Abstract

On the basis of the Rayleigh–Sommerfeld formulas, the analytical propagation equation of nonparaxial controllable dark-hollow beams (CDHBs) in free space is derived. The far-field approaches and the paraxial approximation are dealt with as special cases of our general results. By using the derived formulas, the nonparaxial propagation properties of CDHBs in free space are illustrated and are analyzed with numerical examples. Some detailed comparisons of the results obtained with the paraxial results are made, which show that the f parameter and the propagation distance play an important role in determining the nonparaxiality of the CDHBs.

© 2008 Optical Society of America

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References

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    [CrossRef]
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2007

2006

Z. Mei and D. Zhao, "Controllable elliptical dark-hollow beams," J. Opt. Soc. Am. A 23, 919-925 (2006).
[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]

2005

2004

2003

2002

2000

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

1999

H. Laabs and A. T. Friberg, "Nonparaxial eigenmodes of stable resonators," IEEE J. Quantum Electron. 35, 198-207 (1999).
[CrossRef]

Y. Song, D. Milam, and W. T. Hill, "Long narrow all-light atom guide," Opt. Lett. 24, 1805-1807 (1999).
[CrossRef]

1998

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]

H. Laabs, "Propagation of Hermite-Gaussian beams beyond the paraxial approximation," Opt. Commun. 147, 1-4 (1998).
[CrossRef]

1997

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]

1992

1985

1979

1975

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

1972

IEEE J. Quantum Electron.

H. Laabs and A. T. Friberg, "Nonparaxial eigenmodes of stable resonators," IEEE J. Quantum Electron. 35, 198-207 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

H. Laabs, "Propagation of Hermite-Gaussian beams beyond the paraxial approximation," Opt. Commun. 147, 1-4 (1998).
[CrossRef]

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

Opt. Lett.

Phys. Rev. A

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

Phys. Rev. Lett.

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

R. K. Luneburg, Mathematical Theory of Optics (U. California, Berkeley, 1964).

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Figures (5)

Fig. 1
Fig. 1

Contour graphs of the normalized intensity distribution of a CDHB for different beam parameters. (a) N = 10 , ϵ = 0.8 , w 0 = 1.5 cm ; (b) N = 10 , ϵ = 0.8 , w 0 = 2 cm ; (c) N = 20 , ϵ = 0.8 , w 0 = 1.5 cm ; (d) N = 10 , ϵ = 0.5 , w 0 = 2 cm .

Fig. 2
Fig. 2

Normalized radial intensity distribution of CDHBs with N = 10 and ϵ = 0.8 at the plane z = 0.1 z R for different values of parameter f. The solid curves represent the nonparaxial results, and the dotted curves denote the paraxial result.

Fig. 3
Fig. 3

Normalized radial intensity distribution of CDHBs with N = 10 and ϵ = 0.8 at the plane z = 5 z R for different values of parameter f. The solid curves represent the nonparaxial results, and the dotted curves denote the paraxial result.

Fig. 4
Fig. 4

Normalized radial intensity distribution of CDHBs with N = 10 , ϵ = 0.8 and f = 0.01 at several propagation distances. The solid curves represent the nonparaxial results, and the dotted curves denote the paraxial result.

Fig. 5
Fig. 5

Normalized radial intensity distribution of CDHBs with N = 10 , ϵ = 0.8 and f = 0.1 at several propagation distances. The solid curves represent the nonparaxial results, and the dotted curves denote the paraxial result.

Equations (22)

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E ( ρ 0 , 0 ) = n = 1 N a n [ exp ( n ρ 0 2 w 0 2 ) exp ( n ρ 0 2 v 0 2 ) ] ,
N = 1 , 2 , 3 , ,
a n = ( 1 ) n 1 N ( N n )
E ( ρ , φ , z ) = 1 2 π 0 2 π 0 E ( ρ 0 , φ 0 , 0 ) z ( e i k R R ) ρ 0 d ρ 0 d φ 0 ,
R = ρ 0 2 + ρ 2 2 ρ 0 ρ cos ( φ φ 0 ) + z 2 .
R = r + ρ 0 2 2 ρ 0 ρ cos ( φ φ 0 ) 2 r ,
0 2 π exp ( i x cos θ ) d θ = 2 π J 0 ( x ) ,
0 exp ( p t ) J 0 ( 2 α 1 2 t 1 2 ) d t = p 1 exp ( α p ) ,
E ( ρ , z ) = i k z 2 r 2 exp ( i k r ) n = 1 N a n { ( n w 0 2 i k 2 r ) 1 exp [ ( k ρ 2 r ) 2 n w 0 2 i k 2 r ] ( n v 0 2 i k 2 r ) 1 exp [ ( k ρ 2 r ) 2 n v 0 2 i k 2 r ] } .
R = r ρ 0 ρ cos ( φ φ 0 ) r .
E ( ρ , z ) = i k z 2 r 2 exp ( i k r ) n = 1 N a n { ( n w 0 2 ) 1 exp [ k 2 ρ 2 w 0 2 4 n r 2 ] ( n v 0 2 ) 1 exp [ k 2 ρ 2 v 0 2 4 n r 2 ] } .
E ( ρ , z ) = i ( z z R ) 4 f 2 α 1 2 exp ( i α 1 ) n = 1 N a n [ β 11 1 exp ( π 2 η 2 α 1 2 β 11 ) β 12 1 exp ( π 2 η 2 α 1 2 β 12 ) ] ,
α 1 = [ 4 π 2 η 2 + ( z z R ) 2 4 f 4 ] 1 2 ,
β 11 = n f 2 i 2 α 1 ,
β 12 = n f 2 ϵ 2 i 2 α 1 ,
η = ρ λ .
E ( ρ , z ) = i ( z z R ) 4 f 4 α 1 2 exp ( i α 1 ) n = 1 N a n [ 1 n exp ( π 2 η 2 n α 1 2 f 2 ) ϵ 2 n exp ( ϵ 2 π 2 η 2 n α 1 2 f 2 ) ]
r z + ρ 2 2 z .
E ( ρ , z ) = i 2 α 2 exp ( i α 2 ) exp ( 2 i π 2 η 2 α 2 ) n = 1 N a n { β 21 1 exp [ π 2 η 2 α 2 2 β 21 ] β 22 1 exp [ π 2 η 2 α 2 2 β 22 ] } ,
α 2 = z z R 2 f 2 ,
β 21 = n f 2 i 2 α 2 ,
β 22 = n f 2 ϵ 2 i 2 α 2 .

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