Abstract

A new mathematical model of dark-hollow beams, described as hollow sinh-Gaussian (HsG) beams, has been introduced. The intensity distributions of HsG beams are characterized by a single bright ring along the propagation whose size is determined by the order of beams; the shape of the ring can be controlled by beam width and this leads to the elliptical HsG beams. Propagation characteristics of HsG beams through an ABCD optical system have been researched, they can be regarded as superposition of a series of Hypergeometric-Gaussian (HyGG) beams. As a numerical example, the propagation characteristics of HsG beams in free space have been demonstrated graphically.

© 2012 OSA

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  1. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
    [CrossRef]
  2. M. Yan, J. Yin, and Y. Zhu, “Dark-hollow-beam guiding and splitting of a low-velocity atomic beam,” J. Opt. Soc. Am. B 17(11), 1817–1820 (2000).
    [CrossRef]
  3. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [CrossRef] [PubMed]
  4. R. M. Herman and T. A. Wiggins, “Production and uses of diffraction less beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991).
    [CrossRef]
  5. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000).
    [CrossRef]
  6. S. Topuzoski and Lj. Janicijevic, “Conversion of high-order Laguerre-Gaussian beams into Bessel beams of increased, reduced or zeroth order by use of a helical axicon,” Opt. Commun. 282(17), 3426–3432 (2009).
    [CrossRef]
  7. C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express 17(15), 12891–12899 (2009).
    [CrossRef] [PubMed]
  8. I. A. Litvin, N. A. Khilo, A. Forbes, and V. N. Belyi, “Intra-cavity generation of Bessel-like beams with longitudinally dependent cone angles,” Opt. Express 18(5), 4701–4708 (2010).
    [CrossRef] [PubMed]
  9. A. Carbajal-Dominguez, J. Bernal, A. Martin-Ruiz, and G. M. Niconoff, “Generation of J(0) Bessel beams with controlled spatial coherence featuresJ0,” Opt. Express 18(8), 8400–8405 (2010).
    [CrossRef] [PubMed]
  10. Q. Sun, K. Zhou, G. Fang, Z. Liu, and S. Liu, “Generation of spiraling high-order Bessel beams,” Appl. Phys. B 104(1), 215–221 (2011).
    [CrossRef]
  11. V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel-like beams with z-dependent cone angles,” Opt. Express 18(3), 1966–1973 (2010).
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  12. Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28(13), 1084–1086 (2003).
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  13. 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(6), 1058–1065 (2004).
    [CrossRef] [PubMed]
  14. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
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  15. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32(7), 742–744 (2007).
    [CrossRef] [PubMed]
  16. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32(21), 3053–3055 (2007).
    [CrossRef] [PubMed]
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  19. V. V. Kotlyar and A. A. Kovalev, “Family of hypergeometric laser beams,” J. Opt. Soc. Am. A 25(1), 262–270 (2008).
    [CrossRef] [PubMed]
  20. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt. 13(7), 075703 (2011).
    [CrossRef]
  21. Z. Mei and D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A 22(9), 1898–1902 (2005).
    [CrossRef] [PubMed]
  22. Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32(15), 2076–2078 (2007).
    [CrossRef] [PubMed]
  23. G. Zeng-Hui and L. Bai-Da, “Nonparaxial dark-hollow Gaussian beams,” Chin. Phys. Lett. 23(1), 106–109 (2006).
    [CrossRef]
  24. Z. Mei and D. Zhao, “Nonparaxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A 25(3), 537–542 (2008).
    [CrossRef] [PubMed]
  25. G. Schweiger, R. Nett, B. Özel, and T. Weigel, “Generation of hollow beams by spiral rays in multimode light guides,” Opt. Express 18(5), 4510–4517 (2010).
    [CrossRef] [PubMed]
  26. D. Kuang and Z. Fang, “Microaxicave: inverted microaxicon to generate a hollow beam,” Opt. Lett. 35(13), 2158–2160 (2010).
    [CrossRef] [PubMed]
  27. A. Ito, Y. Kozawa, and S. Sato, “Generation of hollow scalar and vector beams using a spot-defect mirror,” J. Opt. Soc. Am. A 27(9), 2072–2077 (2010).
    [CrossRef] [PubMed]
  28. Y. Zheng, X. Wang, F. Shen, and X. Li, “Generation of dark hollow beam via coherent combination based on adaptive optics,” Opt. Express 18(26), 26946–26958 (2010).
    [CrossRef] [PubMed]
  29. L. W. Casperson, D. G. Hall, and A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14(12), 3341–3348 (1997).
    [CrossRef]
  30. L. W. Casperson and A. A. Tovar, “Hermite–sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15(4), 954–961 (1998).
    [CrossRef]
  31. A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15(9), 2425–2432 (1998).
    [CrossRef] [PubMed]
  32. Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 21(7), 1290–1299 (2004).
    [CrossRef] [PubMed]
  33. S. Konar and S. Jana, “Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity,” Opt. Commun. 236(1-3), 7–20 (2004).
    [CrossRef]
  34. R. P. Chen, H. P. Zheng, and X. X. Chu, “Propagation properties of a sinh-Gaussian beam in a Kerr medium,” Appl. Phys. B 102(3), 695–698 (2011).
    [CrossRef]
  35. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

2011 (3)

Q. Sun, K. Zhou, G. Fang, Z. Liu, and S. Liu, “Generation of spiraling high-order Bessel beams,” Appl. Phys. B 104(1), 215–221 (2011).
[CrossRef]

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt. 13(7), 075703 (2011).
[CrossRef]

R. P. Chen, H. P. Zheng, and X. X. Chu, “Propagation properties of a sinh-Gaussian beam in a Kerr medium,” Appl. Phys. B 102(3), 695–698 (2011).
[CrossRef]

2010 (7)

2009 (2)

S. Topuzoski and Lj. Janicijevic, “Conversion of high-order Laguerre-Gaussian beams into Bessel beams of increased, reduced or zeroth order by use of a helical axicon,” Opt. Commun. 282(17), 3426–3432 (2009).
[CrossRef]

C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express 17(15), 12891–12899 (2009).
[CrossRef] [PubMed]

2008 (4)

2007 (3)

2006 (2)

G. Zeng-Hui and L. Bai-Da, “Nonparaxial dark-hollow Gaussian beams,” Chin. Phys. Lett. 23(1), 106–109 (2006).
[CrossRef]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (3)

2003 (1)

2000 (2)

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000).
[CrossRef]

M. Yan, J. Yin, and Y. Zhu, “Dark-hollow-beam guiding and splitting of a low-velocity atomic beam,” J. Opt. Soc. Am. B 17(11), 1817–1820 (2000).
[CrossRef]

1998 (2)

1997 (2)

L. W. Casperson, D. G. Hall, and A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14(12), 3341–3348 (1997).
[CrossRef]

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

1991 (1)

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Ahmad, M. A.

Arlt, J.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000).
[CrossRef]

Bai-Da, L.

G. Zeng-Hui and L. Bai-Da, “Nonparaxial dark-hollow Gaussian beams,” Chin. Phys. Lett. 23(1), 106–109 (2006).
[CrossRef]

Baykal, Y.

Belyi, V.

Belyi, V. N.

Bernal, J.

Cai, Y.

Carbajal-Dominguez, A.

Casperson, L. W.

Chen, R. P.

R. P. Chen, H. P. Zheng, and X. X. Chu, “Propagation properties of a sinh-Gaussian beam in a Kerr medium,” Appl. Phys. B 102(3), 695–698 (2011).
[CrossRef]

Chu, X. X.

R. P. Chen, H. P. Zheng, and X. X. Chu, “Propagation properties of a sinh-Gaussian beam in a Kerr medium,” Appl. Phys. B 102(3), 695–698 (2011).
[CrossRef]

Dholakia, K.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000).
[CrossRef]

Donegan, J. F.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Fang, G.

Q. Sun, K. Zhou, G. Fang, Z. Liu, and S. Liu, “Generation of spiraling high-order Bessel beams,” Appl. Phys. B 104(1), 215–221 (2011).
[CrossRef]

Fang, Z.

Forbes, A.

Hall, D. G.

He, S.

Herman, R. M.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Ito, A.

Jana, S.

S. Konar and S. Jana, “Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity,” Opt. Commun. 236(1-3), 7–20 (2004).
[CrossRef]

Janicijevic, Lj.

S. Topuzoski and Lj. Janicijevic, “Conversion of high-order Laguerre-Gaussian beams into Bessel beams of increased, reduced or zeroth order by use of a helical axicon,” Opt. Commun. 282(17), 3426–3432 (2009).
[CrossRef]

Karimi, E.

B. Piccirillo, L. Marrucci, E. Karimi, and E. Santamato, “Improved focusing with Hypergeometric-Gaussian type-II optical modes,” Opt. Express 16, 21070–21075 (2008).

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32(21), 3053–3055 (2007).
[CrossRef] [PubMed]

Kazak, N.

Khilo, N.

Khilo, N. A.

Khonina, S. N.

Konar, S.

S. Konar and S. Jana, “Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity,” Opt. Commun. 236(1-3), 7–20 (2004).
[CrossRef]

Kotlyar, V. V.

Kovalev, A. A.

Kozawa, Y.

Kuang, D.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Li, X.

Lin, J.

Lin, Q.

Litvin, I. A.

Liu, J.

Liu, S.

Q. Sun, K. Zhou, G. Fang, Z. Liu, and S. Liu, “Generation of spiraling high-order Bessel beams,” Appl. Phys. B 104(1), 215–221 (2011).
[CrossRef]

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32(15), 2076–2078 (2007).
[CrossRef] [PubMed]

Liu, Z.

Q. Sun, K. Zhou, G. Fang, Z. Liu, and S. Liu, “Generation of spiraling high-order Bessel beams,” Appl. Phys. B 104(1), 215–221 (2011).
[CrossRef]

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32(15), 2076–2078 (2007).
[CrossRef] [PubMed]

Lu, X.

Lunney, J. G.

Marrucci, L.

B. Piccirillo, L. Marrucci, E. Karimi, and E. Santamato, “Improved focusing with Hypergeometric-Gaussian type-II optical modes,” Opt. Express 16, 21070–21075 (2008).

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32(21), 3053–3055 (2007).
[CrossRef] [PubMed]

Martin-Ruiz, A.

Mei, Z.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Nett, R.

Niconoff, G. M.

O’Dwyer, D. P.

Özel, B.

Phelan, C. F.

Piccirillo, B.

B. Piccirillo, L. Marrucci, E. Karimi, and E. Santamato, “Improved focusing with Hypergeometric-Gaussian type-II optical modes,” Opt. Express 16, 21070–21075 (2008).

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32(21), 3053–3055 (2007).
[CrossRef] [PubMed]

Rakovich, Y. P.

Ropot, P.

Santamato, E.

B. Piccirillo, L. Marrucci, E. Karimi, and E. Santamato, “Improved focusing with Hypergeometric-Gaussian type-II optical modes,” Opt. Express 16, 21070–21075 (2008).

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32(21), 3053–3055 (2007).
[CrossRef] [PubMed]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Sato, S.

Schweiger, G.

Shen, F.

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Skidanov, R. V.

Soifer, V. A.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt. 13(7), 075703 (2011).
[CrossRef]

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32(7), 742–744 (2007).
[CrossRef] [PubMed]

Sun, Q.

Q. Sun, K. Zhou, G. Fang, Z. Liu, and S. Liu, “Generation of spiraling high-order Bessel beams,” Appl. Phys. B 104(1), 215–221 (2011).
[CrossRef]

Topuzoski, S.

S. Topuzoski and Lj. Janicijevic, “Conversion of high-order Laguerre-Gaussian beams into Bessel beams of increased, reduced or zeroth order by use of a helical axicon,” Opt. Commun. 282(17), 3426–3432 (2009).
[CrossRef]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Tovar, A. A.

Turunen, J.

Wang, X.

Weigel, T.

Wiggins, T. A.

Yan, M.

Yin, J.

Zeng-Hui, G.

G. Zeng-Hui and L. Bai-Da, “Nonparaxial dark-hollow Gaussian beams,” Chin. Phys. Lett. 23(1), 106–109 (2006).
[CrossRef]

Zhao, D.

Zhao, H.

Zheng, H. P.

R. P. Chen, H. P. Zheng, and X. X. Chu, “Propagation properties of a sinh-Gaussian beam in a Kerr medium,” Appl. Phys. B 102(3), 695–698 (2011).
[CrossRef]

Zheng, Y.

Zhou, K.

Q. Sun, K. Zhou, G. Fang, Z. Liu, and S. Liu, “Generation of spiraling high-order Bessel beams,” Appl. Phys. B 104(1), 215–221 (2011).
[CrossRef]

Zhu, Y.

Zito, G.

Appl. Opt. (1)

Appl. Phys. B (2)

Q. Sun, K. Zhou, G. Fang, Z. Liu, and S. Liu, “Generation of spiraling high-order Bessel beams,” Appl. Phys. B 104(1), 215–221 (2011).
[CrossRef]

R. P. Chen, H. P. Zheng, and X. X. Chu, “Propagation properties of a sinh-Gaussian beam in a Kerr medium,” Appl. Phys. B 102(3), 695–698 (2011).
[CrossRef]

Chin. Phys. Lett. (1)

G. Zeng-Hui and L. Bai-Da, “Nonparaxial dark-hollow Gaussian beams,” Chin. Phys. Lett. 23(1), 106–109 (2006).
[CrossRef]

J. Opt. (1)

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt. 13(7), 075703 (2011).
[CrossRef]

J. Opt. Soc. Am. A (10)

Z. Mei and D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A 22(9), 1898–1902 (2005).
[CrossRef] [PubMed]

A. Ito, Y. Kozawa, and S. Sato, “Generation of hollow scalar and vector beams using a spot-defect mirror,” J. Opt. Soc. Am. A 27(9), 2072–2077 (2010).
[CrossRef] [PubMed]

Z. Mei and D. Zhao, “Nonparaxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A 25(3), 537–542 (2008).
[CrossRef] [PubMed]

L. W. Casperson, D. G. Hall, and A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14(12), 3341–3348 (1997).
[CrossRef]

L. W. Casperson and A. A. Tovar, “Hermite–sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15(4), 954–961 (1998).
[CrossRef]

A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15(9), 2425–2432 (1998).
[CrossRef] [PubMed]

Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 21(7), 1290–1299 (2004).
[CrossRef] [PubMed]

R. M. Herman and T. A. Wiggins, “Production and uses of diffraction less beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991).
[CrossRef]

V. V. Kotlyar and A. A. Kovalev, “Family of hypergeometric laser beams,” J. Opt. Soc. Am. A 25(1), 262–270 (2008).
[CrossRef] [PubMed]

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(6), 1058–1065 (2004).
[CrossRef] [PubMed]

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

Opt. Commun. (3)

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000).
[CrossRef]

S. Topuzoski and Lj. Janicijevic, “Conversion of high-order Laguerre-Gaussian beams into Bessel beams of increased, reduced or zeroth order by use of a helical axicon,” Opt. Commun. 282(17), 3426–3432 (2009).
[CrossRef]

S. Konar and S. Jana, “Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity,” Opt. Commun. 236(1-3), 7–20 (2004).
[CrossRef]

Opt. Express (8)

G. Schweiger, R. Nett, B. Özel, and T. Weigel, “Generation of hollow beams by spiral rays in multimode light guides,” Opt. Express 18(5), 4510–4517 (2010).
[CrossRef] [PubMed]

Y. Zheng, X. Wang, F. Shen, and X. Li, “Generation of dark hollow beam via coherent combination based on adaptive optics,” Opt. Express 18(26), 26946–26958 (2010).
[CrossRef] [PubMed]

C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express 17(15), 12891–12899 (2009).
[CrossRef] [PubMed]

I. A. Litvin, N. A. Khilo, A. Forbes, and V. N. Belyi, “Intra-cavity generation of Bessel-like beams with longitudinally dependent cone angles,” Opt. Express 18(5), 4701–4708 (2010).
[CrossRef] [PubMed]

A. Carbajal-Dominguez, J. Bernal, A. Martin-Ruiz, and G. M. Niconoff, “Generation of J(0) Bessel beams with controlled spatial coherence featuresJ0,” Opt. Express 18(8), 8400–8405 (2010).
[CrossRef] [PubMed]

V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel-like beams with z-dependent cone angles,” Opt. Express 18(3), 1966–1973 (2010).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

B. Piccirillo, L. Marrucci, E. Karimi, and E. Santamato, “Improved focusing with Hypergeometric-Gaussian type-II optical modes,” Opt. Express 16, 21070–21075 (2008).

Opt. Lett. (5)

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Other (1)

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Supplementary Material (4)

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Figures (7)

Fig. 1
Fig. 1

Normalized intensity distributions of HsG beams in free space propagation at different distances. (a)-(d) z = 0, (e)-(h) z = zR. The parameters are λ = 632.8nm, ω = 1mm and z = 4.965m.

Fig. 2
Fig. 2

Normalized radial intensity distributions of HsG beams with different orders in free space propagation at two different distances. (a) z = 0, (b) z = 2zR. The parameters are λ = 632.8nm, ω = 1mm and zR = 4.965m.

Fig. 3
Fig. 3

Comparison of the beam propagation properties with different orders in free space. (a)-(c) n = 5, (d)-(f) n = 10. The parameters are λ = 632.8nm, ω = 1mm and zR = 4.965m.

Fig. 4
Fig. 4

Dynamic simulations of HsG beams in free space propagation. (a) n = 3 (Media 1), (b) n = 10 (Media 2). The parameters are λ = 632.8nm, ω = 1mm, zR = 4.965m.

Fig. 5
Fig. 5

Comparison of the elliptical beam propagation properties for different values of n and ω at two different distances. (a)-(d) z = 0, (e)-(h) z = zR. The parameters are λ = 632.8nm and zR = 4.965m.

Fig. 6
Fig. 6

Normalized three-dimensional intensity distributions of the elliptical HsG beams in free space propagation at different distances. (a)-(c) n = 4, ωx = 1.2mm, ωy = 1.2mm. (d)-(f) n = 10, ωx = 1.2mm, ωy = 1mm. The parameters are λ = 632.8nm and zR = 4.965m.

Fig. 7
Fig. 7

Dynamic simulations of elliptical HsG beams in free space propagation. (a) n = 2, ωx = 1mm, ωy = 1.5mm (Media 3), (b) n = 7, ωx = 1.5mm, ωy = 1mm (Media 4). The parameters are λ = 632.8nm and zR = 4.965m.

Equations (15)

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E n ( r,0 )= sinh n ( r ω )exp( r 2 ω 2 ),
E n ( r,0 )= m=0 n a m b m exp[ ( r+ c m ) 2 ω 2 ] ,
a m = ( -1 ) m 2 n ( n m ), b m =exp[ ( m n 2 ) 2 ], c m =ω( m n 2 ),
U( r,z )= i λB exp(ikz) 0 2π 0 exp{ ik 2B [ Ar ' 2 2rr'cos( θθ' )+D r 2 ] } U 0 ( r',0 )r'dr'dθ' ,
J 0 ( t )= 1 2π 0 2π exp( itcosφ )dφ ,
E n ( r,z )= i λB exp(ikz)exp( ikD r 2 2B ) m=0 n a m b m 0 exp( ikAr ' 2 2B ) × J 0 ( krr' B )exp[ ( r'+ c m ) 2 ω 2 ]r'dr'.
0 t μ exp( a 2 x 2 ) J ν ( pt )dt =Γ( μ+v+1 2 ) p ν 2 v+1 a μ+v+1 Γ( ν+1 ) F 1 1 ( μ+ν+1 2 ,ν+1; p 2 4 a 2 ),
E n ( r,z )= i ω 2 2λB exp(ikz)exp( ik r 2 2q ) m=0 n s=0 a m ( n2m ) s s! Γ( 1+s/2 ) × ( B/ q 0 A+B/ q 0 ) 1+ s 2 F 1 1 ( s 2 , 1; ik r 2 2B( A+B/ q 0 ) ),
q= A+B/ q 0 C+D/ q 0 ,
χ 2 =2B( A+B/ q 0 ).
[ A B C D ]=[ 1 z 0 1 ].
E n ( r,z )= 1 2π exp(ikz) s=0 | HyGG sβ( β=0 ) n ,
| HyGG sβ( β=0 ) n = m=0 n a m ( n2m ) s s! | HyGG sβ( β=0 ) .
| HyGG sβ( β=0 ) =Γ( 1+s/2 ) ( 1+z/ q 0 ) 1s/2 ( z/ q 0 ) s/2 ×exp[ r 2 ω 2 ( 1+z/ q 0 ) ] F 1 1 ( s 2 , 1; r 2 ω 2 ( 1+z/ q 0 )z/ q 0 ).
E n ( x,y,0 )= sinh n [ ( x 2 ω x 2 + y 2 ω y 2 ) 1/2 ]exp( x 2 ω x 2 y 2 ω y 2 ),

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