Abstract

Analytical formula is derived for the M2-factor of coherent and partially coherent dark hollow beams (DHB) in turbulent atmosphere based on the extended Huygens-Fresnel integral and the second-order moments of the Wigner distribution function. Our numerical results show that the M2- factor of a DHB in turbulent atmosphere increases on propagation, which is much different from its invariant properties in free-space, and is mainly determined by the parameters of the beam and the atmosphere. The relative M2-factor of a DHB increases slower than that of Gaussian and flat-topped beams on propagation, which means a DHB is less affected by the atmospheric turbulence than Gaussian and flat-topped beams. Furthermore, the relative M2-factor of a DHB with lower coherence, longer wavelength and larger dark size is less affected by the atmospheric turbulence. Our results will be useful in long-distance free-space optical communications.

© 2009 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).
  2. R. Martinez-Herrero and P. M. Mejias, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18(19), 1669–1671 (1993).
    [PubMed]
  3. R. Martinez-Herrero, P. M. Mejias, and M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagation through hard-edged apertures,” Opt. Lett. 20(2), 124–126 (1995).
    [PubMed]
  4. F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
  5. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999).
  6. B. Zhang, X. Chu, and Q. Li, “Generalized beam-propagation factor of partially coherent beams propagating through hard-edged apertures,” J. Opt. Soc. Am. A 19(7), 1370–1375 (2002).
  7. X. Chu, B. Zhang, and Q. Wen, “Generalized M2 factor of a partially coherent beam propagating through a circular hard-edged aperture,” Appl. Opt. 42(21), 4280–4284 (2003).
    [PubMed]
  8. P. A. Belanger, “Beam quality factor of the LP01 mode of the step-index fiber,” Opt. Eng. 32(9), 2107–2109 (1993).
  9. P. A. Belanger, Y. Champagne, and C. Pare, “Beam-propagation factor of diffracted laser-beams,” Opt. Commun. 105(3-4), 233–242 (1994).
  10. S. A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).
  11. S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
  12. B. Lü, B. Zhang, and H. Ma, “Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine Gaussian beams,” Opt. Lett. 24(10), 640–642 (1999).
  13. Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).
  14. D. Deng, Z. Xia, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution, M2 factor of beams generated by Gaussian mirror resonator,” Optik (Stuttg.) 118(11), 533–536 (2007).
  15. G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (2009).
  16. J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E. Wolf, ed., North-Holland, Amsterdam, 2003, pp.119–204.
  17. X. Wang and M. G. Littman, “Laser cavity for generation of variable-radius rings of light,” Opt. Lett. 18(10), 767–768 (1993).
    [PubMed]
  18. H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
    [PubMed]
  19. J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4-6), 287–292 (1997).
  20. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
    [PubMed]
  21. 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).
    [PubMed]
  22. Z. Liu, J. Dai, X. Sun, and S. Liu, “Generation of hollow Gaussian beam by phase-only filtering,” Opt. Express 16(24), 19926–19933 (2008).
    [PubMed]
  23. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
  24. 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).
  25. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).
  26. Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28(13), 1084–1086 (2003).
    [PubMed]
  27. Z. Mei and D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A 22(9), 1898–1902 (2005).
  28. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32(21), 3179–3181 (2007).
    [PubMed]
  29. D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25(1), 83–87 (2008).
  30. Y. Cai and S. He, “Propagation of hollow Gaussian beams through apertured paraxial optical systems,” J. Opt. Soc. Am. A 23(6), 1410–1418 (2006).
  31. G. Wu, Q. Lou, and J. Zhou, “Analytical vectorial structure of hollow Gaussian beams in the far field,” Opt. Express 16(9), 6417–6424 (2008).
    [PubMed]
  32. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
    [PubMed]
  33. Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90(1), 87–92 (2008).
  34. D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341(1-4), 352–356 (2005).
  35. D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M2 factor of hollow Gaussian beams,” Appl. Opt. 44(33), 7187–7190 (2005).
    [PubMed]
  36. Z. Mei and D. Zhao, “Generalized M2 factor of hard-edged diffracted controllable dark-hollow beams,” Opt. Commun. 263(2), 261–266 (2006).
  37. Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation,” J. Opt. Soc. Am. B 23(7), 1398–1407 (2006).
  38. X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007).
  39. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
    [PubMed]
  40. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001).
  41. L. C. Andrews, and R. L. Phillips, Laser beam propagation in the turbulent atmosphere, 2nd edition, SPIE press, Bellington, 2005.
  42. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002).
  43. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
  44. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
  45. R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
    [PubMed]
  46. Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 8(6), 537–545 (2006).
  47. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
  48. M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
  49. H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).
  50. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
    [PubMed]
  51. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
    [PubMed]
  52. M. H. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281(13), 3395–3402 (2008).
  53. L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  54. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

2009 (2)

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (2009).

Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
[PubMed]

2008 (10)

M. H. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281(13), 3395–3402 (2008).

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).

Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
[PubMed]

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
[PubMed]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).

Z. Liu, J. Dai, X. Sun, and S. Liu, “Generation of hollow Gaussian beam by phase-only filtering,” Opt. Express 16(24), 19926–19933 (2008).
[PubMed]

D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25(1), 83–87 (2008).

G. Wu, Q. Lou, and J. Zhou, “Analytical vectorial structure of hollow Gaussian beams in the far field,” Opt. Express 16(9), 6417–6424 (2008).
[PubMed]

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90(1), 87–92 (2008).

2007 (5)

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32(21), 3179–3181 (2007).
[PubMed]

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).
[PubMed]

D. Deng, Z. Xia, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution, M2 factor of beams generated by Gaussian mirror resonator,” Optik (Stuttg.) 118(11), 533–536 (2007).

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[PubMed]

X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007).

2006 (6)

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 8(6), 537–545 (2006).

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).

Z. Mei and D. Zhao, “Generalized M2 factor of hard-edged diffracted controllable dark-hollow beams,” Opt. Commun. 263(2), 261–266 (2006).

Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation,” J. Opt. Soc. Am. B 23(7), 1398–1407 (2006).

Y. Cai and S. He, “Propagation of hollow Gaussian beams through apertured paraxial optical systems,” J. Opt. Soc. Am. A 23(6), 1410–1418 (2006).

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

2005 (3)

2004 (1)

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).

2003 (2)

2002 (3)

2000 (1)

1999 (2)

1998 (1)

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).

1997 (2)

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).

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4-6), 287–292 (1997).

1996 (1)

S. A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).

1995 (1)

1994 (2)

P. A. Belanger, Y. Champagne, and C. Pare, “Beam-propagation factor of diffracted laser-beams,” Opt. Commun. 105(3-4), 233–242 (1994).

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[PubMed]

1993 (3)

1992 (1)

1991 (1)

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).

Ahmad, M. A.

Alavinejad, M.

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).

Amarande, S. A.

S. A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).

Arias, M.

Baykal, Y.

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90(1), 87–92 (2008).

Belanger, P. A.

P. A. Belanger, Y. Champagne, and C. Pare, “Beam-propagation factor of diffracted laser-beams,” Opt. Commun. 105(3-4), 233–242 (1994).

P. A. Belanger, “Beam quality factor of the LP01 mode of the step-index fiber,” Opt. Eng. 32(9), 2107–2109 (1993).

Borghi, R.

Cai, Y.

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90(1), 87–92 (2008).

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
[PubMed]

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32(21), 3179–3181 (2007).
[PubMed]

X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007).

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 8(6), 537–545 (2006).

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

Y. Cai and S. He, “Propagation of hollow Gaussian beams through apertured paraxial optical systems,” J. Opt. Soc. Am. A 23(6), 1410–1418 (2006).

Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation,” J. Opt. Soc. Am. B 23(7), 1398–1407 (2006).

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28(13), 1084–1086 (2003).
[PubMed]

Champagne, Y.

P. A. Belanger, Y. Champagne, and C. Pare, “Beam-propagation factor of diffracted laser-beams,” Opt. Commun. 105(3-4), 233–242 (1994).

Chávez-Cerda, S.

Chen, Y.

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90(1), 87–92 (2008).

Choi, K.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[PubMed]

Chu, X.

Cincotti, G.

Dai, J.

Dan, Y.

Davidson, F. M.

Deng, D.

D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25(1), 83–87 (2008).

D. Deng, Z. Xia, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution, M2 factor of beams generated by Gaussian mirror resonator,” Optik (Stuttg.) 118(11), 533–536 (2007).

D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341(1-4), 352–356 (2005).

D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M2 factor of hollow Gaussian beams,” Appl. Opt. 44(33), 7187–7190 (2005).
[PubMed]

Eyyuboglu, H. T.

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90(1), 87–92 (2008).

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).

Fan, Z.

Fenichel, H.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[PubMed]

Fu, X.

Ghafary, B.

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).

Gori, F.

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999).

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).

Gutiérrez-Vega, J. C.

He, S.

Heckenberg, N. R.

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).

Iturbe-Castillo, M. D.

Jhe, W.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4-6), 287–292 (1997).

Kashani, F. D.

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).

Kim, K.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4-6), 287–292 (1997).

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).

Lee, H. S.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[PubMed]

Lee, K.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4-6), 287–292 (1997).

Li, Q.

Lin, J.

Lin, Q.

Littman, M. G.

Liu, J.

Liu, S.

Liu, Z.

Lou, Q.

Lu, X.

Lü, B.

Ma, H.

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).

Mahdieh, M. H.

M. H. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281(13), 3395–3402 (2008).

Martinez-Herrero, R.

McDuff, R.

Mei, Z.

Z. Mei and D. Zhao, “Generalized M2 factor of hard-edged diffracted controllable dark-hollow beams,” Opt. Commun. 263(2), 261–266 (2006).

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

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).

Mejias, P. M.

Noh, H.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4-6), 287–292 (1997).

Noriega-Manez, R. J.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).

Pare, C.

P. A. Belanger, Y. Champagne, and C. Pare, “Beam-propagation factor of diffracted laser-beams,” Opt. Commun. 105(3-4), 233–242 (1994).

Ricklin, J. C.

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).

Santarsiero, M.

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999).

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).

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).

Shao, J.

D. Deng, Z. Xia, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution, M2 factor of beams generated by Gaussian mirror resonator,” Optik (Stuttg.) 118(11), 533–536 (2007).

D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M2 factor of hollow Gaussian beams,” Appl. Opt. 44(33), 7187–7190 (2005).
[PubMed]

Sheppard, C. J. R.

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).

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).

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).

Smith, C. P.

Sona, A.

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).

Stewart, B. W.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[PubMed]

Sun, X.

Tian, G.

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).

Vahimaa, P.

Wang, F.

Wang, X.

Wang, Y.

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
[PubMed]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4-6), 287–292 (1997).

Wei, C.

D. Deng, Z. Xia, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution, M2 factor of beams generated by Gaussian mirror resonator,” Optik (Stuttg.) 118(11), 533–536 (2007).

D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M2 factor of hollow Gaussian beams,” Appl. Opt. 44(33), 7187–7190 (2005).
[PubMed]

Wen, Q.

White, A. G.

Wu, G.

Xia, Z.

D. Deng, Z. Xia, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution, M2 factor of beams generated by Gaussian mirror resonator,” Optik (Stuttg.) 118(11), 533–536 (2007).

Xu, S.

Yin, J.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4-6), 287–292 (1997).

Young, C. Y.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).

Yu, H.

Zhang, B.

Zhang, L.

Zhao, C.

Zhao, D.

Z. Mei and D. Zhao, “Generalized M2 factor of hard-edged diffracted controllable dark-hollow beams,” Opt. Commun. 263(2), 261–266 (2006).

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

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).

Zhao, H.

Zhou, G.

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (2009).

Zhou, J.

Appl. Opt. (2)

Appl. Phys. B (2)

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (2009).

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90(1), 87–92 (2008).

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).

J. Opt. A, Pure Appl. Opt. (2)

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 8(6), 537–545 (2006).

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (2)

Opt. Commun. (9)

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4-6), 287–292 (1997).

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).

P. A. Belanger, Y. Champagne, and C. Pare, “Beam-propagation factor of diffracted laser-beams,” Opt. Commun. 105(3-4), 233–242 (1994).

S. A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).

Z. Mei and D. Zhao, “Generalized M2 factor of hard-edged diffracted controllable dark-hollow beams,” Opt. Commun. 263(2), 261–266 (2006).

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).

M. H. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281(13), 3395–3402 (2008).

Opt. Eng. (2)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).

P. A. Belanger, “Beam quality factor of the LP01 mode of the step-index fiber,” Opt. Eng. 32(9), 2107–2109 (1993).

Opt. Express (5)

Opt. Laser Technol. (1)

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).

Opt. Lasers Eng. (1)

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).

Opt. Lett. (11)

Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
[PubMed]

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
[PubMed]

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32(21), 3179–3181 (2007).
[PubMed]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28(13), 1084–1086 (2003).
[PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[PubMed]

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).
[PubMed]

X. Wang and M. G. Littman, “Laser cavity for generation of variable-radius rings of light,” Opt. Lett. 18(10), 767–768 (1993).
[PubMed]

R. Martinez-Herrero and P. M. Mejias, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18(19), 1669–1671 (1993).
[PubMed]

R. Martinez-Herrero, P. M. Mejias, and M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagation through hard-edged apertures,” Opt. Lett. 20(2), 124–126 (1995).
[PubMed]

B. Lü, B. Zhang, and H. Ma, “Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine Gaussian beams,” Opt. Lett. 24(10), 640–642 (1999).

Optik (Stuttg.) (1)

D. Deng, Z. Xia, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution, M2 factor of beams generated by Gaussian mirror resonator,” Optik (Stuttg.) 118(11), 533–536 (2007).

Phys. Lett. A (2)

D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341(1-4), 352–356 (2005).

X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007).

Phys. Rev. A (1)

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[PubMed]

Phys. Rev. Lett. (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).

Other (6)

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E. Wolf, ed., North-Holland, Amsterdam, 2003, pp.119–204.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001).

L. C. Andrews, and R. L. Phillips, Laser beam propagation in the turbulent atmosphere, 2nd edition, SPIE press, Bellington, 2005.

L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Cross line (y = 0) of the normalized intensity distribution of a circular DHB for different values of N and p with w0=1mm

Fig. 2
Fig. 2

M2 -factor of a coherent circular DHB in the source plane (z = 0) versus N and p

Fig. 3
Fig. 3

Normalized M2 -factor of a coherent circular DHB on propagation in turbulent atmosphere for different values of the structure constant ( Cn2 ) of the turbulent atmosphere

Fig. 4
Fig. 4

Normalized M2 -factor of a coherent circular DHB on propagation in turbulent atmosphere for different values of inner scale of the turbulence ( l0 )

Fig. 5
Fig. 5

Normalized M2 -factor of a coherent circular DHB on propagation in turbulent atmosphere for different values of beam order N

Fig. 6
Fig. 6

Normalized M2 -factor of a coherent circular DHB on propagation in turbulent atmosphere for different values of scaling factor p

Fig. 7
Fig. 7

Normalized M2 -factor of a coherent circular DHB on propagation in turbulent atmosphere for different values of wavelengthλ

Fig. 8
Fig. 8

Normalized M2 -factors of coherent Gaussian beam, circular flat-topped beam and circular DHB on propagation in turbulent atmosphere.

Fig. 9
Fig. 9

Normalized M2 -factor of a partially coherent circular DHB on propagation in turbulent atmosphere for different values of the initial transverse coherence width σg

Equations (27)

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

EN(ρ;0)=n=1N(1)n1N(Nn)[exp(nρ2w02)exp(nρ2wp2)],
WN(ρ1',ρ2';0)=EN(ρ1';0)EN*(ρ2';0)                    =n=1Nm=1N(1)n+mN2(Nn)(Nm)[exp(nρ1'2w02)exp(nρ1'2wp2)]                        ×[exp(mρ2'2w02)exp(mρ2'2wp2)]exp[(ρ1'ρ2')22σg2],
W(ρ,ρd;z)=(k2πz)2W(ρ',ρd';0)                      ×exp[ikz(ρρ')·(ρdρd')H(ρd,ρd';z)]d2ρ'd2ρd',
ρ'=(ρ1'+ρ2')2, ρd'=ρ1'ρ2', ρ=(ρ1+ρ2)2, ρd=ρ1ρ2,
W(ρ',ρd';0)=W(ρ1',ρ2';0)=W(ρ'+ρd'2,ρ'ρd'2;0).
H(ρd,ρd';z)=4π2k2z01dξ0[1J0(κ|ρd'ξ+(1ξ)ρd|)]Φn(κ)κdκ,
W(ρ,ρd;z)=(12π)2W(ρ'',ρd+zkκd;0)                      ×exp{iρ·κd+iρ''·κdH(ρd,ρd+zkκd;z)}d2ρ''d2κd,
W(ρ'',ρd+zkκd;0)=n=1Nm=1N(1)n+mN2(Nn)(Nm)                                 {exp[A1ρ''2A2ρ''(ρd+zkκd)A3(ρd+zkκd)2]                                 exp[B1ρ''2B2ρ''(ρd+zkκd)B3(ρd+zkκd)2]                                 exp[C1ρ''2C2ρ''(ρd+zkκd)C3(ρd+zkκd)2]                                 +exp[D1ρ''2D2ρ''(ρd+zkκd)D3(ρd+zkκd)2]},
A1=nw02+mw02, A2=nw02mw02, A3=A14+12σg2, B1=nw02+mwp2,B2=nw02mwp2, B3=B14+12σg2, C1=nwp2+mw02, C2=nwp2mw02,C3=C14+12σg2, D1=nwp2+mwp2, D2=nwp2mwp2, D3=D14+12σg2.
h(ρ,θ;z)=(k2π)2W(ρ,ρd;z)exp(ikθ·ρd)d2ρd,
h(ρ,θ;z)=k216π3n=1Nm=1N(1)n+mN2(Nn)(Nm){1A1exp[a1ρd2b1κd2+c1ρd·κdikθ·ρdiρ·κdH(ρd,ρd+zkκd,z)]d2κdd2ρd1B1exp[a2ρd2b2κd2+c2ρd·κdikθ·ρdiρ·κdH(ρd,ρd+zkκd,z)]d2κdd2ρd1C1exp[a3ρd2b3κd2+c3ρd·κdikθ·ρdiρ·κdH(ρd,ρd+zkκd,z)]d2κdd2ρd+1D1exp[a4ρd2b4κd2+c4ρd·κdikθ·ρdiρ·κdH(ρd,ρd+zkκd,z)]d2κdd2ρd}
a1=A3A224A1,b1=A3z2k2+14A1A22z24A1k2+iA2z2kA1,c1=zA222kA12zA3kiA22A1,a2=B3B224B1,b2=B3z2k2+14B1B22z24B1k2+iB2z2kB1,c2=zB222kB12zB3kiB22B1,a3=C3C224C1,b3=C3z2k2+14C1C22z24C1k2+iC2z2kC1,c3=zC222kC12zC3kiC22C1,a4=D3D224D1,b4=D3z2k2+14D1D22z24D1k2+iD2z2kD1,c4=zD222kD12zD3kiD22D1.
exp(s2x2±qx)dx=πsexp(q24s2),(s>0)
M2(z)=k(ρ2θ2ρ·θ2)1/2            =k[(x2+y2)(θx2+θy2)(xθx+yθy)2]1/2,
<xn1yn2θxm1θym2>=1Pxn1yn2θxm1θym2h(ρ,θ,z)d2ρd2θ,
P=h(ρ,θ,z)d2ρd2θ.
P=πn=1Nm=1N(1)n+mN2(Nn)(Nm)(1A11B11C1+1D1),
<ρ2>=πPn=1Nm=1N(1)n+mN2(Nn)(Nm)[(z2k2+1A12A22z2A12k2)(z2k2+1B12B22z2B12k2)             (z2k2+1C12C22z2C12k2)+(z2k2+1D12D22z2D12k2)]+43π2Tz3,
<θ2>=πPn=1Nm=1N(1)n+mN2(Nn)(Nm)[(1k2A22A12k2)(1k2B22B12k2)            (1k2C22C12k2)+(1k2D22D12k2)]+4π2zT,                           
<ρ·θ>=πPn=1Nm=1N(1)n+mN2(Nn)(Nm)[(zC22k2C12zk2)(zD22k2D12zk2)               (zA22k2A12zk2)+(zB22k2B12zk2)]+2π2z2T,
T=0Φn(κ)κ3dκ.  
δ(s)=12πexp(isx)dx,     
δn(s)=12π(ix)nexp(isx)dx, (n=0, 1, 2) 
f(x)δn(x)dx=(1)nf(n)(0), (n=1, 2)  
          M2(z)=k(<ρ2><θ2><ρ·θ>2)1/2          =k({πPn=1Nm=1N(1)n+mN2(Nn)(Nm)[(z2k2+1A12A22z2A12k2)(z2k2+1B12B22z2B12k2)           (z2k2+1C12C22z2C12k2)+(z2k2+1D12D22z2D12k2)]+43π2Tz3}·{πPn=1Nm=1N(1)n+mN2(Nn)(Nm)           [(1k2A22A12k2)(1k2B22B12k2)(1k2C22C12k2)+(1k2D22D12k2)]+4π2zT}          {πPn=1Nm=1N(1)n+mN2(Nn)(Nm)[(zA22k2A12zk2)+(zB22k2B12zk2)+(zC22k2C12zk2)            (zD22k2D12zk2)]+2π2z2T}2)1/2.                                                                  
Φn(κ)=0.033Cn2κ11/3exp(κ2κm2) 
T=0Φn(κ)κ3dκ=0.1661Cn2l01/3

Metrics