Abstract

Fifth-order corrected expressions for the fields of a radially polarized Laguerre–Gauss (R-TEMn1) laser beams are derived based on perturbative Lax series expansion. When the order of Laguerre polynomial is equal to zero, the corresponding beam reduces to the lowest-order radially polarized beam (R-TEM01). Simulation results show that the accuracy of the fifth-order correction for R-TEMn1 depends not only on the diffraction angle of the beam as R-TEM01 does, but also on the order of the beam.

© 2007 Optical Society of America

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References

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2007 (2)

2006 (2)

2005 (1)

Y. Kozawa and S. Sato, Opt. Lett. 30, 3603 (2005).
[CrossRef]

2004 (1)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

2002 (2)

S. R. Seshadri, Opt. Lett. 27, 1872 (2002).
[CrossRef]

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

1991 (1)

M. O. Scully and M. S. Zubairy, Phys. Rev. A 44, 2656 (1991).
[CrossRef] [PubMed]

1979 (1)

L. W. Davis, Phys. Rev. A 19, 1177 (1979).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Phys. Rev. A (3)

M. O. Scully and M. S. Zubairy, Phys. Rev. A 44, 2656 (1991).
[CrossRef] [PubMed]

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

L. W. Davis, Phys. Rev. A 19, 1177 (1979).
[CrossRef]

Phys. Rev. Lett. (2)

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Other (2)

S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1996), p. 853, formula 7.421.4.

K. T. McDonald, 'Axicon Gaussian laser beams,' puhep1.princeton.edu/~mcdonald/examples/axicon.pdf.

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Figures (1)

Fig. 1
Fig. 1

Relative difference between the third-order and fifth-order corrected fields evaluated by Eq. (20) as a function of the diffraction angle ϵ. Rectangles, R - TEM 01 mode; circles, R - TEM 11 mode; triangles, R - TEM 21 mode.

Equations (20)

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1 ρ ρ ( ρ ψ ρ ) 4 i ψ ζ = ϵ 2 2 ψ ζ 2 .
ψ = ψ 0 + ϵ 2 ψ 2 + ϵ 4 ψ 4 + .
1 ρ ρ ( ρ ψ 0 ρ ) 4 i ψ 0 ζ = 0 ,
1 ρ ρ ( ρ ψ 2 m ρ ) 4 i ψ 2 m ζ = 2 ψ 2 m 2 ζ 2 , m = 1 , 2 , .
ψ 0 = Q exp ( Q ρ 2 ) , ψ 2 = ( Q 2 Q 3 ρ 4 4 ) ψ 0 ,
ψ 4 = ( 3 Q 2 8 3 Q 4 ρ 4 16 Q 5 ρ 6 8 + Q 6 ρ 8 32 ) ψ 0 ,
E = i ω k 2 ( A ) i ω A , B = × A .
ϕ 0 = L n ( x ) Q n ψ 0 ,
1 ρ ρ ( ρ ϕ 2 ρ ) 4 i ϕ 2 ζ = 2 ϕ 0 ζ 2 .
f ( ρ , ζ ) = 0 d y [ y f ¯ ( y , ζ ) J 0 ( y ρ ) ] ,
f ¯ ( y , ζ ) = 0 d ρ [ ρ f ¯ ( ρ , ζ ) J 0 ( y ρ ) ] ,
0 d y [ y J 0 ( y ρ ) y 2 n exp ( y 2 4 Q ) ] = n ! 2 ( 4 Q ) n + 1 exp ( Q ρ 2 ) L n ( Q ρ 2 ) ,
4 i ϕ ¯ 2 ζ y 2 ϕ ¯ 2 = 1 8 n ! 4 n 1 y 2 n + 4 exp ( y 2 4 Q ) .
ϕ ¯ 2 = [ C 0 4 Q + C ( y ) ] exp ( y 2 4 Q ) ,
ϕ 2 = [ ( n + 1 ) L n + 1 ( x ) 1 4 ( n + 1 ) ( n + 2 ) L n + 2 ( x ) ] Q n + 2 exp ( x ) ,
ϕ 4 = [ 15 16 ( n + 1 ) ( n + 2 ) L n + 2 ( x ) 3 8 ( n + 1 ) ( n + 2 ) ( n + 3 ) L n + 3 ( x ) + 1 32 ( n + 1 ) ( n + 2 ) ( n + 3 ) ( n + 4 ) L n + 4 ( x ) ] Q n + 3 exp ( x ) .
E z = i E 0 Q n + 2 exp ( β ) { ϵ 2 ( n + 1 ) L n + 1 ( x ) + ϵ 4 Q ( n + 1 ) ( n + 2 ) [ L n + 2 ( x ) ( n + 3 ) L n + 3 ( x ) 4 ] } ,
E ρ = E 0 exp ( β ) Q n + 2 ρ { ϵ L n 1 ( x ) + ϵ 3 Q ( n + 1 ) [ L n + 1 1 ( x ) 2 ( n + 2 ) L n + 2 1 ( x ) 4 ] + ϵ 5 Q 2 ( n + 1 ) ( n + 2 ) [ 5 L n + 2 1 ( x ) 16 ( n + 3 ) L n + 3 1 ( x ) 4 + ( n + 3 ) ( n + 4 ) L n + 4 1 ( x ) 32 ] }
B φ = E 0 c exp ( β ) Q n + 2 ρ { ϵ L n 1 ( x ) + ϵ 3 Q ( n + 1 ) [ L n + 1 1 ( x ) ( n + 2 ) L n + 2 1 ( x ) 4 ] + ϵ 5 Q 2 ( n + 1 ) ( n + 2 ) [ 15 L n + 2 1 ( x ) 16 3 ( n + 3 ) L n + 3 1 ( x ) 8 + ( n + 3 ) ( n + 4 ) L n + 4 1 ( x ) 32 ] }
F = 0 S z , 5 S z , 3 ρ d ρ 0 S z , 3 ρ d ρ ,

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