Abstract

The exact full-wave generalization of the azimuthally varying scalar real-argument Laguerre–Gauss beam is obtained. The radiation intensity of the resulting azimuthally varying real-argument Laguerre–Gauss wave is determined. The main characteristics of the radiation intensity pattern are discussed. The total power Pn,m is determined, where n is the radial mode number and m is the azimuthal mode number. By the use of 1Pn,m, the characteristics of the quality of the paraxial beam approximation to the real-argument Laguerre–Gauss wave are investigated.

© 2007 Optical Society of America

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References

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  1. J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
    [CrossRef]
  2. S. R. Seshadri, "Quality of paraxial electromagnetic beams," Appl. Opt. 45, 5335-5345 (2006).
    [CrossRef] [PubMed]
  3. S. R. Seshadri, "Dynamics of the linearly polarized fundamental Gaussian light wave," J. Opt. Soc. Am. A 24, 482-492 (2007).
    [CrossRef]
  4. H. Kogelnik and T. Li, "Laser beams and resonators," Appl. Opt. 5, 1550-1567 (1966).
    [CrossRef] [PubMed]
  5. T. Takenaka, M. Yokota, and O. Fukumitsu, "Propagation for light beams beyond the paraxial approximation," J. Opt. Soc. Am. A 2, 826-829 (1985).
    [CrossRef]
  6. S. R. Seshadri, "Complex-argument Laguerre-Gauss beams: transport of mean-squared beam width," Appl. Opt. 44, 7339-7343 (2005).
    [CrossRef] [PubMed]
  7. S. R. Seshadri, "Virtual source for a Laguerre-Gauss beam," Opt. Lett. 27, 1872-1874 (2002).
    [CrossRef]
  8. M. A. Bandres and J. C. Gutierrez-Vega, "Higher-order complex source for elegant Laguerre-Gauss waves," Opt. Lett. 29, 2213-2215 (2004).
    [CrossRef] [PubMed]
  9. S. R. Seshadri, "Radiation pattern of cylindrically symmetric scalar Laguerre-Gauss beams," Opt. Lett. 32, 1159-1161 (2007).
    [CrossRef] [PubMed]
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-290.
  11. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of the Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
    [CrossRef] [PubMed]
  12. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1965), p. 784, Formula 22.9.15.

2007

2006

2005

2004

2002

2000

J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
[CrossRef]

1992

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of the Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

1985

1966

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1965), p. 784, Formula 22.9.15.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of the Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Arlt, J.

J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
[CrossRef]

Bandres, M. A.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of the Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Dholakia, K.

J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
[CrossRef]

Fukumitsu, O.

Gutierrez-Vega, J. C.

Hitomi, T.

J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
[CrossRef]

Kogelnik, H.

Li, T.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-290.

Seshadri, S. R.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of the Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1965), p. 784, Formula 22.9.15.

Takenaka, T.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of the Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-290.

Yokota, M.

Appl. Opt.

Appl. Phys. B

J. Arlt, T. Hitomi, and K. Dholakia, "Atom guiding along Laguerre-Gaussian and Bessel light beams," Appl. Phys. B 71, 549-554 (2000).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rev. A

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of the Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Other

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1965), p. 784, Formula 22.9.15.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-290.

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Figures (4)

Fig. 1
Fig. 1

Radiation intensity pattern of the Laguerre–Gauss wave for k w 0 = 2.980 , ϕ = 0 , m = 1 , and n = 0 , 1, and 2. The radiation intensity pattern has cos 2 ϕ dependence on ϕ.

Fig. 2
Fig. 2

Same as for Fig. 1 for n = 3 , 4, and 5, and the other parameters are the same as for Fig. 1.

Fig. 3
Fig. 3

Radiation intensity pattern of the Laguerre–Gauss wave for ϕ = 0 , m = 1 , n = 2 , and k w 0 = 2.513 (curve a), k w 0 = 3.142 (curve b), and k w 0 = 3.770 (curve c). The radiation intensity pattern has cos 2 ϕ dependence on ϕ.

Fig. 4
Fig. 4

1 P n , m as functions of w 0 λ for m = 1 , for n = 0 (curve a), n = 1 (curve b), and n = 2 (curve c) and for 0.05 < w 0 λ < 1 . P n , m is the power in the azimuthally varying real-argument Laguerre–Gauss wave of mode numbers n and m.

Equations (43)

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F n , m ( 0 ) ( ρ , ϕ , z ) = K n , m ( 0 ) cos m ϕ exp ( i k z ) w 0 w exp [ i ( 2 n + m + 1 ) × tan 1 ( z b ) ] ( 2 ρ 2 w 2 ) m 2 L n m ( 2 ρ 2 w 2 ) exp ( q 2 ρ n 2 ) ,
0 d x x m L n m ( a x ) L p m ( a x ) exp ( a x ) = a ( m + 1 ) ( n + m ) ! δ n p n ! ,
K n , m ( 0 ) = [ 4 n ! P ( 0 ) ε m π ω b ( n + m ) ! ] 1 2 ,
S n , m ( 0 ) ( ρ , ϕ , z ) = K n , m ( 0 ) cos m ϕ exp ( i k z ) ( 1 ) n n ! 2 2 n + m 2 q ( 2 n + 2 + m ) ( q 2 ρ n 2 ) m 2 L n m ( q 2 ρ n 2 ) exp ( q 2 ρ n 2 ) .
F n , m ( 0 ) ( ρ , ϕ , z ) = l = 0 l = n b n l , m S n l , m ( 0 ) ( ρ , ϕ , z ) ,
b n l , m = ( 1 ) n ( n + m ) ! 2 n l ( n + m l ) ! ( n l ) ! l ! × [ n ! ( n + m l ) ! ( n + m ) ! ( n l ) ! ] 1 2 .
S n , m ( ρ , ϕ , z ) = K n , m ( 0 ) cos m ϕ 2 m 2 b exp ( k b ) 0 d η η J m ( η ρ ) ( η 2 w 0 2 ) n ( η w 0 ) m ζ 1 exp [ i ζ ( z i b ) ] ,
ζ = ( k 2 η 2 ) 1 2 .
F n , m ( ρ , ϕ , z ) = l = 0 l = n b n l , m S n l , m ( ρ , ϕ , z ) .
π z , n , m ( ρ , ϕ , z ) = 1 2 Re [ i ω F n , m * ( ρ , ϕ , z ) z F n , m ( ρ , ϕ , z ) ] ,
P n , m = 0 2 π d ϕ 0 d ρ ρ π z , n , m ( ρ , ϕ , z ) .
P n , m = 0 2 π d ϕ 0 π 2 d θ sin θ Φ n , m ( θ , ϕ ) ,
Φ n , m ( θ , ϕ ) = cos 2 m ϕ ( ϵ m π f 0 2 ) 1 2 m [ G n , m ( k w 0 sin θ ) ] 2 exp [ k 2 w 0 2 ( 1 cos θ ) ] ,
G n , m ( k w 0 sin θ ) = ( k w 0 sin θ ) m l = 0 l = n b n l , m ( k 2 w 0 2 sin 2 θ ) ( n l ) ( ( n l ) ! ( n + m l ) ! ) 1 2 ,
Φ 0 , 0 ( θ , ϕ ) = ( 2 π f 0 2 ) 1 exp [ k 2 w 0 2 ( 1 cos θ ) ] .
P 0 , 1 = 1 f 0 2 + ( 0.5 k 2 w 0 2 + f 0 2 ) exp ( k 2 w 0 2 ) ,
P 1 , 1 = 1 8 f 0 2 + 33 f 0 4 45 f 0 6 ( k 6 w 0 6 16 k 4 w 0 4 2 + 5 k 2 w 0 2 8 + 2 + 5 f 0 2 2 12 f 0 4 45 f 0 6 ) exp ( k 2 w 0 2 ) ,
P 2 , 1 = 1 21 f 0 2 + 294 f 0 4 2 , 040 f 0 6 + 6 , 930 f 0 8 9 , 450 f 0 10 + ( k 10 w 0 10 384 + k 8 w 0 8 16 91 k 6 w 0 6 192 + k 4 w 0 4 + 7 k 2 w 0 2 8 + 3 57 f 0 2 4 144 f 0 4 165 f 0 6 + 2 , 520 f 0 8 + 9 , 450 f 0 10 ) exp ( k 2 w 0 2 ) .
RHS = b r , m K r , m ( 0 ) exp ( i k z ) cos m ϕ ( 1 ) r 2 2 r + m 2 q 2 ( r + 1 ) ( m + r ) ! .
LHS = K n , m ( 0 ) exp ( i k z ) cos m ϕ 2 m 2 q 2 ( n + m + 1 ) q * ( 2 n ) I n , r ; m ,
I n , r ; m = 0 d x x m L n m ( 2 q 2 q * 2 x ) L r m ( q 2 x ) exp ( q 2 x ) .
b r , m = K n , m ( 0 ) K r , m ( 0 ) q 2 ( n + m + 1 ) q 2 r q * 2 n ( 1 ) r 2 2 r ( m + r ) ! I n , r ; m .
g ( x , t ) = ( 1 t ) ( m + 1 ) exp [ x t ( 1 t ) ] ,
L p m ( x ) = 1 p ! Lim t = 0 p t p g ( x , t ) .
I n , r ; m = 1 n ! r ! Lim t = 0 Lim s = 0 n t n r s r 1 ( 1 t ) ( m + 1 ) ( 1 s ) ( m + 1 ) × 0 d x x m exp [ x q 2 ( 1 + 2 q * 2 t 1 t + s 1 s ) ] .
I n , r ; m = m ! n ! r ! 1 q 2 ( m + 1 ) Lim t = 0 Lim s = 0 n t n r s r [ 1 t ( 1 2 q * 2 ) 2 q * 2 t s ] ( m + 1 ) .
F s = Lim s = 0 r s r [ 1 t ( 1 2 q * 2 ) 2 q * 2 t s ] ( m + 1 ) ,
F s = ( m + r ) ! m ! 2 r q * 2 r t r [ 1 t ( 1 2 q * 2 ) ] m + 1 + r .
F t = Lim t = 0 n t n t r [ 1 t ( 1 2 q * 2 ) ] m + 1 + r = Lim t = 0 l = 0 l = n n ! l ! ( n l ) ! n l t r t n l l t l 1 [ 1 t ( 1 2 q * 2 ) ] m + 1 + r .
F t = Lim t = 0 l = 0 l = n n ! l ! ( n l ) ! r ! ( r n + l ) ! t r n + l × ( m + r + l ) ! ( m + r ) ! ( 1 2 q * 2 ) l [ 1 t ( 1 2 q * 2 ) ] m + 1 + r + l .
F t = 0 for r > n ,
F t = n ! ( n r ) ! ( m + n ) ! ( m + r ) ! ( 1 2 q * 2 ) n r for r < n .
1 2 q * 2 = q * 2 q 2 ,
I n , r ; m = 0 for r > n ,
I n , r ; m = ( m + n ) ! ( 1 ) n r 2 r q * 2 n r ! ( n r ) ! q 2 ( m + 1 ) q 2 ( n r ) for r < n .
b r , m = 0 for r > n ,
b r , m = ( 1 ) n ( n + m ) ! 2 r ( m + r ) ! r ! ( n r ) ! K n , m ( 0 ) K r , m ( 0 ) for r n .
P 0 , 1 = k 4 w 0 4 2 0 π 2 d θ sin 3 θ exp [ k 2 w 0 2 ( 1 cos θ ) ] ,
P 1 , 1 = k 4 w 0 4 0 π 2 d θ sin 3 θ ( 0.25 k 2 w 0 2 sin 2 θ 1 ) 2 × exp k 2 w 0 2 ( 1 cos θ ) ,
P 2 , 1 = 3 k 4 w 0 4 2 0 π 2 d θ sin 3 θ ( k 4 w 0 4 sin 4 θ 24 k 2 w 0 2 sin 2 θ 2 + 1 ) 2 × exp k 2 w 0 2 ( 1 cos θ ) .
P 0 , 1 = k 4 w 0 4 3 for k w 0 0 ,
P 1 , 1 = 2 k 4 w 0 4 3 for k w 0 0 ,
P 2 , 1 = k 4 w 0 4 for k w 0 0 .

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