Abstract

A group of virtual sources that generate a hollow Gaussian wave are determined on the basis of the superposition of beams. A closed-form expression is derived for the hollow Gaussian wave that in the appropriate limit yields the paraxial hollow Gaussian beam (HGB). From the perturbative series representation of a complex-source-point spherical wave, an infinite series nonparaxial correction expression for a HGB is derived. The infinite series expression of a HGB can provide accuracy up to any order of diffraction angle. The radiation intensity of the hollow Gaussian wave is ascertained, and the radiation intensity pattern is characterized. The total time-averaged power is evaluated. The characteristics of the quality of the paraxial beam approximation to the full hollow Gaussian wave are discussed.

© 2009 Optical Society of America

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  1. 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]
  2. K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602(R) (2001).
    [CrossRef]
  3. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
    [CrossRef] [PubMed]
  4. N. Simpson, K. Dholakia, L. Allen, and M. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52-54 (1997).
    [CrossRef] [PubMed]
  5. L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
    [CrossRef] [PubMed]
  6. X. Wang and M. G. Littman, “Laser cavity for generation of variable-radius rings of light,” Opt. Lett. 18, 767-768 (1993).
    [CrossRef] [PubMed]
  7. I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67-70 (1998).
    [CrossRef]
  8. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15, 2226-2234 (1998).
    [CrossRef]
  9. F. K. Fatemi and M. Bashkansky, “Generation of hollow beams by using a binary spatial light modulator,” Opt. Lett. 31, 864-866 (2006).
    [CrossRef] [PubMed]
  10. Z. J. Liu, H. F. Zhao, J. L. Liu, J. Lin, M. A. Ahmad, and S. T. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32, 2076-2078 (2007).
    [CrossRef] [PubMed]
  11. Y. J. Cai, X. H. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084-1086 (2003).
    [CrossRef] [PubMed]
  12. Y. Cai and S. He, “Propagation of hollow Gaussian beams through apertured paraxial optical systems,” J. Opt. Soc. Am. A 23, 1410-1418 (2006).
    [CrossRef]
  13. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353-1367 (2006).
    [CrossRef] [PubMed]
  14. D. G. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M2 factor of hollow Gaussian beams,” Appl. Opt. 44, 7187-7190 (2005).
    [CrossRef] [PubMed]
  15. D. M. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341, 352-356 (2005).
    [CrossRef]
  16. G. H. Wu, Q. H. Lou, and J. Zhou, “Analytical vectorial structure of hollow Gaussian beams in the far field,” Opt. Express 16, 6417-6424 (2008).
    [CrossRef] [PubMed]
  17. D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11, 2747-2752 (2003).
    [CrossRef] [PubMed]
  18. S. R. Seshadri, “Radiation pattern of cylindrically symmetric scalar Laguerre-Gauss beams,” Opt. Lett. 32, 1159-1161 (2007).
    [CrossRef] [PubMed]
  19. S. R. Seshadri, “Radiation pattern of azimuthally varying scalar Laguerre-Gauss waves,” J. Opt. Soc. Am. A 24, 3348-3353 (2007).
    [CrossRef]
  20. S. Sato and Y. Kozawa, “Hollow vortex beams,” J. Opt. Soc. Am. A 26, 142-146 (2009).
    [CrossRef]
  21. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
    [CrossRef]
  22. M. Couture and P. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355-359 (1981).
    [CrossRef]
  23. S. R. Seshadri, “Nonparaxial corrections for the fundamental Gaussian beam,” J. Opt. Soc. Am. A 19, 2134-2141 (2002).
    [CrossRef]
  24. R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. 28, 774-776 (2003).
    [CrossRef] [PubMed]
  25. K. L. Duan and B. D. Lü, Opt. Express 11, 1474-1480 (2003).
    [CrossRef] [PubMed]
  26. S. R. Seshadri, “Quality of paraxial electromagnetic beams,” Appl. Opt. 45, 5335-5345 (2006).
    [CrossRef] [PubMed]
  27. D. M. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23, 1228-1234 (2006).
    [CrossRef]
  28. D. M. Deng, Q. Guo, L. J. Wu, and X. B. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B 24, 636-643 (2007).
    [CrossRef]
  29. D. M. Deng, Q. Guo, S. Lan, and X. B. Yang, “Application of the multiscale singular perturbation method to nonparaxial beam propagations in free space,” J. Opt. Soc. Am. A 24, 3317-3325 (2007).
    [CrossRef]
  30. Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express 15, 11942-11951 (2007).
    [CrossRef] [PubMed]
  31. D. G. Deng, H. Yu, S. Q. Xu, G. L. Tian, and Z. X. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25, 83-87 (2008).
    [CrossRef]
  32. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
    [CrossRef]
  33. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751-760 (1976).
    [CrossRef]
  34. S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699-700 (1977).
    [CrossRef]
  35. S. R. Seshadri, “Virtual source for the Bessel-Gauss beam,” Opt. Lett. 27, 998-1000 (2002).
    [CrossRef]
  36. S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27, 1872-1874 (2002).
    [CrossRef]
  37. S. R. Seshadri, “Virtual source for a Hermite-Gauss beam,” Opt. Lett. 28, 595-597 (2003).
    [CrossRef] [PubMed]
  38. M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29, 2213-2215 (2004).
    [CrossRef] [PubMed]
  39. Y. C. Zhang, Y. J. Song, Z. R. Chen, J. H. Ji, and Z. X. Shi, “Virtual sources for a cosh-Gaussian beam,” Opt. Lett. 32, 292-294 (2007).
    [CrossRef] [PubMed]
  40. D. M. Deng and Q. Guo, “Elegant Hermite-Laguerre-Gaussian beams,” Opt. Lett. 33, 1225-1227 (2008).
    [CrossRef] [PubMed]
  41. J. D. Joannopoulis, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton Univ. Press, 1995).
  42. E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. 10, 189-371 (1989).
    [CrossRef]
  43. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).
  44. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 287-290.

2009 (1)

2008 (3)

2007 (7)

2006 (5)

2005 (2)

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

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

2004 (1)

2003 (5)

2002 (3)

2001 (2)

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

1999 (1)

D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

1998 (3)

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67-70 (1998).
[CrossRef]

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]

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15, 2226-2234 (1998).
[CrossRef]

1997 (1)

1993 (1)

1989 (1)

E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. 10, 189-371 (1989).
[CrossRef]

1981 (1)

M. Couture and P. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355-359 (1981).
[CrossRef]

1977 (1)

1976 (1)

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

Ahmad, M. A.

Allen, L.

Arlt, J.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Bandres, M. A.

Bashkansky, M.

Belanger, P.

M. Couture and P. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355-359 (1981).
[CrossRef]

Bongs, K.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Borghi, R.

Bryant, P. E.

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Burger, S.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Cai, Y.

Cai, Y. J.

Chen, Z. R.

Couture, M.

M. Couture and P. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355-359 (1981).
[CrossRef]

Deng, D. G.

Deng, D. M.

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

Dettmer, S.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Dholakia, K.

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

N. Simpson, K. Dholakia, L. Allen, and M. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

Duan, K. L.

Ertmer, W.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Fan, Z.

Fan, Z. X.

Fatemi, F. K.

Felsen, L. B.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Fu, X.

Gan, X.

Ganic, D.

Grimm, R.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67-70 (1998).
[CrossRef]

Gu, M.

Guo, Q.

Gutiérrez-Vega, J. C.

He, S.

Hellweg, D.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

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]

Ji, J. H.

Joannopoulis, J. D.

J. D. Joannopoulis, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton Univ. Press, 1995).

Kozawa, Y.

Lan, S.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Lin, J.

Lin, Q.

Littman, M. G.

Liu, J. L.

Liu, S. T.

Liu, Z. J.

Lou, Q. H.

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Lu, X. H.

Lü, B. D.

MacDonald, M.

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 287-290.

Manek, I.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67-70 (1998).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Meade, R. D.

J. D. Joannopoulis, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton Univ. Press, 1995).

Mehta, D.

D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Mei, Z.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Ovchinnikov, Y. B.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67-70 (1998).
[CrossRef]

Padgett, M.

Paterson, L.

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Rief, M.

D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Rozas, D.

Sacks, Z. S.

Santarsiero, M.

Sato, S.

Sengstock, K.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Seshadri, S. R.

Shao, J.

Shi, Z. X.

Shin, S. Y.

Sibbett, W.

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Simmons, R. M.

D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Simpson, N.

Smith, D. A.

D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Song, Y. J.

Spudich, J. A.

D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Swartzlander, G. A.

Tian, G. L.

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, C.

Weniger, E. J.

E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. 10, 189-371 (1989).
[CrossRef]

Winn, J. N.

J. D. Joannopoulis, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton Univ. Press, 1995).

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 287-290.

Wu, G. H.

Wu, L. J.

Xu, S. Q.

Yang, X. B.

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]

Yu, H.

Zhang, Y. C.

Zhao, D.

Zhao, H. F.

Zhou, J.

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]

Appl. Opt. (2)

Comput. Phys. Rep. (1)

E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. 10, 189-371 (1989).
[CrossRef]

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67-70 (1998).
[CrossRef]

Opt. Express (5)

Opt. Lett. (13)

S. R. Seshadri, “Radiation pattern of cylindrically symmetric scalar Laguerre-Gauss beams,” Opt. Lett. 32, 1159-1161 (2007).
[CrossRef] [PubMed]

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

M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29, 2213-2215 (2004).
[CrossRef] [PubMed]

F. K. Fatemi and M. Bashkansky, “Generation of hollow beams by using a binary spatial light modulator,” Opt. Lett. 31, 864-866 (2006).
[CrossRef] [PubMed]

S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27, 1872-1874 (2002).
[CrossRef]

S. R. Seshadri, “Virtual source for a Hermite-Gauss beam,” Opt. Lett. 28, 595-597 (2003).
[CrossRef] [PubMed]

R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. 28, 774-776 (2003).
[CrossRef] [PubMed]

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

D. M. Deng and Q. Guo, “Elegant Hermite-Laguerre-Gaussian beams,” Opt. Lett. 33, 1225-1227 (2008).
[CrossRef] [PubMed]

Y. C. Zhang, Y. J. Song, Z. R. Chen, J. H. Ji, and Z. X. Shi, “Virtual sources for a cosh-Gaussian beam,” Opt. Lett. 32, 292-294 (2007).
[CrossRef] [PubMed]

N. Simpson, K. Dholakia, L. Allen, and M. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

S. R. Seshadri, “Virtual source for the Bessel-Gauss beam,” Opt. Lett. 27, 998-1000 (2002).
[CrossRef]

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

Phys. Lett. A (1)

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

Phys. Rev. A (4)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

M. Couture and P. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355-359 (1981).
[CrossRef]

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]

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Science (2)

D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Other (3)

J. D. Joannopoulis, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton Univ. Press, 1995).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 287-290.

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Figures (4)

Fig. 1
Fig. 1

Normalized time-averaged power flow per unit area in the plane (a) z = 0.3 z R and (b) z = 0.6 z R of the HGB with n = 4 evaluated by Eqs. (15, 17) with s = 0 (solid curve), i.e., the paraxial solution; s = 2 (dashed curve), i.e., up to the fourth-order correction solution; s = 3 (dotted curve), i.e., up to the sixth-order correction solution; and by Eqs. (12, 17) (dashed–dotted curve), i.e., the exact solution.

Fig. 2
Fig. 2

Normalized time-averaged power flow per unit area versus z z R for (a) ρ = 0 , i.e., on-axis, evaluated by Eqs. (16, 17) and (b) ρ = λ evaluated by Eqs. (15, 17) of the HGB with n = 4 and s = 0 (solid curve), i.e., the paraxial solution; s = 2 (dashed curve), i.e., up to the fourth-order correction solution; s = 3 (dotted curve), i.e., up to the sixth-order correction solution; and by Eqs. (12, 17) (dashed–dotted curve), i.e., the exact solution.

Fig. 3
Fig. 3

Radiation intensity pattern of the hollow Gaussian wave for the mode numbers (a) n = 0 , 1 , 2 and (b) n = 3 , 4 , 5 and k w 0 = 2.980 .

Fig. 4
Fig. 4

( 1 P n as functions of w 0 λ for n = 0 , 1 , 2 , 3 and for 0.05 < w 0 λ < 1 . P n is the power of the hollow Gaussian wave of mode number n; λ is the wavelength.

Equations (23)

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E n ( ρ , 0 ) = C 0 ( ρ 2 w 0 2 ) n exp ( ρ 2 w 0 2 ) ,
E n ( ρ , 0 ) = C 0 n ! m = 0 n ( 1 ) m ( m n ) L m ( ρ 2 w 0 2 ) exp ( ρ 2 w 0 2 ) .
( t 2 + 2 z 2 + k 2 ) U n ( ρ , z ) = C 0 n ! m = 0 n ( m n ) S cs ( m ) ( t 2 ) m δ ( ρ ) ρ δ ( z z cs ) ,
U n ( ρ , z ) = 0 + U ̃ n ( α , z ) J 0 ( α ρ ) α d α ,
U ̃ n ( α , z ) = 0 + U n ( ρ , z ) J 0 ( α ρ ) ρ d ρ ,
U n ( ρ , z ) = C 0 n ! m = 0 n ( m n ) i 2 S cs ( m ) 0 + d α α ( α 2 ) m exp [ i ζ ( z z cs ) ] J 0 ( α ρ ) ζ ,
U n p ( ρ , z ) = C 0 n ! m = 0 n ( m n ) ( 1 ) m i 2 k S cs ( m ) exp [ i k ( z z cs ) ] 0 + d α α 2 m + 1 exp [ i α 2 ( z z cs ) ( 2 k ) ] J 0 ( α ρ ) .
0 + α 2 m + 1 exp ( a 2 α 2 ) J 0 ( α β ) d α = m ! 2 a 2 ( m + 1 ) exp ( β 2 4 a 2 ) L m ( β 2 4 a 2 ) ,
U n p ( ρ , z ) = C 0 n ! m = 0 n ( m n ) i 4 k m ! ( 1 ) m S cs ( m ) exp [ i k ( z z cs ) ] ( 2 i k z z cs ) m + 1 exp [ i k ρ 2 2 ( z z cs ) ] L m [ i k ρ 2 2 ( z z cs ) ] .
z cs = i k w 0 2 2 = i z R ,
S cs ( m ) = 2 i z R m ! ( w 0 2 4 ) m exp ( k z R ) .
U n ( ρ , z ) = C 0 n ! m = 0 n ( m n ) ( 1 ) m z R m ! ( w 0 2 4 ) m exp ( k z R ) 0 + d α α 2 m + 1 exp [ i ζ ( z z cs ) ] J 0 ( α ρ ) ζ .
( ρ 2 + 2 z 2 + k 2 ) G ( ρ , z ) = S cs ( m ) δ ( ρ ) 2 π ρ δ ( z z cs )
U n ( ρ , z ) = i C 0 n ! m = 0 n ( m n ) z R m ! ( w 0 2 4 ) m exp ( k z R ) ( t 2 ) m [ exp ( i k R ) R ] .
U n ( ρ , z ) = C 0 exp ( i k z ) exp [ i k ρ 2 2 ( z i z R ) ] s = 0 σ 2 s n ! m = 0 n ( m n ) t = 0 s α t s ( 1 ) m ( 2 s t + m ) ! m ! ( 2 s t ) ! ( i z R z i z R ) s + m + 1 L 2 s t + m [ i k ρ 2 2 ( z i z R ) ] ,
U n ( 0 , z ) = C 0 exp ( i k z ) s = 0 σ 2 s n ! m = 0 n ( m n ) t = 0 s ( 1 ) m α t s ( 2 s t + m ) ! m ! ( 2 s t ) ! ( i z R z i z R ) s + m + 1 .
Π z , n ( ρ , z ) = 1 2 Re [ i ω E n * ( ρ , z ) z E n ( ρ , z ) ] ,
P n = 0 2 π d φ 0 Π z , n ( ρ , z ) ρ d ρ .
P n = 0 2 π 0 π 2 d θ d φ Φ n ( θ , φ ) sin θ ,
Φ n ( θ , φ ) = 1 2 π σ 2 2 2 n ( 2 n ) ! exp [ k 2 w 0 2 ( 1 cos θ ) ] Ψ n 2 ( θ ) ,
P 0 = 1 exp ( k 2 w 0 2 ) ,
P 1 = 1 σ 2 + 3 σ 4 ( 3 2 + 2 σ 2 + 3 σ 4 k 2 w 0 2 1 8 k 4 w 0 4 ) exp ( k 2 w 0 2 ) ,
P 2 = 1 2 3 σ 2 + 5 σ 4 45 σ 6 + 105 σ 8 + ( 7 24 + 2 3 σ 2 25 4 σ 4 60 σ 6 105 σ 8 + 13 6 k 2 w 0 2 13 16 k 4 w 0 4 + 1 12 k 6 w 0 6 1 384 k 8 w 0 8 ) exp ( k 2 w 0 2 ) .

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