Abstract

Based on the definition of fractional Fourier transform (FRT), the propagation properties of an anomalous hollow beam (AHB) through the FRT have been investigated in detail. An analytical formula is derived for the FRT of an AHB. By using the derived formula, the properties of an AHB in the FRT plane are illustrated numerically. The results show that the properties of an AHB in the FRT plane are closely related to the parameters of the beam and the fractional order p. The derived formula provides an effective and convenient way for analyzing and calculating the FRT of an AHB.

© 2009 Optical Society of America

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    [CrossRef]
  2. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shinmizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
    [CrossRef]
  3. H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
    [CrossRef] [PubMed]
  4. J. Yin, Y. Zhu, W. Jhe, and Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509-513 (1998).
    [CrossRef]
  5. L. Zhang, X. Lu, X. Chen, and S. He, “Generation of a dark-hollow beam inside a cavity,” Chin. Phys. Lett. 21, 298-301 (2004).
    [CrossRef]
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    [CrossRef]
  7. R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932-942 (1991).
    [CrossRef]
  8. X. Wang and M. G. Littman, “Laser cavity for generation of variable-radius rings of light,” Opt. Lett. 18, 767-768 (1993).
    [CrossRef] [PubMed]
  9. H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
    [CrossRef] [PubMed]
  10. S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
    [CrossRef] [PubMed]
  11. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
    [CrossRef]
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    [CrossRef]
  13. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  14. X. Lu, X. Chen, L. Zhang, and D. Xue, “High-order Bessel-Gaussian beam and its propagation properties,” Chin. Phys. Lett. 20, 2155-2157 (2003).
    [CrossRef]
  15. Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084-6 (2003).
    [CrossRef] [PubMed]
  16. Y. Cai and Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A 21, 1058-1065 (2004).
    [CrossRef]
  17. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett. 94, 134802 (2005).
    [CrossRef] [PubMed]
  18. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32, 3179-3181 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]
  20. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181-2186 (1993).
    [CrossRef]
  21. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875-1881 (1993).
    [CrossRef]
  22. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522-2531 (1993).
    [CrossRef]
  23. J. Hua, L. Liu, and G. Li, “Observing the fractional Fourier transform by free-space Fresnel direction,” Appl. Opt. 36, 512-513 (1997).
    [CrossRef] [PubMed]
  24. D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, and H. M. Ozktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424-31 (1995).
    [CrossRef]
  25. Y. Zhang, B. Dong, B. Gu, and G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114-1120 (1998).
    [CrossRef]
  26. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388-1390 (1994).
    [CrossRef] [PubMed]
  27. Y. Cai and Q. Lin, “Properties of a flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
    [CrossRef]
  28. Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beams,” Opt. Commun. 217, 7-13 (2003).
    [CrossRef]
  29. C. Zheng, “Fractional Fourier transform for a hollow Gaussian beam,” Phys. Lett. A 335, 156-161 (2006).
    [CrossRef]
  30. D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik. 114, 504-508 (2003).
    [CrossRef]
  31. Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27, 1672-1674 (2002).
    [CrossRef]
  32. Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
    [CrossRef]
  33. Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
    [CrossRef]
  34. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24, 1937-1944 (2007).
    [CrossRef]

2007

2006

C. Zheng, “Fractional Fourier transform for a hollow Gaussian beam,” Phys. Lett. A 335, 156-161 (2006).
[CrossRef]

2005

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

2004

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
[CrossRef]

L. Zhang, X. Lu, X. Chen, and S. He, “Generation of a dark-hollow beam inside a cavity,” Chin. Phys. Lett. 21, 298-301 (2004).
[CrossRef]

Y. Cai and Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A 21, 1058-1065 (2004).
[CrossRef]

2003

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084-6 (2003).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

X. Lu, X. Chen, L. Zhang, and D. Xue, “High-order Bessel-Gaussian beam and its propagation properties,” Chin. Phys. Lett. 20, 2155-2157 (2003).
[CrossRef]

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik. 114, 504-508 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Properties of a flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beams,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

2002

2000

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
[CrossRef]

1999

1998

J. Yin, Y. Zhu, W. Jhe, and Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509-513 (1998).
[CrossRef]

Y. Zhang, B. Dong, B. Gu, and G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114-1120 (1998).
[CrossRef]

1997

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shinmizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

J. Hua, L. Liu, and G. Li, “Observing the fractional Fourier transform by free-space Fresnel direction,” Appl. Opt. 36, 512-513 (1997).
[CrossRef] [PubMed]

1996

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

1995

1994

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388-1390 (1994).
[CrossRef] [PubMed]

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

1993

1991

1987

V. I. Balykin and V. S. Letokhov, “The possibility of deep laser focusing of an atomic beam into the A-region,” Opt. Commun. 64, 151-156 (1987).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

1980

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

Arlt, J.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
[CrossRef]

Balykin, V. I.

V. I. Balykin and V. S. Letokhov, “The possibility of deep laser focusing of an atomic beam into the A-region,” Opt. Commun. 64, 151-156 (1987).
[CrossRef]

Bitran, Y.

Cai, Y.

Chen, X.

L. Zhang, X. Lu, X. Chen, and S. He, “Generation of a dark-hollow beam inside a cavity,” Chin. Phys. Lett. 21, 298-301 (2004).
[CrossRef]

X. Lu, X. Chen, L. Zhang, and D. Xue, “High-order Bessel-Gaussian beam and its propagation properties,” Chin. Phys. Lett. 20, 2155-2157 (2003).
[CrossRef]

Choi, K.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Dholakia, K.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
[CrossRef]

Dong, B.

Dorsch, R. G.

Fenichel, H.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Gao, W. J.

J. P. Yin, W. J. Gao, and Y. F. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics,Vol. 45, E.Wolf, ed. (North-Holland, 2003), pp. 119-204.
[CrossRef]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Gu, B.

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

He, S.

L. Zhang, X. Lu, X. Chen, and S. He, “Generation of a dark-hollow beam inside a cavity,” Chin. Phys. Lett. 21, 298-301 (2004).
[CrossRef]

Herman, R. M.

Hill, W. T.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shinmizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Hua, J.

Ito, H.

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

Jhe, W.

J. Yin, Y. Zhu, W. Jhe, and Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509-513 (1998).
[CrossRef]

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

Jing, F.

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik. 114, 504-508 (2003).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shinmizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Lee, H. S.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Letokhov, V. S.

V. I. Balykin and V. S. Letokhov, “The possibility of deep laser focusing of an atomic beam into the A-region,” Opt. Commun. 64, 151-156 (1987).
[CrossRef]

Li, G.

Li, J.

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Lin, Q.

Littman, M. G.

Liu, H.

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik. 114, 504-508 (2003).
[CrossRef]

Liu, L.

Lohmann, A. W.

Lu, X.

L. Zhang, X. Lu, X. Chen, and S. He, “Generation of a dark-hollow beam inside a cavity,” Chin. Phys. Lett. 21, 298-301 (2004).
[CrossRef]

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084-6 (2003).
[CrossRef] [PubMed]

X. Lu, X. Chen, L. Zhang, and D. Xue, “High-order Bessel-Gaussian beam and its propagation properties,” Chin. Phys. Lett. 20, 2155-2157 (2003).
[CrossRef]

Mao, H.

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik. 114, 504-508 (2003).
[CrossRef]

Marksteiner, S.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Mendlovic, D.

Milam, D.

Namias, V.

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

Ohtsu, M.

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

Ozaktas, H. M.

Ozktas, H. M.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Pellat-Finet, P.

Rolston, S. L.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Sakaki, K.

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, “Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500-4503 (1996).
[CrossRef] [PubMed]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shinmizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Savage, C. M.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Shinmizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shinmizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shinmizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Song, Y.

Stewart, B. W.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shinmizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Wang, F.

Wang, X.

Wang, Y.

J. Yin, Y. Zhu, W. Jhe, and Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509-513 (1998).
[CrossRef]

Wei, X.

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik. 114, 504-508 (2003).
[CrossRef]

Wiggins, T. A.

Wu, J.

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Wu, Y. K.

Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Xue, D.

X. Lu, X. Chen, L. Zhang, and D. Xue, “High-order Bessel-Gaussian beam and its propagation properties,” Chin. Phys. Lett. 20, 2155-2157 (2003).
[CrossRef]

Yang, G.

Yin, J.

J. Yin, Y. Zhu, W. Jhe, and Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509-513 (1998).
[CrossRef]

Yin, J. P.

J. P. Yin, W. J. Gao, and Y. F. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics,Vol. 45, E.Wolf, ed. (North-Holland, 2003), pp. 119-204.
[CrossRef]

Zalevsky, Z.

Zhang, L.

L. Zhang, X. Lu, X. Chen, and S. He, “Generation of a dark-hollow beam inside a cavity,” Chin. Phys. Lett. 21, 298-301 (2004).
[CrossRef]

X. Lu, X. Chen, L. Zhang, and D. Xue, “High-order Bessel-Gaussian beam and its propagation properties,” Chin. Phys. Lett. 20, 2155-2157 (2003).
[CrossRef]

Zhang, Y.

Zhao, D.

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik. 114, 504-508 (2003).
[CrossRef]

Zheng, C.

C. Zheng, “Fractional Fourier transform for a hollow Gaussian beam,” Phys. Lett. A 335, 156-161 (2006).
[CrossRef]

Zhu, Q.

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik. 114, 504-508 (2003).
[CrossRef]

Zhu, Y.

J. Yin, Y. Zhu, W. Jhe, and Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58, 509-513 (1998).
[CrossRef]

Zhu, Y. F.

J. P. Yin, W. J. Gao, and Y. F. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics,Vol. 45, E.Wolf, ed. (North-Holland, 2003), pp. 119-204.
[CrossRef]

Zoller, P.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50, 2680-2690 (1994).
[CrossRef] [PubMed]

Appl. Opt.

Chin. Phys. Lett.

X. Lu, X. Chen, L. Zhang, and D. Xue, “High-order Bessel-Gaussian beam and its propagation properties,” Chin. Phys. Lett. 20, 2155-2157 (2003).
[CrossRef]

L. Zhang, X. Lu, X. Chen, and S. He, “Generation of a dark-hollow beam inside a cavity,” Chin. Phys. Lett. 21, 298-301 (2004).
[CrossRef]

J. Inst Math. Appl.

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

J. Opt. A, Pure Appl. Opt.

Y. Cai and Q. Lin, “Properties of a flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Y. Zhang, B. Dong, B. Gu, and G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114-1120 (1998).
[CrossRef]

R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932-942 (1991).
[CrossRef]

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

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

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

D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, and H. M. Ozktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424-31 (1995).
[CrossRef]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A 21, 1058-1065 (2004).
[CrossRef]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24, 1937-1944 (2007).
[CrossRef]

Opt. Commun.

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beams,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
[CrossRef]

V. I. Balykin and V. S. Letokhov, “The possibility of deep laser focusing of an atomic beam into the A-region,” Opt. Commun. 64, 151-156 (1987).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Opt. Lett.

Optik.

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik. 114, 504-508 (2003).
[CrossRef]

Phys. Lett. A

C. Zheng, “Fractional Fourier transform for a hollow Gaussian beam,” Phys. Lett. A 335, 156-161 (2006).
[CrossRef]

Phys. Rev. A

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Figures (6)

Fig. 1
Fig. 1

Optical system for performing the FRT: (a) One-lens system, (b) two-lens system.

Fig. 2
Fig. 2

Normalized 3D intensity distribution and corresponding line ( y = 0 ) of an AHB for w 0 x = 2 mm and w 0 y = 1 mm .

Fig. 3
Fig. 3

Normalized intensity distribution of an AHB in the FRT plane at different fractional orders: (a) p = 0.98 , (b) p = 0.995 , (c) p = 1 . (d) The evolution of the intensity distribution from p = 0.5 to p = 1.5 .

Fig. 4
Fig. 4

Evolution of the intensity distribution of an AHB in the FRT plane at different beam waist sizes.

Fig. 5
Fig. 5

Evolution of the intensity distribution of an AHB in the FRT plane at different lens focal lengths.

Fig. 6
Fig. 6

Evolution of the on-axis intensity of an AHB on the FRT plane versus the fractional order p.

Equations (10)

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E p ( u ) = 1 i λ f sin φ + E ( r ) exp [ i π ( | r | 2 + | u | 2 ) λ f tan φ ] exp [ 2 π i r u λ f sin φ ] d 2 r ,
E p ( x 1 , y 1 ) = 1 i λ f sin φ + + E ( x , y ) exp [ i π ( x 2 + y 2 + x 1 2 + y 1 2 ) λ f tan φ ] exp [ 2 π i ( x 1 x + y 1 y ) λ f sin φ ] d x d y ,
E ( x 1 , y 1 , z = 0 ) = ( 2 + 8 x 1 2 w 0 x 2 + 8 y 1 2 w 0 y 2 ) exp ( x 1 2 w 0 x 2 y 1 2 w 0 y 2 ) ,
E ( x 1 , y 1 ) = 2 i k f sin φ × 1 ( k w 0 x 2 cos φ 2 i f sin φ ) 2 ( 2 f sin φ + i k w 0 y 2 cos φ ) 2 × 1 ( 2 w 0 x 2 + i k f tan φ ) ( 2 w 0 y 2 + i k f tan φ ) × ( α + β x 1 2 + γ y 1 2 ) × exp [ i k 2 f tan φ ( x 1 2 + y 1 2 ) ] × exp [ 1 2 k 2 ( w 0 x 2 2 f 2 sin 2 φ + i k f w 0 x 2 sin φ cos φ x 1 2 + w 0 y 2 2 f 2 sin 2 φ + i k f w 0 y 2 sin φ cos φ y 1 2 ) ] ,
α = 48 f 4 sin 4 φ 32 i k f 3 sin 3 φ cos φ ( w 0 x 2 + w 0 y 2 ) + k 4 w 0 x 4 w 0 y 4 cos 4 φ + 4 k 2 f 2 sin 2 φ cos 2 φ ( w 0 x 4 + 4 w 0 x 2 w 0 y 2 + w 0 y 4 ) ,
β = 4 k 2 w 0 x 2 ( 2 f sin φ + i k w 0 y 2 cos φ ) 2 ,
γ = 4 k 2 w 0 y 2 ( 2 f sin φ + i k w 0 x 2 cos φ ) 2 .
+ exp ( a x 2 ) d x = π a ,
+ x exp ( a x 2 ) d x = 0 ,
+ x 2 exp ( a x 2 ) d x = 1 2 a π a .

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