Abstract

The concept of vectorial Laguerre-Bessel-Gaussian (LBG) beams is proposed. On the basis of vectorial Rayleigh-Sommerfeld formulas, the analytical formulas for the nonparaxial propagation of vectorial LBG beams are derived and applied to study the nonparaxial propagation properties of vectorial LBG beams. The far field and paraxial approximation are dealt with as special cases of our general results. Some detailed comparisons of the obtained results with the paraxial results are made, which show the propagation of paraxial and nonparaxial LBG beams is all instable in the near field and the f parameter plays the important role in determining the nonparaxiality of vectorial LBG beams. The beam parameter α also affects the propagation behavior of nonparaxial LBG beams. Under certain conditions, the obtained results can be reduced to those of the cases for vectorial Laguerre-Gaussian and Bessel Gaussian beams.

© 2007 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
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  11. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).
  12. A. Ciattoni, B. Crosignani, and P. D. Porto, "Vectorial analytical description of propagation of a highly nonparxial beam," Opt. Commun. 202, 17-20 (2002).
    [CrossRef]

2005

2003

2002

A. Ciattoni, B. Crosignani, and P. D. Porto, "Vectorial analytical description of propagation of a highly nonparxial beam," Opt. Commun. 202, 17-20 (2002).
[CrossRef]

2000

1998

1997

1992

1985

T. Takenaka, M. Yokota, and O. Fukumitsu, "Propagation of light beams beyond the paraxial approximation," J. Opt. Soc. Am. A 2, 836-829 (1985).

1970

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1970).

J. Opt. Soc. Am. A

Opt. Commun.

A. Ciattoni, B. Crosignani, and P. D. Porto, "Vectorial analytical description of propagation of a highly nonparxial beam," Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Opt. Express

Phys. Rev. A

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1970).

Other

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

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Figures (7)

Fig. 1.
Fig. 1.

Three-dimensional intensity distributions and corresponding contour graphs of LBG beams with n=3, m=3, q=0 for various values of the coefficient α at the plane z=0. (a) α=0, (b) α=0.5, (c) α=1, (d) α=2, (e) α=3.5, (f) α=5.

Fig. 2.
Fig. 2.

Normalized intensity distributions of nonparaxial vectorial LBG beams at various values of the propagation distance z for f=0.02.

Fig. 3.
Fig. 3.

Normalized intensity distributions of nonparaxial vectorial LBG beams at various values of the propagation distance z for f=0.2.

Fig. 4.
Fig. 4.

Three-dimensional intensity distributions and corresponding contour graphs of nonparaxial (a) and paraxial (b) LBG beams with n=3, m=3, q=0 for f=0.2 at the plane z=15z R .

Fig. 5.
Fig. 5.

Normalized intensity distributions of a LBG beam with n=3, m=3, q=0 at the plane z=15z R for various values of α when f=0.2. The solid curves denote the nonparaxial results I, the dash curves express I x and the dotted curves are the paraxial results I p . (a) α=0, (b) α=0.5, (c) α=1, (d) α=2, (e) α=3.5, (f) α=5.

Fig. 6.
Fig. 6.

Normalized intensity distributions of a LG beam with n=3, m=3 at the plane z=15z R . (a) f=0.02, (b) f=0.2.

Fig. 7.
Fig. 7.

Normalized intensity distributions of a BG beam with q=0, α=3.5 at the plane z=15z R . (a) f=0.02, (b) f=0.2.

Equations (58)

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E x ( x 0 , y 0 , 0 ) = ( 2 ρ 0 w 0 ) m L n m ( 2 ρ 0 2 w 0 2 ) J q ( α ρ 0 w 0 ) exp ( ρ 0 2 w 0 2 ) exp [ i ( m q ) θ 0 ] ,
E y ( x 0 , y 0 , 0 ) = 0 ,
E x ( r ) = 1 2 π + E x ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
E y ( r ) = 1 2 π + E y ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
E z ( r ) = 1 2 π + [ E x ( x 0 , y 0 , 0 ) G ( r , r 0 ) x + E y ( x 0 , y 0 , 0 ) G ( r , r 0 ) y ] d x 0 d y 0 ,
G ( r , r 0 ) = exp ( ik r r 0 ) r r 0 .
G ( r , r 0 ) 1 r exp [ ik ( r + x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r ) ] ,
E x ( ρ , θ , z ) = ( i ) m q + 1 kz r exp ( ikr ) r exp [ i ( m q ) θ ]
× 0 ( 2 w 0 ρ 0 ) m L n m ( 2 ρ 0 2 w 0 2 ) exp ( g ρ 0 2 ) J q ( α ρ 0 w 0 ) J m q ( k ρ ρ 0 r ) ρ 0 d ρ 0 ,
E y ( ρ , θ , z ) = 0 ,
E z ( ρ , θ , z ) = ( i ) m q + 1 k ρ cos θ r exp ( ikr ) r exp [ i ( m q ) θ ] 0 ( 2 w 0 ρ 0 ) m L n m ( 2 ρ 0 2 w 0 2 )
× exp ( g ρ 0 2 ) J q ( α ρ 0 w 0 ) J m q ( k ρ ρ 0 r ) ρ 0 d ρ 0 + k 2 r exp ( ikr ) r 0 ( 2 w 0 ρ 0 ) m ,
× L n m ( 2 ρ 0 2 w 0 2 ) exp ( g ρ 0 2 ) J q ( α ρ 0 w 0 ) { ( i ) m q exp [ i ( m q 1 ) θ ]
× J m q 1 ( k ρ ρ 0 r ) + ( i ) m q + 2 exp [ i ( m q + 1 ) θ ] J m q + 1 ( k ρ ρ 0 r ) } ρ 0 2 d ρ 0
g = 1 w 0 2 ik 2 r .
E x ( ρ , θ , z ) = ( i ) m q + 1 2 k r z exp ( i k r ) r exp [ i ( m q ) θ ] l 1 l 2 l 3 n F 1 ( l 1 , l 2 , l 3 ) 2 2 l 3 + m
× α 2 l 1 + q ( ρ r ) 2 l 2 + m q g 1 ( l 1 + l 2 + l 3 + m + 1 ) Γ ( l 1 + l 2 + l 3 + m + 1 ) f 2 l 2 + m q + 2
E y ( ρ , θ , z ) = 0 ,
E z ( ρ , θ , z ) = ( i ) m q + 1 2 k r ρ cos θ exp ( i k r ) r exp [ i ( m q ) θ ] l 1 l 2 l 3 n F 1 ( l 1 , l 2 , l 3 )
× α 2 l 1 + q ( ρ r ) 2 l 2 + m q 2 2 l 3 + m f 2 l 2 + m q + 2 g 1 ( l 1 + l 2 + l 3 + m + 1 ) Γ ( l 1 + l 2 + l 3 + m + 1 )
+ ( i ) m q k 2 r exp ( i k r ) r exp [ i ( m q 1 ) θ ] l 1 l 2 l 3 n F 1 ( l 1 , l 2 , l 3 ) ( m q + l 2 ) 1
× α 2 l 1 + q ( ρ r ) 2 l 2 + m q 1 2 2 l 3 + m f 2 l 2 + m q + 2 g 1 ( l 1 + l 2 + l 3 + m + 1 ) Γ ( l 1 + l 2 + l 3 + m + 1 )
+ ( i ) m q + 2 4 k 2 r exp ( i k r ) r exp [ i ( m q + 1 ) θ ] l 1 l 2 l 3 n F 1 ( l 1 , l 2 , l 3 ) ( m q + l 2 + 1 )
× α 2 l 1 + q ( ρ r ) 2 l 2 + m q + 1 2 2 l 3 + m f 2 l 2 + m q + 4 g 1 ( l 1 + l 2 + l 3 + m + 2 ) Γ ( l 1 + l 2 + l 3 + m + 2 )
F 1 ( l 1 , l 2 , l 3 ) = ( 1 ) l 1 + l 2 + l 3 ( n + m ) ! 2 2 l 1 + 2 l 2 + m l 1 ! l 2 ! l 3 ! ( q + l 1 ) ! ( m q + l 2 ) ! ( n l 3 ) ! ( m + l 3 ) ! ,
g 1 = 1 i 2 k f 2 r ,
f = 1 k w 0
G ( r , r 0 ) 1 r exp [ i k ( r x x 0 + y y 0 r ) ] .
E x ( ρ , θ , z ) = ( i ) m q + 1 2 kr z exp ( ikr ) r exp [ i ( m q ) θ ] l 1 l 2 l 3 n F 1 ( l 1 , l 2 , l 3 ) 2 2 l 3 + m ,
× α 2 l 1 + q ( ρ r ) 2 l 2 + m q Γ ( l 1 + l 2 + l 3 + m + 1 ) f 2 l 2 + m q + 2 ,
E y ( ρ , θ , z ) = 0 ,
E z ( ρ , θ , z ) = ( i ) m q + 1 2 kr ρ cos θ exp ( ikr ) r exp [ i ( m q ) θ ] l 1 l 2 l 3 n F 1 ( l 1 , l 2 , l 3 ) α 2 l 1 + q
× ( ρ r ) 2 l 2 + m q 2 2 l 3 + m f 2 l 2 + m q + 2 Γ ( l 1 + l 2 + l 3 + m + 1 ) + ( i ) m q k 2 r exp ( ikr ) r
× exp [ i ( m q 1 ) θ ] l 1 l 2 l 3 n F 1 ( l 1 , l 2 , l 3 ) ( m q + l 2 ) 1 α 2 l 2 + q ( ρ r ) 2 l 2 + m q 1 2 2 l 3 + m f 2 l 2 + m q + 2
× Γ ( l 1 + l 2 + l 3 + m + 1 ) + ( 1 ) m q + 2 4 k 2 r exp ( ikr ) r exp [ i ( m q + 1 ) θ ]
× l 1 l 2 l 3 n F 1 ( l 1 , l 2 , l 3 ) ( m q + l 2 + 1 ) α 2 l 1 + q ( ρ r ) 2 l 2 + m q + 1 2 2 l 3 + m f 2 l 2 + m q + 4 Γ ( l 1 + l 2 + l 3 + m + 2 )
r z + x 2 + y 2 2 z ,
E xp ( ρ , θ , z ) = ( i ) m q + 1 2 kz exp [ ik ( z + ρ 2 2 z ) ] exp [ i ( m q ) θ ] l 1 l 2 l 3 n F 1 ( l 1 , l 2 , l 3 ) 2 2 l 3 + m
× α 2 l 1 + q ( ρ 2 ) 2 l 2 + m q g p ( l 1 + l 2 + l 3 + m + 1 ) Γ ( l 1 + l 2 + l 3 + m + 1 ) f 2 l 2 + m q + 2 ,
g p = 1 i 2 kf 2 z .
E x ( ρ , θ , z ) = ( i ) m + 1 2 kr z exp ( ikr ) r exp ( imθ ) l 2 l 3 n F 2 ( l 2 , l 3 ) 2 2 l 3 + m
× ( ρ r ) 2 l 2 + m g 1 ( l 2 + l 3 + m + 1 ) Γ ( l 2 + l 3 + m + 1 ) f 2 l 2 + m + 2 ,
E y ( ρ , θ , z ) = 0 ,
E z ( ρ , θ , z ) = ( i ) m + 1 2 kr ρ cos θ exp ( ikr ) r exp [ im θ ] l 2 l 3 n F 2 ( l 2 , l 3 ) ( ρ r ) 2 l 2 + m 2 2 l 3 + m f 2 l 2 + m + 2
× g 1 ( l 2 + l 3 + m +1 ) Γ ( l 2 + l 3 + m + 1 ) + ( i ) m k 2 r exp ( ikr ) r exp [ i ( m 1 ) θ ]
× l 2 l 3 n F 2 ( l 2 , l 3 ) ( m + l 2 ) 1 ( ρ r ) 2 l 2 + m 1 2 2 l 3 + m f 2 l 2 + m + 2 g 1 ( l 2 + l 3 + m + 1 ) Γ ( l 2 + l 3 + m + 1 ) ,
+ ( i ) m + 2 4 k 2 r exp ( ikr ) r exp [ i ( m + 1 ) θ ] l 2 l 3 F 2 ( l 2 , l 3 ) ( m + l 2 + 1 )
× ( ρ r ) 2 l 2 + m + 1 2 2 l 3 + m f 2 l 2 + m + 4 g 1 ( l 2 + l 3 + m + 2 ) Γ ( l 2 + l 3 + m + 2 )
F 2 ( l 2 , l 3 ) = ( 1 ) l 2 + l 3 ( n + m ) ! 2 2 l 2 + m l 2 ! l 3 ! ( m + l 3 ) ! ( n l 3 ) ! ( m + l 3 ) ! .
E x ( ρ , θ , z ) = ( i ) q + 1 2 kr z exp ( ikr ) r exp ( iq θ ) l 1 l 2 F 3 ( l 1 , l 2 )
× α 2 l 1 + q ( ρ r ) 2 l 2 q g 1 ( l 1 + l 2 + 1 ) Γ ( l 1 + l 2 + 1 ) f 2 l 2 q + 2 ,
E y ( ρ , θ , z ) = 0 ,
E z ( ρ , θ , z ) = ( i ) q + 1 2 kr ρ cos θ exp ( ikr ) r exp ( iq θ ) l 1 l 2 F 3 ( l 1 , l 2 ) α 2 l 1 + q ( ρ r ) 2 l 2 q
× 1 f 2 l 2 q + 2 g 1 ( l 1 + l 2 + 1 ) Γ ( l 1 + l 2 + 1 ) + ( i ) q k 2 r exp ( ikr ) r exp [ i ( q + 1 ) θ ]
× l 1 l 2 F 3 ( l 1 , l 2 ) ( q + l 2 ) 1 α 2 l 1 + q ( ρ r ) 2 l 2 q 1 1 f 2 l 2 q + 2 g 1 ( l 1 + l 2 + 1 ) Γ ( l 1 + l 2 + 1 ) ,
+ ( i ) q + 2 4 k 2 r exp ( ikr ) r exp [ i ( q 1 ) θ ] l 1 l 2 F 3 ( l 1 , l 2 ) ( q + l 2 + 1 ) α 2 l 1 + q
( ρ r ) 2 l 2 q + 1 1 f 2 l 2 q + 4 g 1 ( l 1 + l 2 + 2 ) Γ ( l 1 + l 2 + 2 )
F 3 ( l 1 , l 2 ) = ( 1 ) l 1 + l 2 2 2 l 1 + 2 l 2 l 1 ! l 2 ! ( q + l 1 ) ! ( q + l 2 ) ! .

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