Abstract

The exact vector integral solution for all the electromagnetic field components of a general flattened Gaussian laser mode is derived by using the angular spectrum method. This solution includes the pure and annular Gaussian modes as special cases. The integrals are of the form of Gegenbauer’s finite integral and are computed analytically for each case, yielding fields satisfying the Maxwell equations exactly in the form of quickly converging Fourier–Gegenbauer series.

© 2006 Optical Society of America

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  1. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  2. H. Hora, Physics of Laser Driven Plasmas (Wiley, 1981).
  3. L. W. Davis, Phys. Rev. A 19, 1177 (1979).
    [CrossRef]
  4. J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
    [CrossRef]
  5. L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
    [CrossRef] [PubMed]
  6. H. Hora, Laser Plasma Physics: Forces and the Nonlinearity Principle (SPIE, 2000).
  7. H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, in Proc. SPIE, 3886, 145 (2000).
  8. S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, Phys. Rev. Lett. 95, 035004 (2005).
    [CrossRef] [PubMed]
  9. B. W. Boreham and H. Hora, Phys. Rev. Lett. 42, 776 (1979).
    [CrossRef]
  10. B. Quesnel and P. Mora, Phys. Rev. E 58, 3719 (1998).
    [CrossRef]
  11. A. Maltsev and T. Ditmire, Phys. Rev. Lett. 90, 053002 (2003).
    [CrossRef] [PubMed]
  12. H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, in Proceedings of the 13th International Conference on Laser Interactions and Related Plasma Phenomena, G.Miley and E.M.Campbell, eds., AIP Proc. No. 406 (AIP, 1997), p. 495.
    [CrossRef]
  13. S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, Phys. Plasmas 12, 013102 (2005).
    [CrossRef]
  14. S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
    [CrossRef]
  15. P. Varga and P. Török, Opt. Commun. 152, 108 (1998).
    [CrossRef]
  16. S. DeSilestri, P. Laporta, V. Magni, and O. Svelto, IEEE J. Quantum Electron. QE-24, 1172 (1988).
    [CrossRef]
  17. Y. Li, Opt. Lett. 27, 1007 (2002).
    [CrossRef]
  18. R. Borghi, J. Opt. Soc. Am. A 18, 1627 (2001).
    [CrossRef]
  19. H. Mao and D. Zhao, J. Opt. Soc. Am. A 22, 647 (2005).
    [CrossRef]
  20. Y. Li, Opt. Commun. 206, 225 (2002).
    [CrossRef]
  21. F. Gori, Opt. Commun. 107, 335 (1994).
    [CrossRef]
  22. R. Potvliege, J. Opt. Soc. Am. A 17, 1043 (2000).
    [CrossRef]
  23. M. Santarsiero and R. Borghi, J. Opt. Soc. Am. A 16, 188 (1999).
    [CrossRef]
  24. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. S. Spagnolo, J. Opt. Soc. Am. A 13, 1385 (1996).
    [CrossRef]
  25. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1980).
  26. R. Kant, J. Mod. Opt. 40, 337 (1993).
    [CrossRef]
  27. M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions, 12th ed. (Dover, 1972).

2005 (3)

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, Phys. Plasmas 12, 013102 (2005).
[CrossRef]

H. Mao and D. Zhao, J. Opt. Soc. Am. A 22, 647 (2005).
[CrossRef]

2004 (1)

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
[CrossRef]

2003 (1)

A. Maltsev and T. Ditmire, Phys. Rev. Lett. 90, 053002 (2003).
[CrossRef] [PubMed]

2002 (2)

Y. Li, Opt. Commun. 206, 225 (2002).
[CrossRef]

Y. Li, Opt. Lett. 27, 1007 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

1998 (2)

P. Varga and P. Török, Opt. Commun. 152, 108 (1998).
[CrossRef]

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

1996 (1)

1994 (1)

F. Gori, Opt. Commun. 107, 335 (1994).
[CrossRef]

1993 (1)

R. Kant, J. Mod. Opt. 40, 337 (1993).
[CrossRef]

1990 (1)

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

1989 (1)

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

1988 (1)

S. DeSilestri, P. Laporta, V. Magni, and O. Svelto, IEEE J. Quantum Electron. QE-24, 1172 (1988).
[CrossRef]

1979 (2)

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

B. W. Boreham and H. Hora, Phys. Rev. Lett. 42, 776 (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. 66, 2800 (1989).
[CrossRef]

Ambrosini, D.

Bagini, V.

Banerjee, S.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

Barton, J. P.

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

Bonnaud, G.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
[CrossRef]

Boreham, B. W.

B. W. Boreham and H. Hora, Phys. Rev. Lett. 42, 776 (1979).
[CrossRef]

Borghi, R.

Castillo, R.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, in Proc. SPIE, 3886, 145 (2000).

Cicchitelli, L.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

Davis, L. W.

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

DeSilestri, S.

S. DeSilestri, P. Laporta, V. Magni, and O. Svelto, IEEE J. Quantum Electron. QE-24, 1172 (1988).
[CrossRef]

Ditmire, T.

A. Maltsev and T. Ditmire, Phys. Rev. Lett. 90, 053002 (2003).
[CrossRef] [PubMed]

Gori, F.

Hauser, T.

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, in Proceedings of the 13th International Conference on Laser Interactions and Related Plasma Phenomena, G.Miley and E.M.Campbell, eds., AIP Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Ho, Y. K.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, in Proc. SPIE, 3886, 145 (2000).

Hoelss, M.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, in Proc. SPIE, 3886, 145 (2000).

Hora, H.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

B. W. Boreham and H. Hora, Phys. Rev. Lett. 42, 776 (1979).
[CrossRef]

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, in Proc. SPIE, 3886, 145 (2000).

H. Hora, Physics of Laser Driven Plasmas (Wiley, 1981).

H. Hora, Laser Plasma Physics: Forces and the Nonlinearity Principle (SPIE, 2000).

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, in Proceedings of the 13th International Conference on Laser Interactions and Related Plasma Phenomena, G.Miley and E.M.Campbell, eds., AIP Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Kando, M.

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, Phys. Plasmas 12, 013102 (2005).
[CrossRef]

Kant, R.

R. Kant, J. Mod. Opt. 40, 337 (1993).
[CrossRef]

Kato, Y.

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, in Proceedings of the 13th International Conference on Laser Interactions and Related Plasma Phenomena, G.Miley and E.M.Campbell, eds., AIP Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Kitagawa, Y.

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, in Proceedings of the 13th International Conference on Laser Interactions and Related Plasma Phenomena, G.Miley and E.M.Campbell, eds., AIP Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Kotaki, H.

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, Phys. Plasmas 12, 013102 (2005).
[CrossRef]

Laporta, P.

S. DeSilestri, P. Laporta, V. Magni, and O. Svelto, IEEE J. Quantum Electron. QE-24, 1172 (1988).
[CrossRef]

Lax, M.

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

Li, Y.

Y. Li, Opt. Lett. 27, 1007 (2002).
[CrossRef]

Y. Li, Opt. Commun. 206, 225 (2002).
[CrossRef]

Loubere, R.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
[CrossRef]

Louisell, W. H.

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

Magni, V.

S. DeSilestri, P. Laporta, V. Magni, and O. Svelto, IEEE J. Quantum Electron. QE-24, 1172 (1988).
[CrossRef]

Maksimchuk, A.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

Malka, V.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
[CrossRef]

Maltsev, A.

A. Maltsev and T. Ditmire, Phys. Rev. Lett. 90, 053002 (2003).
[CrossRef] [PubMed]

Mao, H.

Masuda, S.

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, Phys. Plasmas 12, 013102 (2005).
[CrossRef]

McKnight, W. B.

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

Michel, P.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
[CrossRef]

Mima, K.

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, in Proceedings of the 13th International Conference on Laser Interactions and Related Plasma Phenomena, G.Miley and E.M.Campbell, eds., AIP Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Mora, P.

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

Nakajima, K.

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, Phys. Plasmas 12, 013102 (2005).
[CrossRef]

Osman, F.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, in Proc. SPIE, 3886, 145 (2000).

Ovadia, J.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
[CrossRef]

Pacileo, A. M.

Postle, R.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

Potvliege, R.

Quesnel, B.

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

Riazuelo, G.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
[CrossRef]

Santarsiero, M.

Scheid, W.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, in Proc. SPIE, 3886, 145 (2000).

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, in Proceedings of the 13th International Conference on Laser Interactions and Related Plasma Phenomena, G.Miley and E.M.Campbell, eds., AIP Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Sepke, S.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

Shah, R.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

Spagnolo, G. S.

Svelto, O.

S. DeSilestri, P. Laporta, V. Magni, and O. Svelto, IEEE J. Quantum Electron. QE-24, 1172 (1988).
[CrossRef]

Tikhonchuk, V. T.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
[CrossRef]

Török, P.

P. Varga and P. Török, Opt. Commun. 152, 108 (1998).
[CrossRef]

Umstadter, D.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

Valenzuela, A.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

Varga, P.

P. Varga and P. Török, Opt. Commun. 152, 108 (1998).
[CrossRef]

Walraet, F.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
[CrossRef]

Wang, J. W.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, in Proc. SPIE, 3886, 145 (2000).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1980).

Weber, S.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
[CrossRef]

Yamanaka, T.

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, in Proceedings of the 13th International Conference on Laser Interactions and Related Plasma Phenomena, G.Miley and E.M.Campbell, eds., AIP Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

Zhao, D.

IEEE J. Quantum Electron. (1)

S. DeSilestri, P. Laporta, V. Magni, and O. Svelto, IEEE J. Quantum Electron. QE-24, 1172 (1988).
[CrossRef]

J. Appl. Phys. (1)

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

J. Mod. Opt. (1)

R. Kant, J. Mod. Opt. 40, 337 (1993).
[CrossRef]

J. Opt. Soc. Am. A (5)

Laser Part. Beams (1)

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, Laser Part. Beams 22, 189 (2004).
[CrossRef]

Opt. Commun. (3)

P. Varga and P. Török, Opt. Commun. 152, 108 (1998).
[CrossRef]

Y. Li, Opt. Commun. 206, 225 (2002).
[CrossRef]

F. Gori, Opt. Commun. 107, 335 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Plasmas (1)

S. Masuda, M. Kando, H. Kotaki, and K. Nakajima, Phys. Plasmas 12, 013102 (2005).
[CrossRef]

Phys. Rev. A (3)

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

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

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

Phys. Rev. E (1)

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

Phys. Rev. Lett. (3)

A. Maltsev and T. Ditmire, Phys. Rev. Lett. 90, 053002 (2003).
[CrossRef] [PubMed]

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, Phys. Rev. Lett. 95, 035004 (2005).
[CrossRef] [PubMed]

B. W. Boreham and H. Hora, Phys. Rev. Lett. 42, 776 (1979).
[CrossRef]

Other (6)

H. Hora, Physics of Laser Driven Plasmas (Wiley, 1981).

H. Hora, Laser Plasma Physics: Forces and the Nonlinearity Principle (SPIE, 2000).

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, in Proc. SPIE, 3886, 145 (2000).

H. Hora, W. Scheid, T. Hauser, Y. Kato, Y. Kitagawa, K. Mima, and T. Yamanaka, in Proceedings of the 13th International Conference on Laser Interactions and Related Plasma Phenomena, G.Miley and E.M.Campbell, eds., AIP Proc. No. 406 (AIP, 1997), p. 495.
[CrossRef]

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1980).

M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions, 12th ed. (Dover, 1972).

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Equations (23)

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

E x ( x , y , z = 0 ) = E 0 N = 0 A N ( r 2 w 0 2 ) N e ( r 2 w 0 2 ) ,
E x a = 0 1 e ( b 2 ϵ 2 ) e i m k 0 z J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b ,
E z a = i x ¯ 0 1 e ( b 2 ϵ 2 ) e i m k 0 z m J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b ,
B x a = x y ¯ 2 0 1 e ( b 2 ϵ 2 ) e i m k 0 z m J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b ,
B y a = [ 0 1 e ( b 2 ϵ 2 ) m e i m k 0 z J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b x ¯ 2 0 1 e ( b 2 ϵ 2 ) e i m k 0 z m J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b ] ,
B z a = i y ¯ 0 1 e ( b 2 ϵ 2 ) e i m k 0 z J 0 ( k 0 r b ) L N ( b 2 ϵ 2 ) b d b ,
E x ( x , y , z ) = 1 2 ( I 1 + x 2 y 2 k 0 r 3 I 2 + y 2 r 2 I 3 ) ,
E y ( x , y , z ) = 1 2 x y k 0 r 3 ( 2 I 2 k 0 r I 3 ) ,
E z ( x , y , z ) = 1 2 x r I 4 .
I 1 = 0 1 e ζ 2 ( m + m 2 ) sin ( ϕ m ) J 0 ( Λ ) L N ( ζ 2 ) d m ,
I 2 = 0 1 e ζ 2 sin ( ϕ m ) J 1 ( Λ ) L N ( ζ 2 ) 1 m 2 d m ,
I 3 = 0 1 e ζ 2 sin ( ϕ m ) J 0 ( Λ ) L N ( ζ 2 ) ( 1 m 2 ) d m ,
I 4 = 0 1 e ζ 2 κ ( m ) cos ( ϕ m ) J 1 ( Λ ) L N ( ζ 2 ) d m ,
α 1 α f d ( cos θ , sin θ ) sin θ J n ( v sin θ sin α ) exp ( i u cos θ sin 2 α ) d θ ,
f d ( cos θ , sin θ ) = s = 0 a s , N d C s 1 2 ( cos θ ) ,
a s , N d = N s 0 1 e ξ 2 ϵ 2 L N ( 1 ξ 2 ϵ 2 ) ξ d C s 1 2 ( ξ ) d ξ ,
I d = 2 s = 0 a s , N d i s C s 1 2 ( z ρ ) j s ( k 0 ρ ) ,
a s , N d = ( 2 s + 1 s ) [ a s 1 , N d + 1 ( s 1 2 s 3 ) a s 2 , N d ] ,
a 0 , N d = 1 4 e 1 ϵ 2 m = 0 N k = 0 m ( N m ) ( 1 m ! ) ( m k ) × ( i ) μ ϵ μ + 2 m 2 k 2 γ ( 2 m + μ 2 , 1 ϵ 2 ) ,
a 0 , 0 d = 1 4 ( i ) d + 1 ϵ d 1 e 1 ϵ 2 γ ( d + 1 2 , 1 ϵ 2 ) ,
E x = s = 0 i s { ( a s , N 1 + a s , N 2 ) C s 1 2 ( z ρ ) j s ( k 0 ρ ) a s , N 0 x 2 + z 2 k 0 ρ 3 [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) ( z k 0 ρ 2 ) C s 3 2 ( z ρ ) j s ( k 0 ρ ) ] + a s , N 0 y 2 ρ 2 [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) z k 0 ρ 2 C s 3 2 ( z ρ ) j s ( k 0 ρ ) ] a s , N 0 y 2 z k 0 ρ 4 [ C s 3 2 ( z ρ ) j s ( k 0 ρ ) 3 z k 0 ρ 2 C s 5 2 ( z ρ ) j s ( k 0 ρ ) ] } ,
E y = ( x y ρ 2 ) s = 0 i s { a s , N 0 1 k 0 ρ [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) ( z k 0 ρ 2 ) C s 3 2 ( z ρ ) j s ( k 0 ρ ) ] a s , N 0 [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) z k 0 ρ 2 C s 3 2 ( z ρ ) j s ( k 0 ρ ) ] + a s , N 0 z k 0 ρ 2 [ C s 3 2 ( z ρ ) j s ( k 0 ρ ) 3 z k 0 ρ 2 C s 5 2 ( z ρ ) j s ( k 0 ρ ) ] } ,
E z = i ( x ρ ) s = 1 ( a s , N 0 + a s , N 1 ) i s [ C s 1 2 ( z ρ ) j s ( k 0 ρ ) z k 0 ρ 2 C s 1 3 2 ( z ρ ) j s ( k 0 ρ ) ] ,

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