Abstract

Based on the operator transformation technique, the multiple complex point sources required to generate a coherent superposition of waves are introduced and a closed-form analytical expression is derived for this composite wave. From the expression of the composite wave, the paraxial approximation and the nonparaxial corrections of all orders for the corresponding paraxial beam are determined. The paraxial composite beam uniformly represents off-axis Gaussian beams (GBs), sin(cos)-GBs, sinh(cosh)-GBs, nth-order modified Bessel–GBs, and Bessel–GBs with topological charge.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Strohschein, H. J. J. Seguin, and C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. 37, 1045-1048 (1998).
    [CrossRef]
  2. A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A 18, 1897-1904 (2001).
    [CrossRef]
  3. J. P. Yin, W. J. Gao, and Y. F. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E.Wolf, ed. (North-Holland, 2003), pp.119-204.
    [CrossRef]
  4. K. C. Zhu, G. Q. Zhou, X. G. Li, X. J. Zheng, and H. Q. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16, 21315-21320 (2008).
    [CrossRef] [PubMed]
  5. G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
    [CrossRef]
  6. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751-760 (1976).
    [CrossRef]
  7. 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]
  8. M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355-359 (1981).
    [CrossRef]
  9. S. R. Seshadri, “Virtual source for the Bessel-Gauss beam,” Opt. Lett. 27, 998-100 (2002).
    [CrossRef]
  10. S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27, 1872-1874 (2002).
    [CrossRef]
  11. S. R. Seshadri, “Virtual source for a Hermite-Gauss beam,” Opt. Lett. 28, 595-597 (2003).
    [CrossRef] [PubMed]
  12. M. A. Bandres and J. C. Gutierrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29, 2213-2215 (2004).
    [CrossRef] [PubMed]
  13. D. M. Deng and Q. Guo, “Elegant Hermite-Laguerre-Gaussian beams,” Opt. Lett. 33, 1225-1227 (2008).
    [CrossRef] [PubMed]
  14. 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]
  15. L. G. Wang, L. Q. Wang, and S. Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088-1094 (2009).
    [CrossRef]
  16. B. Lü and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17, 2005-2009 (2000).
    [CrossRef]
  17. G. H. Wu, Q. H. Lou, J. Zhou, J. X. Dong, and Y. R. Wei, “Beam combination of a radial laser array: flat-topped beam,” Opt. Laser Technol. 40, 890-894 (2008).
    [CrossRef]
  18. L. W. Casperson, D. G. Hall, and A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14, 3341-3348 (1997).
    [CrossRef]
  19. L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954-961 (1998).
    [CrossRef]
  20. A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,”J. Opt. Soc. Am. A 15, 2425-2432 (1998).
    [CrossRef]
  21. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  22. Y. J. Li, H. Lee, and E. Wolf, “New generalized Bessel-Gauss beams,” J. Opt. Soc. Am. A 21, 640-646 (2004).
    [CrossRef]
  23. S. R. Seshadri, “Scalar modified Bessel-Gauss beams and waves,” J. Opt. Soc. Am. A 24, 2837-2842 (2007).
    [CrossRef]
  24. S. R. Seshadri, “Electromagnetic modified Bessel-Gauss beams and waves,” J. Opt. Soc. Am. A 25, 1-8 (2008).
    [CrossRef]

2009 (1)

L. G. Wang, L. Q. Wang, and S. Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088-1094 (2009).
[CrossRef]

2008 (4)

2007 (2)

2004 (2)

2003 (1)

2002 (2)

2001 (1)

2000 (1)

1998 (3)

1997 (1)

1987 (1)

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

1981 (1)

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

1977 (1)

1976 (1)

1971 (1)

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

Bandres, M. A.

Belanger, P. A.

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

Capjack, C. E.

Casperson, L. W.

Chen, Z. R.

Couture, M.

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

Deng, D. M.

Deschamps, G. A.

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

Dong, J. X.

G. H. Wu, Q. H. Lou, J. Zhou, J. X. Dong, and Y. R. Wei, “Beam combination of a radial laser array: flat-topped beam,” Opt. Laser Technol. 40, 890-894 (2008).
[CrossRef]

Felsen, L. B.

Gao, W. J.

J. P. Yin, W. J. Gao, and Y. F. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E.Wolf, ed. (North-Holland, 2003), pp.119-204.
[CrossRef]

Gori, F.

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

Guattari, G.

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

Guo, Q.

Gutierrez-Vega, J. C.

Hall, D. G.

Ji, J. H.

Lee, H.

Li, X. G.

Li, Y. J.

Lou, Q. H.

G. H. Wu, Q. H. Lou, J. Zhou, J. X. Dong, and Y. R. Wei, “Beam combination of a radial laser array: flat-topped beam,” Opt. Laser Technol. 40, 890-894 (2008).
[CrossRef]

Lü, B.

Ma, H.

Padovani, C.

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

Seguin, H. J. J.

Seshadri, S. R.

Shi, Z. X.

Shin, S. Y.

Song, Y. J.

Strohschein, J. D.

Tang, H. Q.

Tovar, A. A.

Wang, L. G.

L. G. Wang, L. Q. Wang, and S. Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088-1094 (2009).
[CrossRef]

Wang, L. Q.

L. G. Wang, L. Q. Wang, and S. Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088-1094 (2009).
[CrossRef]

Wei, Y. R.

G. H. Wu, Q. H. Lou, J. Zhou, J. X. Dong, and Y. R. Wei, “Beam combination of a radial laser array: flat-topped beam,” Opt. Laser Technol. 40, 890-894 (2008).
[CrossRef]

Wolf, E.

Wu, G. H.

G. H. Wu, Q. H. Lou, J. Zhou, J. X. Dong, and Y. R. Wei, “Beam combination of a radial laser array: flat-topped beam,” Opt. Laser Technol. 40, 890-894 (2008).
[CrossRef]

Yin, J. P.

J. P. Yin, W. J. Gao, and Y. F. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E.Wolf, ed. (North-Holland, 2003), pp.119-204.
[CrossRef]

Zhang, Y. C.

Zheng, X. J.

Zhou, G. Q.

Zhou, J.

G. H. Wu, Q. H. Lou, J. Zhou, J. X. Dong, and Y. R. Wei, “Beam combination of a radial laser array: flat-topped beam,” Opt. Laser Technol. 40, 890-894 (2008).
[CrossRef]

Zhu, K. C.

Zhu, S. Y.

L. G. Wang, L. Q. Wang, and S. Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088-1094 (2009).
[CrossRef]

Zhu, Y. F.

J. P. Yin, W. J. Gao, and Y. F. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E.Wolf, ed. (North-Holland, 2003), pp.119-204.
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

Opt. Commun. (2)

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

L. G. Wang, L. Q. Wang, and S. Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088-1094 (2009).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

G. H. Wu, Q. H. Lou, J. Zhou, J. X. Dong, and Y. R. Wei, “Beam combination of a radial laser array: flat-topped beam,” Opt. Laser Technol. 40, 890-894 (2008).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. A (1)

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

Other (1)

J. P. Yin, W. J. Gao, and Y. F. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E.Wolf, ed. (North-Holland, 2003), pp.119-204.
[CrossRef]

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 (3)

Fig. 1
Fig. 1

Normalized intensity and phase distributions (inset) of the coherent composite wave (12) propagating respectively at z = 0 (left) and z = z R plane (right) in free space, where M = 60 , ϕ m = 2 m π M , Φ m = 4 ϕ m , and u = v = 2 i w 0 , which relates to a paraxial BG with topological charge index n = 4 .

Fig. 2
Fig. 2

Comparison of the normalized intensity respectively associated with the paraxial BGB (dotted-dashed curve), the BGB with added nonparaxial corrections (to ε 4 ) (dotted curve), and the corresponding Bessel–Gaussian wave (solid curve) at the given planes. The parameters are denoted in the figures.

Fig. 3
Fig. 3

Discrepancy between the intensity of the paraxial BGB and of the corresponding Bessel–Gaussian wave at z = z R 5 (solid curve) and z = 4 z R (dotted curve) for n = 1 , w 0 = 2 λ .

Equations (17)

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

2 E + k 2 E = 0 ,
T ̂ ( x , y , R ) = 1 M m = 0 M 1 T ̂ m ,
E G p ( ρ , z ) = Q exp ( i k z Q ρ 2 ) ,
E p ( ρ , z ) = T ̂ E G p ( ρ , z ) = Q M m = 0 M 1 exp ( i k z Q ρ m 2 + i n Φ m )
r m = ( x u cos ϕ m ) 2 + ( y v sin ϕ m ) 2 ;
exp ( Q ρ m 2 + i n Φ m ) = exp [ i n ϕ m Q ( ρ 2 + β 2 4 ) Q β ρ cos ( θ ϕ m ) ] = exp [ Q ( ρ 2 + β 2 4 ) ] l = I l ( Q β ρ ) exp [ i l θ + i ( n l ) ϕ m ] ,
E p ( x , y , z ) = Q A B ( Q ) l = I n + l M ( Q β ρ ) exp [ i k z Q ρ 2 + i ( n + l M ) ϕ ] Q A B ( Q ) I n ( Q β ρ ) exp ( i k z Q ρ 2 + i n ϕ ) ,
E p ( x , y , z ) i n Q A B 1 ( Q ) J n ( Q β ρ ) exp ( i k z Q ρ 2 + i n ϕ ) ,
( 2 + k 2 ) E G = S c s δ ( x ) δ ( y ) δ ( z i z R ) ,
E G ( x , y , z ) = S c s exp ( i k R ) 4 π R ,
T ̂ ( 2 + k 2 ) E G = ( 2 + k 2 ) E = S c s T ̂ δ ( x ) δ ( y ) δ ( z i z R ) .
S c s T ̂ δ ( x ) δ ( y ) δ ( z i z R ) = S c s M m = 0 M 1 exp ( i Φ m ) δ ( x u cos ϕ m ) δ ( y v sin ϕ m ) δ ( z i z R ) ,
E ( x , y , z ) = T ̂ E G ( ρ , z ) = S c s 4 π M m = 0 M 1 exp ( i Φ m + i k R m ) R m ,
E G ( ρ , z ) = E G p ( ρ , z ) + n = 1 ε 2 n E G 2 n ( ρ , z ) ,
E G 2 n ( ρ , z ) = ( Q ρ ) 2 n L n n ( Q ρ 2 ) E G p ( ρ , z ) ,
L n n ( x ) = ( 2 n ) ! m = 0 n ( x ) m m ! ( n m ) ! ( n + m ) ! .
E ( x , y , z ) = T ̂ E G ( ρ , z ) = 1 M m = 0 M 1 exp ( i Φ m ) { E G p ( ρ m , z ) + n = 1 ε 2 n E G 2 n ( ρ m , z ) } .

Metrics