Abstract

Analytic expressions for the fields of a tightly focused radially polarized Gaussian laser beam are derived, accurate to ϵ5, where ϵ is the associated diffraction angle. The fields satisfy Maxwell’s equations, and the calculated beam power based on them is significantly different from that of the paraxial-approximation fields.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B: Photophys. Laser Chem. 72, 109 (2001).
  2. R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  3. M. O. Scully and M. S. Zubairy, Phys. Rev. A 44, 2656 (1991).
    [CrossRef] [PubMed]
  4. Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
    [CrossRef] [PubMed]
  5. K. T. McDonald, puhep1.princeton.edu/~mcdonald/examples/axicon.pdf.
  6. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  7. L. W. Davis, Phys. Rev. A 19, 1177 (1979).
    [CrossRef]
  8. Y. Kozawa and S. Sato, Opt. Lett. 30, 3036 (2005).
    [CrossRef]
  9. Y. Kozawa and S. Sato, Opt. Lett. 31, 820 (2006).
    [CrossRef] [PubMed]
  10. E. Esarey, P. Sprangle, and J. Krall, Phys. Rev. E 52, 5443 (1995).
    [CrossRef]
  11. P. Serafim, P. Sprangle, and B. Hafizi, IEEE Trans. Plasma Sci. 28, 1155 (2000).
    [CrossRef]
  12. Y. I. Salamin, Phys. Rev. A 73, 043402 (2006).
    [CrossRef]
  13. A. Wünsche, J. Opt. Soc. Am. A 9, 765 (1992).
    [CrossRef]
  14. R. Borghi and M. Santarsiero, Opt. Lett. 28, 774 (2003).
    [CrossRef] [PubMed]
  15. J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
    [CrossRef]
  16. K. T. McDonald, hep.princeton.edu/~mcdonald/accel/gaussian.ps, hep.princeton.edu/~mcdonald/accel/gaussian2.ps.
  17. Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. E 67, 016501 (2003).
    [CrossRef]
  18. Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. ST Accel. Beams 5, 101301 (2002).
    [CrossRef]

2006

2005

2003

R. Borghi and M. Santarsiero, Opt. Lett. 28, 774 (2003).
[CrossRef] [PubMed]

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. E 67, 016501 (2003).
[CrossRef]

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

2002

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

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. ST Accel. Beams 5, 101301 (2002).
[CrossRef]

2001

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B: Photophys. Laser Chem. 72, 109 (2001).

2000

P. Serafim, P. Sprangle, and B. Hafizi, IEEE Trans. Plasma Sci. 28, 1155 (2000).
[CrossRef]

1995

E. Esarey, P. Sprangle, and J. Krall, Phys. Rev. E 52, 5443 (1995).
[CrossRef]

1992

1991

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

1989

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[CrossRef]

1979

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

1975

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

Alexander, D. R.

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[CrossRef]

Barton, J. P.

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[CrossRef]

Borghi, R.

Davis, L. W.

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

Dorn, R.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B: Photophys. Laser Chem. 72, 109 (2001).

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B: Photophys. Laser Chem. 72, 109 (2001).

Esarey, E.

E. Esarey, P. Sprangle, and J. Krall, Phys. Rev. E 52, 5443 (1995).
[CrossRef]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B: Photophys. Laser Chem. 72, 109 (2001).

Hafizi, B.

P. Serafim, P. Sprangle, and B. Hafizi, IEEE Trans. Plasma Sci. 28, 1155 (2000).
[CrossRef]

Keitel, C. H.

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. E 67, 016501 (2003).
[CrossRef]

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

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. ST Accel. Beams 5, 101301 (2002).
[CrossRef]

Kozawa, Y.

Krall, J.

E. Esarey, P. Sprangle, and J. Krall, Phys. Rev. E 52, 5443 (1995).
[CrossRef]

Lax, M.

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

Leuchs, G.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B: Photophys. Laser Chem. 72, 109 (2001).

Louisell, W. H.

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

McDonald, K. T.

K. T. McDonald, hep.princeton.edu/~mcdonald/accel/gaussian.ps, hep.princeton.edu/~mcdonald/accel/gaussian2.ps.

K. T. McDonald, puhep1.princeton.edu/~mcdonald/examples/axicon.pdf.

McKnight, W. B.

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

Mocken, G. R.

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. E 67, 016501 (2003).
[CrossRef]

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. ST Accel. Beams 5, 101301 (2002).
[CrossRef]

Quabis, S.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B: Photophys. Laser Chem. 72, 109 (2001).

Salamin, Y. I.

Y. I. Salamin, Phys. Rev. A 73, 043402 (2006).
[CrossRef]

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. E 67, 016501 (2003).
[CrossRef]

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

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. ST Accel. Beams 5, 101301 (2002).
[CrossRef]

Santarsiero, M.

Sato, S.

Scully, M. O.

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

Serafim, P.

P. Serafim, P. Sprangle, and B. Hafizi, IEEE Trans. Plasma Sci. 28, 1155 (2000).
[CrossRef]

Sprangle, P.

P. Serafim, P. Sprangle, and B. Hafizi, IEEE Trans. Plasma Sci. 28, 1155 (2000).
[CrossRef]

E. Esarey, P. Sprangle, and J. Krall, Phys. Rev. E 52, 5443 (1995).
[CrossRef]

Wünsche, A.

Zubairy, M. S.

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

Appl. Phys. B: Photophys. Laser Chem.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B: Photophys. Laser Chem. 72, 109 (2001).

IEEE Trans. Plasma Sci.

P. Serafim, P. Sprangle, and B. Hafizi, IEEE Trans. Plasma Sci. 28, 1155 (2000).
[CrossRef]

J. Appl. Phys.

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rev. A

Y. I. Salamin, Phys. Rev. A 73, 043402 (2006).
[CrossRef]

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]

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

Phys. Rev. E

E. Esarey, P. Sprangle, and J. Krall, Phys. Rev. E 52, 5443 (1995).
[CrossRef]

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. E 67, 016501 (2003).
[CrossRef]

Phys. Rev. Lett.

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]

Phys. Rev. ST Accel. Beams

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, Phys. Rev. ST Accel. Beams 5, 101301 (2002).
[CrossRef]

Other

K. T. McDonald, hep.princeton.edu/~mcdonald/accel/gaussian.ps, hep.princeton.edu/~mcdonald/accel/gaussian2.ps.

K. T. McDonald, puhep1.princeton.edu/~mcdonald/examples/axicon.pdf.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Surface plots of the amplitudes e r , e z , and c b θ of E ¯ r E 0 , E ¯ z E 0 , and c B ¯ θ E 0 , respectively, in the plane z = 0 . For all plots ϵ = 0.75 ( w 0 0.424 λ ) .

Fig. 2
Fig. 2

(a) and (b) Intersections of surfaces similar to those of Figs. 1a, 1b, respectively, with the y = 0 plane. (c) Percentage correction to the calculated power when the nonparaxial terms are included in the fields.

Equations (20)

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

2 Ψ 2 i k Ψ z = 0 .
2 Ψ 4 i Ψ ζ + ϵ 2 2 Ψ ζ 2 = 0 , 2 1 ρ ρ ( ρ ρ ) .
Ψ = Ψ 0 + ϵ 2 Ψ 2 + ϵ 4 Ψ 4 + .
2 Ψ 0 4 i Ψ 0 ζ = 0 ,
2 Ψ 2 m 4 i Ψ 2 m ζ + 2 Ψ 2 m 2 ζ 2 = 0 , m = 1 , 2 , .
Ψ 0 = f e f ρ 2 , Ψ 2 = ( f 2 ρ 4 f 3 4 ) Ψ 0 ,
Ψ 4 = ( 3 f 2 8 3 ρ 4 f 4 16 ρ 6 f 5 8 + ρ 8 f 6 32 ) Ψ 0 ,
f = i ζ + i = e i ψ G 1 + ζ 2 , ψ G = tan 1 ζ .
E = i ω A i c k ( A ) , B = × A .
E r c = E 0 e f ρ 2 e i η { ϵ [ ρ f 2 ] + ϵ 3 [ ρ f 3 2 + ρ 3 f 4 ρ 5 f 5 4 ] + ϵ 5 [ 3 ρ f 4 8 3 ρ 3 f 5 8 + 17 ρ 5 f 6 16 3 ρ 7 f 7 8 + ρ 9 f 8 32 ] } ,
E z c = i E 0 e f ρ 2 e i η { ϵ 2 [ f 2 ρ 2 f 3 ] + ϵ 4 [ f 3 2 + ρ 2 f 4 2 5 ρ 4 f 5 4 + ρ 6 f 6 4 ] } ,
B θ c = E 0 c e f ρ 2 e i η { ϵ [ ρ f 2 ] + ϵ 3 [ ρ f 3 2 + ρ 3 f 4 2 ρ 5 f 5 4 ] + ϵ 5 [ 3 ρ f 4 8 + 3 ρ 3 f 5 8 + 3 ρ 5 ρ 6 16 ρ 7 f 7 4 + ρ 9 f 8 32 ] } .
E r = E { ϵ [ ρ C 2 ] + ϵ 3 [ ρ C 3 2 + ρ 3 C 4 ρ 5 C 5 4 ] + ϵ 5 [ 3 ρ C 4 8 3 ρ 3 C 5 8 + 17 ρ 5 C 6 16 3 ρ 7 C 7 8 + ρ 9 C 8 32 ] } ,
E z = E { ϵ 2 [ S 2 ρ 2 S 3 ] + ϵ 4 [ S 3 2 + ρ 2 S 4 2 5 ρ 4 S 5 4 + ρ 6 S 6 4 ] } ,
B θ = E c { ϵ [ ρ C 2 ] + ϵ 3 [ ρ C 3 2 + ρ 3 C 4 2 ρ 5 C 5 4 ] + ϵ 5 [ 3 ρ C 4 8 + 3 ρ 3 C 5 8 + 3 ρ 5 C 6 16 ρ 7 C 7 4 + ρ 9 C 8 32 ] } .
E = E 0 e r 2 w 2 , w = w 0 1 + ζ 2 ,
C n = ( w 0 w ) n cos ( ψ + n ψ G ) , n = 2 , 3 , ,
S n = ( w 0 w ) n sin ( ψ + n ψ G ) ,
ψ = ψ 0 + ω t k z k r 2 2 R , R = z + z r 2 z ,
P = P 0 [ 1 + 3 ( ϵ 2 ) 2 + 9 ( ϵ 2 ) 4 ] , P 0 = π w 0 2 2 E 0 2 c μ 0 ( ϵ 2 ) 2 ,

Metrics