Abstract

Based on the vectorial Raleigh–Sommerfeld diffraction integral, the nonparaxial propagation of vectorial hollow Gaussian beams (HGBs) in free space is studied. The far-field and paraxial cases can be treated as special cases of our general results. The typical numerical examples are given to illustrate our analytical results and comparisons between the different approximations present that the f parameter still plays an important role in determining the nonparaxiality of vectorial diffracted HGBs.

© 2008 Optical Society of America

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References

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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, 4713-4716 (1997).
    [CrossRef]
  2. Yu. B. Ovchinnikov, I. Manek, and R. Grimm, "Surface trap for Cs atoms based on evanescent-wave cooling," Phys. Rev. Lett. 79, 2225-2228 (1997).
    [CrossRef]
  3. Y. Song, D. Milam, and W. T. Hill, "Long, narrow all-light atom guide," Opt. Lett. 24, 1805-1807 (1999).
    [CrossRef]
  4. H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, "Atomic funnel with evanescent light," Phys. Rev. A 56, 712-718 (1997).
    [CrossRef]
  5. W. L. Power, L. Allen, M. Babiker, and V. E. Lembessis, "Atomic motion in light beams possessing orbital angular momentum," Phys. Rev. A 52, 479-488 (1995).
    [CrossRef] [PubMed]
  6. H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
    [CrossRef] [PubMed]
  7. 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, 2076-2078 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]
  9. Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beam and its propagation," Opt. Lett. 28, 1084-1086 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. Y. Cai and L. Zhang, "Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties," J. Opt. Soc. Am. B 23, 1398-1407 (2006).
    [CrossRef]
  13. Y. Cai and L. Zhang, "Propagation of a hollow Gaussian beam through a paraxial misaligned optical system," Opt. Commun. 265, 607-615 (2006).
    [CrossRef]
  14. Y. Cai, C. Chen, and F. Wang, "Modified hollow Gaussian beam and its paraxial propagation," Opt. Commun. 278, 34-41 (2007).
    [CrossRef]
  15. Y. Cai and S. He, "Propagation of various dark hollow beams in a turbulent atmosphere," Opt. Express 14, 1353-1367 (2006).
    [CrossRef] [PubMed]
  16. C. Zheng, "Fractional Fourier transform for a hollow Gaussian beam," Phys. Lett. A 355, 156-161 (2006).
    [CrossRef]
  17. 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, 7187-7190 (2005).
    [CrossRef] [PubMed]
  18. D. Deng, "Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture," Phys. Lett. A 341, 352-356 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. J. D. Joannopoulis, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).
  24. R. K. Luneberg, Mathematical Theory of Optics (U. California Press, 1964).
  25. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1953).

2007 (3)

P. Liu and B. Lu, "Phase singularities of the transverse field component of high numerical aperture dark-hollow Gaussian beams in the focal region," Opt. Commun. 272, 1-8 (2007).
[CrossRef]

Y. Cai, C. Chen, and F. Wang, "Modified hollow Gaussian beam and its paraxial propagation," Opt. Commun. 278, 34-41 (2007).
[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, 2076-2078 (2007).
[CrossRef] [PubMed]

2006 (7)

C. Zheng, "Fractional Fourier transform for a hollow Gaussian beam," Phys. Lett. A 355, 156-161 (2006).
[CrossRef]

D. Deng, "Nonparaxial propagation of radially polarized light beams," J. Opt. Soc. Am. A 23, 1228-1234 (2006).
[CrossRef]

Z. Gao and B. Lu, "Nonparaxial dark-hollow Gaussian beams," Chin. Phys. Lett. 23, 106-109 (2006).
[CrossRef]

Y. Cai and L. Zhang, "Propagation of a hollow Gaussian beam through a paraxial misaligned optical system," Opt. Commun. 265, 607-615 (2006).
[CrossRef]

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

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]

Y. Cai and L. Zhang, "Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties," J. Opt. Soc. Am. B 23, 1398-1407 (2006).
[CrossRef]

2005 (2)

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, 7187-7190 (2005).
[CrossRef] [PubMed]

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

2004 (2)

2003 (2)

1999 (1)

1997 (3)

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, 4713-4716 (1997).
[CrossRef]

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, "Surface trap for Cs atoms based on evanescent-wave cooling," Phys. Rev. Lett. 79, 2225-2228 (1997).
[CrossRef]

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, "Atomic funnel with evanescent light," Phys. Rev. A 56, 712-718 (1997).
[CrossRef]

1995 (1)

W. L. Power, L. Allen, M. Babiker, and V. E. Lembessis, "Atomic motion in light beams possessing orbital angular momentum," Phys. Rev. A 52, 479-488 (1995).
[CrossRef] [PubMed]

1994 (1)

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Appl. Opt. (1)

Chin. Phys. Lett. (1)

Z. Gao and B. Lu, "Nonparaxial dark-hollow Gaussian beams," Chin. Phys. Lett. 23, 106-109 (2006).
[CrossRef]

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

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

Opt. Commun. (3)

P. Liu and B. Lu, "Phase singularities of the transverse field component of high numerical aperture dark-hollow Gaussian beams in the focal region," Opt. Commun. 272, 1-8 (2007).
[CrossRef]

Y. Cai and L. Zhang, "Propagation of a hollow Gaussian beam through a paraxial misaligned optical system," Opt. Commun. 265, 607-615 (2006).
[CrossRef]

Y. Cai, C. Chen, and F. Wang, "Modified hollow Gaussian beam and its paraxial propagation," Opt. Commun. 278, 34-41 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Lett. A (2)

C. Zheng, "Fractional Fourier transform for a hollow Gaussian beam," Phys. Lett. A 355, 156-161 (2006).
[CrossRef]

D. 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 (3)

H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, "Atomic funnel with evanescent light," Phys. Rev. A 56, 712-718 (1997).
[CrossRef]

W. L. Power, L. Allen, M. Babiker, and V. E. Lembessis, "Atomic motion in light beams possessing orbital angular momentum," Phys. Rev. A 52, 479-488 (1995).
[CrossRef] [PubMed]

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, "Holographic nondiverging hollow beam," Phys. Rev. A 49, 4922-4927 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (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, 4713-4716 (1997).
[CrossRef]

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, "Surface trap for Cs atoms based on evanescent-wave cooling," Phys. Rev. Lett. 79, 2225-2228 (1997).
[CrossRef]

Other (3)

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

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, 1964).

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

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Figures (5)

Fig. 1
Fig. 1

Transversal irradiance distribution of the HGBs for nonparaxial propagation in free space at the plane (a) z = 0.1 z R , (b) z = 0.4 z R , (c) z = 0.7 z R , and (d) z = z R .

Fig. 2
Fig. 2

Transversal irradiance distribution of the HGBs with different beam order m for nonparaxial propagation in free space at the plane z = z R (a) m = 0 , (b) m = 3 , (c) m = 7 , and (d) m = 10

Fig. 3
Fig. 3

Transversal irradiance distribution of the HGBs at the plane z = z R for nonparaxial and paraxial approximation (a) m = 5 , f = 0.1 and (b) m = 5 , f = 0.5 .

Fig. 4
Fig. 4

Transversal irradiance distribution of the HGBs at the plane z = 5 z R for nonparaxial and paraxial approximation (a) m = 5 , f = 0.1 and (b) m = 5 , f = 0.5

Fig. 5
Fig. 5

Transversal irradiance distribution of the HGBs at the plane z = 10 z R for nonparaxial and far-field approximation (a) m = 5 , f = 0.1 and (b) m = 5 , f = 0.5 .

Equations (21)

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E m x ( x 0 , y 0 , 0 ) = G 0 ( x 0 2 + y 0 2 ω 0 2 ) m exp ( x 0 2 + y 0 2 ω 0 2 ) ,
m = 0 , 1 , 2 , ,
E m y ( x 0 , y 0 , 0 ) = 0 ,
E m x ( r ) = 1 2 π E m x ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
E m y ( r ) = 1 2 π E m y ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
E m z ( r ) = 1 2 π [ E m x ( x 0 , y 0 , 0 ) G ( r , r 0 ) x + E m y ( x 0 , y 0 , 0 ) G ( r , r 0 ) y ] d x 0 d y 0 ,
G ( r , r 0 ) = exp ( i k r r 0 ) r r 0 .
r r 0 r + x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r .
E m x ( x , y , z ) = i k G 0 z r exp ( i k r ) r ω 0 2 m 0 ρ 0 2 m exp ( g ρ 0 2 ) J 0 ( k h ρ 0 ) ρ 0 d ρ 0 ,
E m y ( x , y , z ) = 0 ,
E m z ( x , y , z ) = i k G 0 x r exp ( i k r ) r ω 0 2 m 0 ρ 0 2 m exp ( g ρ 0 2 ) J 0 ( k h ρ 0 ) ρ 0 d ρ 0 , k G 0 x r exp ( i k r ) r ω 0 2 m 1 x 2 + y 2 0 ρ 0 2 m exp ( g ρ 0 2 ) J 1 ( k h ρ 0 ) ρ 0 2 d ρ 0 ,
0 exp ( p t ) t q 2 + n J q ( 2 a 1 2 t 1 2 ) d t = n ! a q 2 p ( n + q + 1 ) exp ( a p ) L n q ( a p ) ,
E m x ( x , y , z ) = i k G 0 z m ! 2 r 2 ω 0 2 m exp ( i k r ) g ( m + 1 ) exp ( k 2 h 2 4 g ) L m 0 ( k 2 h 2 4 g ) ,
E m y ( x , y , z ) = 0 ,
E m z ( x , y , z ) = k G 0 x m ! 2 r 2 ω 0 2 m exp ( i k r ) g ( m + 1 ) exp ( k 2 h 2 4 g ) × [ i L m 0 ( k 2 h 2 4 g ) k h L m 1 ( k 2 h 2 4 g ) 2 g x 2 + y 2 ] .
E m x p ( x , y , z ) = i G 0 m ! z R z ( 1 i z R z ) m + 1 exp [ i k z ( z R z ) 2 ( x 2 + y 2 ) ω 0 2 1 i z R z ] × L m 0 ( ( z R z ) 2 ( x 2 + y 2 ) ω 0 2 1 i z R z ) ,
E m y p ( x , y , z ) = 0 ,
E m z p ( x , y , z ) = 2 G 0 m ! f ( z R z ) 2 ( 1 i z R z ) m + 1 x ω 0 exp [ i k z ( z R z ) 2 ( x 2 + y 2 ) ω 0 2 1 i z R z ] × { i L m 0 ( ( z R z ) 2 ( x 2 + y 2 ) ω 0 2 1 i z R z ) z R z 1 i z R z L m 1 ( ( z R z ) 2 ( x 2 + y 2 ) ω 0 2 1 i z R z ) } ,
E m x f ( x , y , z ) = i G 0 m ! z R z exp [ i k z ( z R z ) 2 ( x 2 + y 2 ) ω 0 2 ] L m 0 ( ( z R z ) 2 ( x 2 + y 2 ) ω 0 2 ) ,
E m y f ( x , y , z ) = 0 ,
E m z f ( x , y , z ) = 2 G 0 m ! f ( z R z ) 2 x ω 0 exp [ i k z ( z R z ) 2 x 2 + y 2 ω 0 2 ] × { i L m 0 ( ( z R z ) 2 x 2 + y 2 ω 0 2 ) z R z L m 1 ( ( z R z ) 2 x 2 + y 2 ω 0 2 ) } .

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