Abstract

Based on the vector angular spectrum representation of electromagnetic beams and the method of stationary phase, the analytical vectorial structure of the controllable dark-hollow beam in the far field is derived. Analytical expressions of the energy flux for the TE term, the TM term, and the controllable dark-hollow beam in the far field are presented. The normalized energy flux distributions of the TE term, the TM term, and the controllable dark-hollow beam in the far field are illustrated and analyzed with numerical examples, and the effects of the different parameters on the normalized energy flux distributions are discussed in detail.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. J. Yin, Y. Zhu, W. Wang, Y. Wang, and W. Jhe, “Optical potential for atom guidance in a dark hollow laser beam,” J. Opt. Soc. Am. B 15, 25-33 (1998).
    [CrossRef]
  3. X. Xu, Y. Wang, and W. Jhe, “Theory of atom guidance in a hollow laser beam: dressed-atom approach,” J. Opt. Soc. Am. B 17, 1039-1050 (2002).
    [CrossRef]
  4. 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]
  5. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353-1367 (2006).
    [CrossRef] [PubMed]
  6. D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25, 83-87 (2008).
    [CrossRef]
  7. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155-1166 (1996).
  8. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
    [CrossRef]
  9. K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29-34 (2002).
    [CrossRef]
  10. Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084-1086 (2003).
    [CrossRef] [PubMed]
  11. Z. Mei and D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A 22, 1898-1902 (2005).
    [CrossRef]
  12. Z. Mei and D. Zhao, “Generalized beam propagation factor of hard-edged diffracted controllable dark-hollow beams,” Opt. Commun. 263, 261-266 (2006).
    [CrossRef]
  13. Z. Mei and D. Zhao, “Nonparaxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A 25, 537-542 (2008).
    [CrossRef]
  14. Z. Mei and D. Zhao, “Controllable elliptical dark-hollow beams,” J. Opt. Soc. Am. A 23, 919-925 (2006).
    [CrossRef]
  15. Z. Mei and D. Zhao, “Decentered controllable elliptical dark-hollow beams,” Opt. Commun. 259, 415-423 (2006).
    [CrossRef]
  16. R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18, 1678-1680 (2001).
    [CrossRef]
  17. H. Guo, J. Chen, and S. Zhuang, “Vector plane wave spectrum of an arbitrary polarized electromagnetic wave,” Opt. Express 14, 2095-2100 (2006).
    [CrossRef] [PubMed]
  18. G. Zhou, “Analytical vectorial structure of Laguerre-Gaussian beam in the far field,” Opt. Lett. 31, 2616-2618 (2006).
    [CrossRef] [PubMed]
  19. D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32, 2711-2713 (2007).
    [CrossRef] [PubMed]
  20. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
  21. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195-1201 (1972).
    [CrossRef]
  22. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon Press, New York, 1970).

2008 (2)

2007 (1)

2006 (7)

2005 (1)

2003 (1)

2002 (2)

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29-34 (2002).
[CrossRef]

X. Xu, Y. Wang, and W. Jhe, “Theory of atom guidance in a hollow laser beam: dressed-atom approach,” J. Opt. Soc. Am. B 17, 1039-1050 (2002).
[CrossRef]

2001 (1)

2000 (1)

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

1998 (1)

1997 (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]

1996 (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155-1166 (1996).

1972 (1)

Arlt, J.

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

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155-1166 (1996).

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon Press, New York, 1970).

Bosch, S.

Cai, Y.

Carnicer, A.

Carter, W. H.

Chen, J.

Deng, D.

Dholakia, K.

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

Fan, Z.

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155-1166 (1996).

Guo, H.

Guo, Q.

He, S.

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, 4713-4716 (1997).
[CrossRef]

Jhe, W.

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, 4713-4716 (1997).
[CrossRef]

Lin, Q.

Liu, T.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Lu, X.

Mandel, L.

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

Martínez-Herrero, R.

Mei, Z.

Mejías, P. M.

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155-1166 (1996).

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, 4713-4716 (1997).
[CrossRef]

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155-1166 (1996).

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, 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, 4713-4716 (1997).
[CrossRef]

Spagnolo, G. S.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155-1166 (1996).

Sun, X.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Tang, H.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29-34 (2002).
[CrossRef]

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, 4713-4716 (1997).
[CrossRef]

Wang, W.

Wang, X.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Wang, Y.

Wolf, E.

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

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon Press, New York, 1970).

Xu, S.

Xu, X.

Yin, J.

Yu, H.

Zhao, D.

Zhou, G.

Zhu, K.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Zhu, Y.

Zhuang, S.

J. Mod. Opt. (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155-1166 (1996).

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (3)

Opt. Commun. (4)

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

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Z. Mei and D. Zhao, “Generalized beam propagation factor of hard-edged diffracted controllable dark-hollow beams,” Opt. Commun. 263, 261-266 (2006).
[CrossRef]

Z. Mei and D. Zhao, “Decentered controllable elliptical dark-hollow beams,” Opt. Commun. 259, 415-423 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

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, 4713-4716 (1997).
[CrossRef]

Other (2)

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

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon Press, New York, 1970).

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 (6)

Fig. 1
Fig. 1

Normalized energy flux distribution in the reference plane z = 1000 λ . N = 2 , Ω = 0.2 , and w 0 = 1 λ . (a) TE term. (b) TM term. (c) CDHB.

Fig. 2
Fig. 2

Normalized energy flux distribution in the reference plane z = 1000 λ . N = 2 , Ω = 0.2 , and w 0 = 5 λ . (a) TE term. (b) TM term. (c) CDHB.

Fig. 3
Fig. 3

Normalized energy flux distribution in the reference plane z = 1000 λ . N = 2 , Ω = 0.8 , and w 0 = 1 λ . (a) TE term. (b) TM term. (c) CDHB.

Fig. 4
Fig. 4

Normalized energy flux distribution in the reference plane z = 1000 λ . N = 2 , Ω = 0.8 , and w 0 = 5 λ . (a) TE term. (b) TM term. (c) CDHB.

Fig. 5
Fig. 5

Normalized energy flux distribution in the reference plane z = 1000 λ . N = 3 , Ω = 0.8 , and w 0 = 5 λ . (a) TE term. (b) TM term. (c) CDHB.

Fig. 6
Fig. 6

Normalized energy flux distribution in the reference plane z = 1000 λ . N = 4 , Ω = 0.8 , and w 0 = 5 λ . (a) TE term. (b) TM term. (c) CDHB

Equations (39)

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

( E x ( ρ 0 , 0 ) E y ( ρ 0 , 0 ) ) = ( n = 1 N a n [ exp ( n ρ 0 2 w 0 2 ) exp ( n ρ 0 2 v 0 2 ) ] 0 ) ,
a n = ( 1 ) n 1 N ( N n ) ,
Ω = v 0 w 0 ,
E ( ρ , z ) = A ( p , q ) exp [ i k ( p x + q y + γ z ) ] d p d q ,
A ( p , q ) = A x ( p , q ) ( e x p γ e z ) ,
A x ( p , q ) = 1 λ 2 E x ( ρ 0 , 0 ) exp [ i k ( p x 0 + q y 0 ) ] d x 0 d y 0 = 1 4 π f 2 n = 1 N a n n [ exp ( b 2 4 n f 2 ) Ω 2 exp ( Ω 2 b 2 4 n f 2 ) ] ,
e 1 = q b e x p b e y , e 2 = p γ b e x + q γ b e y b e z ,
s × e 1 = e 2 , e 1 × e 2 = s , e 2 × s = e 1 ,
A ( p , q ) = [ A ( p , q ) . e 1 ] e 1 + [ A ( p , q ) . e 2 ] e 2 .
E ( ρ , z ) = E TE ( ρ , z ) + E TM ( ρ , z ) ,
E TE ( ρ , z ) = [ A ( p , q ) e 1 ] e 1 exp [ i k ( p x + q y + γ z ) ] d p d q ,
E TM ( ρ , z ) = [ A ( p , q ) e 2 ] e 2 exp [ i k ( p x + q y + γ z ) ] d p d q .
E TE ( ρ , z ) = 1 4 π f 2 n = 1 N a n n [ exp ( b 2 4 n f 2 ) Ω 2 exp ( Ω 2 b 2 4 n f 2 ) ] q b 2 ( q e x p e y ) exp [ i k ( p x + q y + γ z ) ] d p d q ,
E TM ( ρ , z ) = 1 4 π f 2 n = 1 N a n n [ exp ( b 2 4 n f 2 ) Ω 2 exp ( Ω 2 b 2 4 n f 2 ) ] ( p 2 b 2 e x + p q b 2 e y p γ e z ) exp [ i k ( p x + q y + γ z ) ] d p d q .
H ( ρ , z ) = H TE ( ρ , z ) + H TM ( ρ , z ) ,
H TE ( ρ , z ) = ( ɛ 0 μ 0 ) 1 2 4 π f 2 n = 1 N a n n [ exp ( b 2 4 n f 2 ) Ω 2 exp ( Ω 2 b 2 4 n f 2 ) ] q b 2 ( p γ e x + q γ e y b 2 e z ) exp [ i k ( p x + q y + γ z ) ] d p d q ,
H TM ( ρ , z ) = ( ɛ 0 μ 0 ) 1 2 4 π f 2 n = 1 N a n n [ exp ( b 2 4 n f 2 ) Ω 2 exp ( Ω 2 b 2 4 n f 2 ) ] p γ b 2 ( q e x p e y ) exp [ i k ( p x + q y + γ z ) ] d p d q ,
E TE ( ρ , z ) = i λ r j = 1 m α j ( τ j β j δ j 2 ) 1 2 M ( p j , q j ) exp [ i k r F ( p j , q j , x , y ) ] , k r ,
M ( p j , q j ) = 1 4 π f 2 n = 1 N a n n [ exp ( p j 2 + q j 2 4 n f 2 ) Ω 2 exp ( Ω 2 ( p j 2 + q j 2 ) 4 n f 2 ) ] q j p j 2 + q j 2 ( q j e x p j e y )
F ( p j , q j , x , y ) = p j x + q j y + ( 1 p j 2 q j 2 ) 1 2 z r .
F ( p , q , x , y ) p p = p j q = q j = 0 ,
F ( p , q , x , y ) q p = p j q = q j = 0 ,
F ( p , q , x , y ) = p x + q y + ( 1 p 2 q 2 ) 1 2 z r .
x r p j ( 1 p j 2 q j 2 ) 1 2 z r = 0 ,
y r q j ( 1 p j 2 q j 2 ) 1 2 z r = 0 .
p 1 = x r ,
q 1 = y r .
τ 1 = F 2 ( p , q , x , y ) p 2 p = p 1 q = q 1 = ( q 2 1 ) z r ( 1 p 2 q 2 ) 3 2 p = p 1 q = q 1 = 1 x 2 z 2 ,
β 1 = F 2 ( p , q , x , y ) q 2 p = p 1 q = q 1 = ( p 2 1 ) z r ( 1 p 2 q 2 ) 3 2 p = p 1 q = q 1 = 1 y 2 z 2 ,
δ 1 = F 2 ( p , q , x , y ) p q p = p 1 q = q 1 = p q z r ( 1 p 2 q 2 ) 3 2 p = p 1 q = q 1 = x y z 2 ,
α 1 = 1 .
E TE ( ρ , z ) = i z r y z ρ 2 r 2 exp ( i k r ) ( y e x x e y ) n = 1 N a n n [ exp ( ρ 2 4 n f 2 r 2 ) Ω 2 exp ( Ω 2 ρ 2 4 n f 2 r 2 ) ] ,
E TM ( ρ , z ) = i z r x ρ 2 r 2 exp ( i k r ) ( x z e x + y z e y ρ 2 e z ) n = 1 N a n n [ exp ( ρ 2 4 n f 2 r 2 ) Ω 2 exp ( Ω 2 ρ 2 4 n f 2 r 2 ) ] .
H TE ( ρ , z ) = i ( ε 0 μ 0 ) 1 2 z r y z ρ 2 r 3 exp ( i k r ) ( x z e x + y z e y ρ 2 e z ) n = 1 N a n n [ exp ( ρ 2 4 n f 2 r 2 ) Ω 2 exp ( Ω 2 ρ 2 4 n f 2 r 2 ) ] ,
H TM ( ρ , z ) = i ( ε 0 μ 0 ) 1 2 z r x ρ 2 r exp ( i k r ) ( y e x x e y ) n = 1 N a n n [ exp ( ρ 2 4 n f 2 r 2 ) Ω 2 exp ( Ω 2 ρ 2 4 n f 2 r 2 ) ] .
S z TE = 1 2 Re [ E TE ( ρ , z ) × H TE * ( ρ , z ) ] z = ( ε 0 μ 0 ) 1 2 z r 2 y 2 z 3 2 ρ 2 r 5 { n = 1 N a n n [ exp ( ρ 2 4 n f 2 r 2 ) Ω 2 exp ( Ω 2 ρ 2 4 n f 2 r 2 ) ] } 2 ,
S z TM = 1 2 Re [ E TM ( ρ , z ) × H TM * ( ρ , z ) ] z = ( ε 0 μ 0 ) 1 2 z r 2 x 2 z 2 ρ 2 r 3 { n = 1 N a n n [ exp ( ρ 2 4 n f 2 r 2 ) Ω 2 exp ( Ω 2 ρ 2 4 n f 2 r 2 ) ] } 2 ,
S z = S z TE + S z TM = ( ε 0 μ 0 ) 1 2 z r 2 z 2 ρ 2 r 3 ( x 2 + z 2 r 2 y 2 ) { n = 1 N a n n [ exp ( ρ 2 4 n f 2 r 2 ) Ω 2 exp ( Ω 2 ρ 2 4 n f 2 r 2 ) ] } 2 .
S z = ( ε 0 μ 0 ) 1 2 z r 2 z 2 r 3 { n = 1 N a n n [ exp ( ρ 2 4 n f 2 r 2 ) Ω 2 exp ( Ω 2 ρ 2 4 n f 2 r 2 ) ] } 2 .

Metrics