Abstract

It is shown that the approach proposed by Lax et al. [Phys. Rev. A 11, 1365 (1975)] for studying the propagation of an electromagnetic beam beyond the paraxial approximation can be efficiently employed even when the beam under consideration presents a very nonparaxial character. The method that we present consists of applying a nonlinear resummation scheme, the so-called δ transformation, to the divergent perturbative series arising from the Lax scheme. Numerical results pertinent to the evaluation of transverse and longitudinal components of the electric field are presented for the particular case of vectorial Gaussian beams, showing the effectiveness of the method.

© 2003 Optical Society of America

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References

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  1. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  2. A. Wünsche, J. Opt. Soc. Am. A 9, 765 (1992).
    [CrossRef]
  3. S. R. Seshadri, J. Opt. Soc. Am. A 19, 2134 (2002).
    [CrossRef]
  4. G. A. Baker and P. R. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge U. Press, Cambridge, 1996).
    [CrossRef]
  5. P. R. Graves-Morris, D. E. Roberts, and A. Salam, J. Comp. Appl. Math. 122, 51 (2000).
    [CrossRef]
  6. E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989).
    [CrossRef]
  7. U. D. Jentschura, J. Becher, E. J. Weniger, and G. Soff, Phys. Rev. Lett. 85, 2446 (2000).
    [CrossRef] [PubMed]
  8. E. J. Weniger, Phys. Rev. A 56, 5165 (1997).
    [CrossRef]
  9. E. J. Weniger, Phys. Rev. Lett. 77, 2859 (1996).
    [CrossRef] [PubMed]
  10. R. Simon, E. C. G. Sudarshan, and N. Mukunda, J. Opt. Soc. Am. A 3, 536 (1986).
    [CrossRef]
  11. T. Takenaka, M. Yokota, and O. Fukumitsu, J. Opt. Soc. Am. A 2, 826 (1985).
    [CrossRef]
  12. R. Borghi, A. Ciattoni, and M. Santarsiero, J. Opt. Soc. Am. A 19, 1207 (2002).
    [CrossRef]

2002

2000

U. D. Jentschura, J. Becher, E. J. Weniger, and G. Soff, Phys. Rev. Lett. 85, 2446 (2000).
[CrossRef] [PubMed]

P. R. Graves-Morris, D. E. Roberts, and A. Salam, J. Comp. Appl. Math. 122, 51 (2000).
[CrossRef]

1997

E. J. Weniger, Phys. Rev. A 56, 5165 (1997).
[CrossRef]

1996

E. J. Weniger, Phys. Rev. Lett. 77, 2859 (1996).
[CrossRef] [PubMed]

1992

1989

E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989).
[CrossRef]

1986

1985

1975

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

Baker, G. A.

G. A. Baker and P. R. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge U. Press, Cambridge, 1996).
[CrossRef]

Becher, J.

U. D. Jentschura, J. Becher, E. J. Weniger, and G. Soff, Phys. Rev. Lett. 85, 2446 (2000).
[CrossRef] [PubMed]

Borghi, R.

Ciattoni, A.

Fukumitsu, O.

Graves-Morris, P. R.

P. R. Graves-Morris, D. E. Roberts, and A. Salam, J. Comp. Appl. Math. 122, 51 (2000).
[CrossRef]

G. A. Baker and P. R. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge U. Press, Cambridge, 1996).
[CrossRef]

Jentschura, U. D.

U. D. Jentschura, J. Becher, E. J. Weniger, and G. Soff, Phys. Rev. Lett. 85, 2446 (2000).
[CrossRef] [PubMed]

Lax, M.

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

Louisell, W. H.

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

McKnight, W. B.

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

Mukunda, N.

Roberts, D. E.

P. R. Graves-Morris, D. E. Roberts, and A. Salam, J. Comp. Appl. Math. 122, 51 (2000).
[CrossRef]

Salam, A.

P. R. Graves-Morris, D. E. Roberts, and A. Salam, J. Comp. Appl. Math. 122, 51 (2000).
[CrossRef]

Santarsiero, M.

Seshadri, S. R.

Simon, R.

Soff, G.

U. D. Jentschura, J. Becher, E. J. Weniger, and G. Soff, Phys. Rev. Lett. 85, 2446 (2000).
[CrossRef] [PubMed]

Sudarshan, E. C. G.

Takenaka, T.

Weniger, E. J.

U. D. Jentschura, J. Becher, E. J. Weniger, and G. Soff, Phys. Rev. Lett. 85, 2446 (2000).
[CrossRef] [PubMed]

E. J. Weniger, Phys. Rev. A 56, 5165 (1997).
[CrossRef]

E. J. Weniger, Phys. Rev. Lett. 77, 2859 (1996).
[CrossRef] [PubMed]

E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989).
[CrossRef]

Wünsche, A.

Yokota, M.

Comput. Phys. Rep.

E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989).
[CrossRef]

J. Comp. Appl. Math.

P. R. Graves-Morris, D. E. Roberts, and A. Salam, J. Comp. Appl. Math. 122, 51 (2000).
[CrossRef]

J. Opt. Soc. Am. A

Phys. Rev. A

E. J. Weniger, Phys. Rev. A 56, 5165 (1997).
[CrossRef]

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

Phys. Rev. Lett.

E. J. Weniger, Phys. Rev. Lett. 77, 2859 (1996).
[CrossRef] [PubMed]

U. D. Jentschura, J. Becher, E. J. Weniger, and G. Soff, Phys. Rev. Lett. 85, 2446 (2000).
[CrossRef] [PubMed]

Other

G. A. Baker and P. R. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge U. Press, Cambridge, 1996).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Behavior of relative error as a function of z/λ for the on-axis amplitude of a Gaussian beam with w=λ/2. The Weniger truncation orders are (dashed curve) n=2, (dotted curve) n=5, and (solid curve) n=14.

Fig. 2
Fig. 2

Behavior of the normalized amplitude of the (a) transverse and (b) longitudinal electric field components as a function of normalized transverse coordinate x/λ for a Gaussian beam spot size w/λ=1/10, at the plane z=λ and for y=0. The Weniger approximant orders are (cirles) n=5, (squares) n=10, and (triangles) n=20. Solid curves, exact solution obtained through the plane-wave expansion.

Tables (1)

Tables Icon

Table 1 Resummation of Lax Series for the Transverse Component of a Gaussian Beam with w=λ/2 at z=λ and r=2λ

Equations (8)

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

××E=k2E,·E=0.
2F+2ikzF=-z2F,·F+zFz+ikFz=0.
Er;0=xˆψ0r;
2ψ+2ikzψ=-z2ψ,xψ+zϕ+ikϕ=0,
ψ2mr,z=i2kmp=1mcpmzpzm+pψ0r,z,
cpm=2m!mp-1!m-1!m+p!,
ϕ1=ikxψ0,ϕ2m+1=ikxψ2m+ikzϕ2m-1,    m1.
δn=j=0n-1jnj1+jn-1sjaj+1j=0n-1jnj1+jn-11aj+1,

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