Abstract

Based on the scalar diffraction theory and the fact that a hard-edged aperture function can be expanded into a finite sum of complex Gaussian functions, an approximate analytical solution for Bessel–Gaussian (BG) beams propagating through a double-apertured fractional Fourier transform (FrFT) system is derived in the cylindrical coordinate. By using the approximate analytical formulas, the propagation properties of BG beams passing through a double-apertured FrFT optical system have been studied in detail by some typical numerical examples. The results indicate that the double-apertured FrFT optical system provides a convenient way for controlling the properties of the BG beams by properly choosing the optical parameters.

© 2012 Optical Society of America

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References

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  1. V. Namias, “The fractional order Fourier transform and its applications to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  2. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  3. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  4. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  5. 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]
  6. A. Torre, “The fractional Fourier transform and some of its applications to optics,” Prog. Opt. 43, 531–596 (2002).
    [CrossRef]
  7. Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7–13 (2003).
    [CrossRef]
  8. 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]
  9. G. Zhou, “Fractional Fourier transform of Lorentz-Gaussian beams,” J. Opt. Soc. Am. A 26, 350–355 (2009).
    [CrossRef]
  10. G. Zhou, “Fractional Fourier transform of Ince-Gaussian beams,” J. Opt. Soc. Am. A 26, 2586–2591 (2009).
    [CrossRef]
  11. K. Wang and C. Zhao, “Fractional Fourier transform for an anomalous hollow beam,” J. Opt. Soc. Am. A 26, 2571–2576 (2009).
    [CrossRef]
  12. C. Zhao, K. Huang, and X. Lu, “Propagation properties of Bessel and Bessel-Gaussian beams in a fractional Fourier transform optical system,” Opt. Laser Technol. 42, 280–284 (2010).
    [CrossRef]
  13. C. Zhao and Y. Cai, “Propagation of a general-type beam through a truncated fractional Fourier transform optical system,” J. Opt. Soc. Am. A 27, 637–647 (2010).
    [CrossRef]
  14. Y. Gao, B. Zhu, D. Liu, and Z. Lin, “Fractional Fourier transform of flat-topped multi-Gaussian beams,” J. Opt. Soc. Am. A 27, 358–365 (2010).
    [CrossRef]
  15. 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]
  16. D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225–235 (2004).
    [CrossRef]
  17. B. Tang and M. Xu, “Fractional Fourier transform for beams generated by Gaussian mirror resonator,” J. Mod. Opt. 56, 1276–1282 (2009).
    [CrossRef]
  18. J. Zhang, X. Lu, and Z. Lin, “Propagation and transformation of Bessel beam through the fractional Fourier transform system with two hard-edged apertures,” Optik 122, 307–310 (2011).
    [CrossRef]
  19. K. Wang and C. Zhao, “Analytical solution for an anomalous hollow beam in a fractional Fourier transforming optical system with a hard aperture,” Opt. Laser Technol. 44, 1232–1239 (2012).
    [CrossRef]
  20. D. Deng, “Propagation of Airy beams through a hard-edged aperture,” Appl. Phys. B 107, 195–200 (2012).
    [CrossRef]
  21. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  22. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef]
  23. P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun. 156, 359–366 (1998).
    [CrossRef]
  24. V. Bagini, F. Frezza, and M. Santarsiero, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
    [CrossRef]
  25. A. April, “Bessel-Gauss beams as rigorous solutions of the Helmholtz equation,” J. Opt. Soc. Am. A 28, 2100–2107 (2011).
    [CrossRef]
  26. X. Ji and B. Lü, “Focal shift and focal switch of Bessel-Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39, 562–568 (2007).
    [CrossRef]
  27. 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]
  28. C. Zhao, L. Wang, and X. Lu, “Radiation forces of highly focused Bessel-Gaussian beams on a dielectric sphere,” Optik 119, 477–480 (2008).
    [CrossRef]
  29. J. J. Wen and M. A. Breazeale, “Diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
    [CrossRef]
  30. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1953).

2012 (2)

K. Wang and C. Zhao, “Analytical solution for an anomalous hollow beam in a fractional Fourier transforming optical system with a hard aperture,” Opt. Laser Technol. 44, 1232–1239 (2012).
[CrossRef]

D. Deng, “Propagation of Airy beams through a hard-edged aperture,” Appl. Phys. B 107, 195–200 (2012).
[CrossRef]

2011 (2)

A. April, “Bessel-Gauss beams as rigorous solutions of the Helmholtz equation,” J. Opt. Soc. Am. A 28, 2100–2107 (2011).
[CrossRef]

J. Zhang, X. Lu, and Z. Lin, “Propagation and transformation of Bessel beam through the fractional Fourier transform system with two hard-edged apertures,” Optik 122, 307–310 (2011).
[CrossRef]

2010 (3)

2009 (4)

2008 (1)

C. Zhao, L. Wang, and X. Lu, “Radiation forces of highly focused Bessel-Gaussian beams on a dielectric sphere,” Optik 119, 477–480 (2008).
[CrossRef]

2007 (1)

X. Ji and B. Lü, “Focal shift and focal switch of Bessel-Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39, 562–568 (2007).
[CrossRef]

2004 (1)

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225–235 (2004).
[CrossRef]

2003 (4)

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, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7–13 (2003).
[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]

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]

2002 (1)

A. Torre, “The fractional Fourier transform and some of its applications to optics,” Prog. Opt. 43, 531–596 (2002).
[CrossRef]

1998 (2)

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]

P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

1996 (1)

V. Bagini, F. Frezza, and M. Santarsiero, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[CrossRef]

1993 (3)

1988 (1)

J. J. Wen and M. A. Breazeale, “Diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

1987 (2)

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

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

1980 (1)

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

April, A.

Bagini, V.

V. Bagini, F. Frezza, and M. Santarsiero, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[CrossRef]

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “Diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Cai, Y.

Chen, X.

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]

Deng, D.

D. Deng, “Propagation of Airy beams through a hard-edged aperture,” Appl. Phys. B 107, 195–200 (2012).
[CrossRef]

Dong, B.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Erdelyi, A.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1953).

Frezza, F.

V. Bagini, F. Frezza, and M. Santarsiero, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[CrossRef]

Gao, Y.

Gori, F.

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

Gu, B.

Guattari, G.

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

Huang, K.

C. Zhao, K. Huang, and X. Lu, “Propagation properties of Bessel and Bessel-Gaussian beams in a fractional Fourier transform optical system,” Opt. Laser Technol. 42, 280–284 (2010).
[CrossRef]

Ji, X.

X. Ji and B. Lü, “Focal shift and focal switch of Bessel-Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39, 562–568 (2007).
[CrossRef]

Jing, F.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225–235 (2004).
[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]

Lin, Q.

Lin, Z.

J. Zhang, X. Lu, and Z. Lin, “Propagation and transformation of Bessel beam through the fractional Fourier transform system with two hard-edged apertures,” Optik 122, 307–310 (2011).
[CrossRef]

Y. Gao, B. Zhu, D. Liu, and Z. Lin, “Fractional Fourier transform of flat-topped multi-Gaussian beams,” J. Opt. Soc. Am. A 27, 358–365 (2010).
[CrossRef]

Liu, D.

Liu, H.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225–235 (2004).
[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]

Lohmann, A. W.

Lu, X.

J. Zhang, X. Lu, and Z. Lin, “Propagation and transformation of Bessel beam through the fractional Fourier transform system with two hard-edged apertures,” Optik 122, 307–310 (2011).
[CrossRef]

C. Zhao, K. Huang, and X. Lu, “Propagation properties of Bessel and Bessel-Gaussian beams in a fractional Fourier transform optical system,” Opt. Laser Technol. 42, 280–284 (2010).
[CrossRef]

C. Zhao, L. Wang, and X. Lu, “Radiation forces of highly focused Bessel-Gaussian beams on a dielectric sphere,” Optik 119, 477–480 (2008).
[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]

Lü, B.

X. Ji and B. Lü, “Focal shift and focal switch of Bessel-Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39, 562–568 (2007).
[CrossRef]

Magnus, W.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1953).

Mao, H.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225–235 (2004).
[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]

Mendlovic, D.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Namias, V.

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

Oberhettinger, F.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1953).

Ozaktas, H. M.

Pääkkönen, P.

P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

Padovani, C.

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

Santarsiero, M.

V. Bagini, F. Frezza, and M. Santarsiero, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[CrossRef]

Tang, B.

B. Tang and M. Xu, “Fractional Fourier transform for beams generated by Gaussian mirror resonator,” J. Mod. Opt. 56, 1276–1282 (2009).
[CrossRef]

Torre, A.

A. Torre, “The fractional Fourier transform and some of its applications to optics,” Prog. Opt. 43, 531–596 (2002).
[CrossRef]

Turunen, J.

P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

Wang, K.

K. Wang and C. Zhao, “Analytical solution for an anomalous hollow beam in a fractional Fourier transforming optical system with a hard aperture,” Opt. Laser Technol. 44, 1232–1239 (2012).
[CrossRef]

K. Wang and C. Zhao, “Fractional Fourier transform for an anomalous hollow beam,” J. Opt. Soc. Am. A 26, 2571–2576 (2009).
[CrossRef]

Wang, L.

C. Zhao, L. Wang, and X. Lu, “Radiation forces of highly focused Bessel-Gaussian beams on a dielectric sphere,” Optik 119, 477–480 (2008).
[CrossRef]

Wang, S.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225–235 (2004).
[CrossRef]

Wei, X.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225–235 (2004).
[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]

Wen, J. J.

J. J. Wen and M. A. Breazeale, “Diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Xu, M.

B. Tang and M. Xu, “Fractional Fourier transform for beams generated by Gaussian mirror resonator,” J. Mod. Opt. 56, 1276–1282 (2009).
[CrossRef]

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.

Zhang, J.

J. Zhang, X. Lu, and Z. Lin, “Propagation and transformation of Bessel beam through the fractional Fourier transform system with two hard-edged apertures,” Optik 122, 307–310 (2011).
[CrossRef]

Zhang, L.

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, C.

K. Wang and C. Zhao, “Analytical solution for an anomalous hollow beam in a fractional Fourier transforming optical system with a hard aperture,” Opt. Laser Technol. 44, 1232–1239 (2012).
[CrossRef]

C. Zhao, K. Huang, and X. Lu, “Propagation properties of Bessel and Bessel-Gaussian beams in a fractional Fourier transform optical system,” Opt. Laser Technol. 42, 280–284 (2010).
[CrossRef]

C. Zhao and Y. Cai, “Propagation of a general-type beam through a truncated fractional Fourier transform optical system,” J. Opt. Soc. Am. A 27, 637–647 (2010).
[CrossRef]

K. Wang and C. Zhao, “Fractional Fourier transform for an anomalous hollow beam,” J. Opt. Soc. Am. A 26, 2571–2576 (2009).
[CrossRef]

C. Zhao, L. Wang, and X. Lu, “Radiation forces of highly focused Bessel-Gaussian beams on a dielectric sphere,” Optik 119, 477–480 (2008).
[CrossRef]

Zhao, D.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225–235 (2004).
[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]

Zhou, G.

Zhu, B.

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]

Appl. Phys. B (1)

D. Deng, “Propagation of Airy beams through a hard-edged aperture,” Appl. Phys. B 107, 195–200 (2012).
[CrossRef]

Chin. Phys. Lett. (1)

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]

J. Acoust. Soc. Am. (1)

J. J. Wen and M. A. Breazeale, “Diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. Inst. Math. Appl. (1)

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

J. Mod. Opt. (2)

B. Tang and M. Xu, “Fractional Fourier transform for beams generated by Gaussian mirror resonator,” J. Mod. Opt. 56, 1276–1282 (2009).
[CrossRef]

V. Bagini, F. Frezza, and M. Santarsiero, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[CrossRef]

J. Opt. Soc. Am. A (11)

A. April, “Bessel-Gauss beams as rigorous solutions of the Helmholtz equation,” J. Opt. Soc. Am. A 28, 2100–2107 (2011).
[CrossRef]

C. Zhao and Y. Cai, “Propagation of a general-type beam through a truncated fractional Fourier transform optical system,” J. Opt. Soc. Am. A 27, 637–647 (2010).
[CrossRef]

Y. Gao, B. Zhu, D. Liu, and Z. Lin, “Fractional Fourier transform of flat-topped multi-Gaussian beams,” J. Opt. Soc. Am. A 27, 358–365 (2010).
[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]

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

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

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]

G. Zhou, “Fractional Fourier transform of Lorentz-Gaussian beams,” J. Opt. Soc. Am. A 26, 350–355 (2009).
[CrossRef]

G. Zhou, “Fractional Fourier transform of Ince-Gaussian beams,” J. Opt. Soc. Am. A 26, 2586–2591 (2009).
[CrossRef]

K. Wang and C. Zhao, “Fractional Fourier transform for an anomalous hollow beam,” J. Opt. Soc. Am. A 26, 2571–2576 (2009).
[CrossRef]

Opt. Commun. (4)

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

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225–235 (2004).
[CrossRef]

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

P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

Opt. Laser Technol. (3)

K. Wang and C. Zhao, “Analytical solution for an anomalous hollow beam in a fractional Fourier transforming optical system with a hard aperture,” Opt. Laser Technol. 44, 1232–1239 (2012).
[CrossRef]

X. Ji and B. Lü, “Focal shift and focal switch of Bessel-Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39, 562–568 (2007).
[CrossRef]

C. Zhao, K. Huang, and X. Lu, “Propagation properties of Bessel and Bessel-Gaussian beams in a fractional Fourier transform optical system,” Opt. Laser Technol. 42, 280–284 (2010).
[CrossRef]

Optik (3)

J. Zhang, X. Lu, and Z. Lin, “Propagation and transformation of Bessel beam through the fractional Fourier transform system with two hard-edged apertures,” Optik 122, 307–310 (2011).
[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]

C. Zhao, L. Wang, and X. Lu, “Radiation forces of highly focused Bessel-Gaussian beams on a dielectric sphere,” Optik 119, 477–480 (2008).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Prog. Opt. (1)

A. Torre, “The fractional Fourier transform and some of its applications to optics,” Prog. Opt. 43, 531–596 (2002).
[CrossRef]

Other (1)

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1953).

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Figures (6)

Fig. 1.
Fig. 1.

Double-apertured optical system for performing the FrFT.

Fig. 2.
Fig. 2.

Intensity distribution of BG beams for different orders. (a) n=0; (b) n=5.

Fig. 3.
Fig. 3.

Normalized intensity distribution of BG beams in the FrFT plane for different aperture radii.

Fig. 4.
Fig. 4.

Normalized intensity distribution of BG beams in the FrFT plane against different fractional orders.

Fig. 5.
Fig. 5.

Normalized intensity distribution of BG beams in the FrFT plane against different beam orders.

Fig. 6.
Fig. 6.

Normalized intensity distribution of BG beams in the FrFT plane against different focal length.

Equations (16)

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

E2(ρ,θ,z)=kexp(ikz)izexp(ik2zρ2)0E1(r0,θ)exp(ik2zr02)Jn(kρzr0)r0dr0,
E1(r0,θ)=E1(r0,θ,z=0)circ(r0b1).
circ(r0b1)={1,r0b10,r0>b1.
circ(r0b1)=m=1MAmexp(Bmb12r02),
E1(r0,θ,z=0)=E0Jn(αr0)exp(r02w02)exp(inθ),
E2(ρ,z)=kexp(ikz)izexp(ik2zρ2)exp(inθ)×m=1MAm0exp[(Bmb12ik2z+1w02)r02]Jn(αr0)Jn(kρzr0)r0dr0.
0exp(q2t2)Jν(at)Jν(ct)tdt=12q2exp(a2+c24q2)Iν(ac2q2),
E2(ρ,z)=kexp(ikz)izexp(ik2zρ2)exp(inθ)m=1MAm12q2exp(α2+(βρ)24q2)In(αβ2q2ρ),
E2(ρ,z)=E2(ρ,z)l=1LAlexp(Blb22ρ2).
Ep(r)=0E2(ρ,θ)exp[iπλftanϕ(r2+ρ2)]2πJn(krfsinϕρ)ρdρ.
0texp(S1t2)Iν(S2t)Jν(St)dt=12S1exp(S22S24S1)Jν(S2S2S1),
Ep(r)=2πkizexp(ikz)exp(inθ1)exp(iπr2λftanϕ)×m=1Ml=1LAmAl12q2exp(α24q2)12S1exp(S224S1)exp(S324S1r2)Jn(S2S32S1r),
S1=β24q2+Blb22ik2ziπλftanϕ,
S2=αβ2q2,
S3=kfsinϕ.
I(r)=|Ep(r)|2.

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