Abstract

A simple method is presented which, when incorporated into a simple symbolic computation code, can be used to derive the analytic perturbative Lax series for electromagnetic fields of a focused radially polarized Gaussian laser beam, up to an arbitrary power of ϵ. Here ϵ is the associated Gaussian beam diffraction angle. The method is expected to be useful, when applied with a suitable nonlinear resummation scheme, in obtaining a precise field description for focusing the output beam from an axicon.

© 2007 Optical Society of America

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  1. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).
  2. R. Dorn, S. Quabit, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  3. B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
    [CrossRef] [PubMed]
  4. P. X. Wang, Y. K. Ho, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, Appl. Phys. Lett. 78, 2253 (2001).
    [CrossRef]
  5. B. Quesnel and P. Mora, Phys. Rev. E 58, 3719 (1998).
    [CrossRef]
  6. F. V. Hartemann and A. K. Kerman, Phys. Rev. Lett. 76, 624 (1996).
    [CrossRef] [PubMed]
  7. K. T. McDonald (2000), puhep1.princeton.edu/~mcdonald/examples/axicon.pdf.
  8. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  9. L. W. Davis, Phys. Rev. A 19, 1177 (1979).
    [CrossRef]
  10. T. Takenaka, M. Yokota, and O. Fukumitsu, J. Opt. Soc. Am. A 2, 826 (1985).
    [CrossRef]
  11. J. P. Barton and D. R. Alexander, J. Appl. Phys. 6, 2800 (1989).
    [CrossRef]
  12. A. Wönsche, J. Opt. Soc. Am. A 9, 765 (1992).
    [CrossRef]
  13. R. Borghi, A. Ciattoni, and M. Santarsiero, J. Opt. Soc. Am. A 19, 1207 (2002).
    [CrossRef]
  14. R. Borghi and M. Santarsiero, Opt. Lett. 28, 774 (2003).
    [CrossRef] [PubMed]
  15. H. Luo and S. Y. Liu, Phys. Rev. E 75, 038501 (2007).
    [CrossRef]
  16. Y. I. Salamin, Opt. Lett. 31, 2619 (2006).
    [CrossRef] [PubMed]
  17. Y. I. Salamin, New J. Phys. 8, 133 (2006).
    [CrossRef]
  18. E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989).
    [CrossRef]
  19. Y. Kozawa and S. Sato, Opt. Lett. 30, 3063 (2005).
    [CrossRef] [PubMed]
  20. Y. Kozawa and S. Sato, Opt. Lett. 31, 820 (2006).
    [CrossRef] [PubMed]
  21. J. Cizek, J. Zamastil, and L. Skala, J. Math. Phys. 44, 962 (2003).
    [CrossRef]

2007 (1)

H. Luo and S. Y. Liu, Phys. Rev. E 75, 038501 (2007).
[CrossRef]

2006 (3)

2005 (1)

2003 (4)

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

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

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

J. Cizek, J. Zamastil, and L. Skala, J. Math. Phys. 44, 962 (2003).
[CrossRef]

2002 (1)

2001 (2)

P. X. Wang, Y. K. Ho, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, Appl. Phys. Lett. 78, 2253 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).

1998 (1)

B. Quesnel and P. Mora, Phys. Rev. E 58, 3719 (1998).
[CrossRef]

1996 (1)

F. V. Hartemann and A. K. Kerman, Phys. Rev. Lett. 76, 624 (1996).
[CrossRef] [PubMed]

1992 (1)

1989 (2)

J. P. Barton and D. R. Alexander, J. Appl. Phys. 6, 2800 (1989).
[CrossRef]

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

1985 (1)

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]

Alexander, D. R.

J. P. Barton and D. R. Alexander, J. Appl. Phys. 6, 2800 (1989).
[CrossRef]

Barton, J. P.

J. P. Barton and D. R. Alexander, J. Appl. Phys. 6, 2800 (1989).
[CrossRef]

Blue, B. E.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Borghi, R.

Cao, N.

P. X. Wang, Y. K. Ho, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, Appl. Phys. Lett. 78, 2253 (2001).
[CrossRef]

Ciattoni, A.

Cizek, J.

J. Cizek, J. Zamastil, and L. Skala, J. Math. Phys. 44, 962 (2003).
[CrossRef]

Clayton, C. E.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Davis, L. W.

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

Decker, F. J.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Dorn, R.

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

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).

Esarey, E.

P. X. Wang, Y. K. Ho, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, Appl. Phys. Lett. 78, 2253 (2001).
[CrossRef]

Fukumitsu, O.

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).

Hartemann, F. V.

F. V. Hartemann and A. K. Kerman, Phys. Rev. Lett. 76, 624 (1996).
[CrossRef] [PubMed]

Ho, Y. K.

P. X. Wang, Y. K. Ho, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, Appl. Phys. Lett. 78, 2253 (2001).
[CrossRef]

Hogan, M. J.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Huang, C.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Iverson, R.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Joshi, C.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Katsouleas, T. C.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Kerman, A. K.

F. V. Hartemann and A. K. Kerman, Phys. Rev. Lett. 76, 624 (1996).
[CrossRef] [PubMed]

Kong, Q.

P. X. Wang, Y. K. Ho, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, Appl. Phys. Lett. 78, 2253 (2001).
[CrossRef]

Kozawa, Y.

Lax, M.

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

Leuchs, G.

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

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).

Liu, S. Y.

H. Luo and S. Y. Liu, Phys. Rev. E 75, 038501 (2007).
[CrossRef]

Louisell, W. H.

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

Lu, W.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Luo, H.

H. Luo and S. Y. Liu, Phys. Rev. E 75, 038501 (2007).
[CrossRef]

Marsh, K. A.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

McDonald, K. T.

K. T. McDonald (2000), 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]

Mora, P.

B. Quesnel and P. Mora, Phys. Rev. E 58, 3719 (1998).
[CrossRef]

Mori, W. B.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Muggli, P.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Nishida, Y.

P. X. Wang, Y. K. Ho, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, Appl. Phys. Lett. 78, 2253 (2001).
[CrossRef]

O'Connell, C. L.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).

Quabit, S.

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

Quesnel, B.

B. Quesnel and P. Mora, Phys. Rev. E 58, 3719 (1998).
[CrossRef]

Salamin, Y. I.

Santarsiero, M.

Sato, S.

Sessler, A. M.

P. X. Wang, Y. K. Ho, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, Appl. Phys. Lett. 78, 2253 (2001).
[CrossRef]

Siemann, R.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Skala, L.

J. Cizek, J. Zamastil, and L. Skala, J. Math. Phys. 44, 962 (2003).
[CrossRef]

Takenaka, T.

Walz, D.

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Wang, P. X.

P. X. Wang, Y. K. Ho, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, Appl. Phys. Lett. 78, 2253 (2001).
[CrossRef]

Weniger, E. J.

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

Wönsche, A.

Yokota, M.

Zamastil, J.

J. Cizek, J. Zamastil, and L. Skala, J. Math. Phys. 44, 962 (2003).
[CrossRef]

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).

Appl. Phys. Lett. (1)

P. X. Wang, Y. K. Ho, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, Appl. Phys. Lett. 78, 2253 (2001).
[CrossRef]

Comput. Phys. Rep. (1)

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

J. Appl. Phys. (1)

J. P. Barton and D. R. Alexander, J. Appl. Phys. 6, 2800 (1989).
[CrossRef]

J. Math. Phys. (1)

J. Cizek, J. Zamastil, and L. Skala, J. Math. Phys. 44, 962 (2003).
[CrossRef]

J. Opt. Soc. Am. A (3)

New J. Phys. (1)

Y. I. Salamin, New J. Phys. 8, 133 (2006).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (2)

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. E (2)

B. Quesnel and P. Mora, Phys. Rev. E 58, 3719 (1998).
[CrossRef]

H. Luo and S. Y. Liu, Phys. Rev. E 75, 038501 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

F. V. Hartemann and A. K. Kerman, Phys. Rev. Lett. 76, 624 (1996).
[CrossRef] [PubMed]

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

B. E. Blue, C. E. Clayton, C. L. O'Connell, F. J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, R. Siemann, and D. Walz, Phys. Rev. Lett. 90, (2003).
[CrossRef] [PubMed]

Other (1)

K. T. McDonald (2000), puhep1.princeton.edu/~mcdonald/examples/axicon.pdf.

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Figures (1)

Fig. 1
Fig. 1

Beam power versus s = 1 k w 0 , evaluated by Eq. (16) (solid curve) and Eq. (17) with n = 2 (dashed curve) and n = 7 (dotted curve). The inset shows beam power versus n based on Eq. (17) at s = 1 π , with the horizontal dashed line showing the value from Eq. (16).

Equations (19)

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

2 A + k 2 A = 0 ,
E = ( i k ) ( A ) i k A , H ϵ = × A ,
A = A ( r ) e z , A ( r ) = A 0 ψ ( r ) e i k z + i ω t ,
2 ψ 2 i k ψ z = 0 .
( 2 ξ 2 + 2 η 2 2 i ζ ) ψ = s 2 2 ψ ζ 2 ,
ψ = n = 0 s 2 n ψ 2 n ,
( 2 ξ 2 + 2 η 2 2 i ζ ) ψ 0 = 0 ,
( 2 ξ 2 + 2 η 2 2 i ζ ) ψ 2 n + 2 = 2 ψ 2 n ζ 2 , n 0 .
ψ 0 = i Q e i Q ρ 2 , Q = 1 ( i + 2 ζ ) , ρ = ξ 2 + η 2 .
exp [ i k ( z 2 + r 2 ) 1 2 + i k z + i k r 2 ( 2 z ) ]
e i k ( z 2 + r 2 ) 1 2 + i k z + i k r 2 ( 2 z ) = n = 0 a 2 n ( ρ , l 2 z ) s 2 n ;
ψ 2 n = C 2 n ψ 0 , C 2 n = a 2 n ( ρ , Q ) + ( n + 1 ) i Q C 2 n 2 ,
E r = E 0 e i Q ρ 2 i k z + i ω t n = 0 ϵ 2 n + 1 E 2 n + 1 ,
E z = i E 0 e i Q ρ 2 i k z + i ω t n = 1 ϵ 2 n E 2 n ,
H θ ϵ = E 0 e i Q ρ 2 i k z + i ω t n = 0 ϵ 2 n + 1 H 2 n + 1 ,
P = c 8 π Re [ E × H * ] d σ .
P = P 0 j = 0 α j , α j = ( j + 1 ) ! 2 j + 1 ( j 2 + 3 j + 2 ) s 2 j ,
p = p 0 + j = 0 n ( 1 ) j C j n ( j + 1 ) n 1 p j p 0 α j j = 0 n ( 1 ) j C j n ( j + 1 ) n 1 1 α j ,
p n = j = 0 n α j ,

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