(ξ)=SH(-ξ),SH0(ξ)=SH(ξ),
{PHa(ξ)fa}(ua)=exp(i2πξua)fa(ua),
{SHa(ξ)fa}(ua)=fa(ua+ξ).
PHa(ξ)=n=0(i2πξ)nn!Uan=F-an=0(i2πξ)nn!UnFa=F-aPH(ξ)Fa.
SHa(ξ)=F-aSH(ξ)Fa.
exp(A)exp(B)=exp(A+B)exp([A, B]/2),
PHa(ξ)=exp(iπξ2 sin α cos α)PH(ξ cos α)SH(ξ sin α),
SHa(ξ)=exp(iπξ2 sin α cos α)SH(ξ cos α)×PH(-ξ sin α).
M(M)=exp[-iπ ln M(UD+DU)],
{M(M)f}(u)=1/Mf(u/M).
M(M)=F-1M(1/M)F,
UD+DU=F-1[-(UD+DU)]F.
M(M)=M001/M=1/M00M-1.
Ma(M)=exp[-iπ ln M(UaDa+DaUa)],
M0(M)=M(M),M1(M)=M(1/M),
{Ma(M)fa}(ua)=1/Mfa(ua/M).
Ma(M)=F-aM(M)Fa.
Ma(M)
=cos α-sin αsin αcos αM001/Mcos αsin α-sin αcos α
=M cos2 α+sin2 α/M(M-1/M)sin α cos α(M-1/M)sin α cos αM sin2 α+cos2 α/M,
Ma(M)=F-aM(M)Fa=F-1-aM(1/M)F1+a.
Q(q)=exp(-iπqU2),
R(r)=exp(-iπrD2).
{Q(q)f}(u)=exp(-iπqu2)f(u),
{R(r)f}(u)=exp(-iπ/4)1/r exp(iπu2/r)*f(u).
R(r)=F-1Q(r)F,
Q(q)=10-q1=10q1-1,
R(r)=1r01=1-r01-1.
Qa(q)=exp(-iπqUa2),
Ra(r)=exp(-iπrDa2),
Q0(q)=Q(q),Q1(q)=R(q),
Q-1(q)=R(q),R0(r)=R(r),
R1(r)=Q(r),R-1(r)=Q(r).
{Qa(q)fa}(ua)=exp(-iπqua2)fa(ua),
{Ra(r)fa}(ua)=exp(-iπ/4)1/r exp(iπua2/r)*fa(ua).
Qa(q)=F-aQ(q)Fa,
Ra(r)=F-aR(r)Fa.
Qa(q)=cos α-sin αsin αcos α10-q1cos αsin α-sin αcos α=1+q sin α cos αq sin2 α-q cos2 α1-q sin α cos α,
Qa(q)=1-tan α0110-q cos2 α11tan α01.
Qa(q)=R(-tan α)Q(q cos2 α)R(tan α).
Ra(r)=1-r sin α cos αr cos2 α-r sin2 α1+r sin α cos α,
Ra(r)=R(cot α)Q(r sin2 α)R(-cot α).
Qa(q)=Q(cot α)R(q sin2 α)Q(-cot α),
Ra(r)=Q(-tan α)R(r cos2 α)Q(tan α).
exp{-iπq[cos2 αU2+sin α cos α(UD+DU)+sin2 αD2]}=exp(iπ tan αD2)exp(-iπq cos2 αU2)×exp(-iπ tan αD2).
exp{-iπq[cos2 αU2+sin α cos α(UD+DU)+sin2 αD2]}=exp(-iπ cot αU2)exp(-iπq sin2 αD2)×exp(iπ cot αU2),
exp{-iπq[sin2 αU2-sin α cos α(UD+DU)+cos2 αD2]}=exp(-iπ cot αD2)exp(-iπq sin2 αU2)×exp(iπ cot αD2),
exp{-iπq[sin2 αU2-sin α cos α(UD+DU)+cos2 αD2]}=exp(iπ tan αU2)exp(-iπq cos2 αD2)×exp(-iπ tan αU2).
ABCD
=101λR21ABCD10-1λR11
=A-BλR1BAλR2-DλR1+B1-1λ2R1R2BλR2+D,
a=1πsin-12BM-1/M,
R1=Bλ(M cos2 α+sin2 α/M-A),
R2=-Bλ(M sin2 α+cos2 α/M-D).
a=1πsin-12λ(d1+d2-d1d2/f)M-1/M,
R1=d1+d2-d1d2/fM cos2 α+sin2 α/M-1+d2/f,
R2=-d1-d2+d1d2/fM sin2 α+cos2 α/M-1+d1/f.
Ma(M)=Q1-DBR(B)Q1-AB
=RA-1BQ(-B)RD-1B,

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