Abstract

A theoretical analysis aimed at investigating the divergent character of perturbative series involved in the study of free-space nonparaxial propagation of vectorial optical beams is proposed. Our analysis predicts a factorial divergence for such series and provides a theoretical framework within which the results of recently published numerical experiments concerning nonparaxial propagation of vectorial Gaussian beams find a meaningful interpretation in terms of the decoding operated on such series by the Weniger transformation.

© 2011 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. Y. Salamin, Opt. Lett. 31, 2619 (2006).
    [CrossRef] [PubMed]
  3. R. Borghi and M. Santarsiero, Opt. Lett. 28, 774 (2003).
    [CrossRef] [PubMed]
  4. E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989).
    [CrossRef]
  5. J. Li, W. Zang, and J. Tian, Opt. Express 17, 4959 (2009).
    [CrossRef] [PubMed]
  6. E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, Phys. Rep. 446, 1 (2007).
    [CrossRef]
  7. C. Brezinski, M. Redivo-Zaglia, and E. J. Weniger, Appl. Numer. Math. 60, 1183 (2010).
    [CrossRef]
  8. There is a nice quotation of a letter from Euler to Goldbach (1745) in that, translated from Latin, reads, “The sum of any given series is the value of the specific finite expression whose expansion gave rise to that same series.”
  9. National Institute of Standards and Technology, Digital Library of Mathematical Functions (National Institute of Standards and Technology, 2010), http://dlmf.nist.gov/.
  10. R. Borghi, Appl. Numer. Math. 60, 1242 (2010).
    [CrossRef]
  11. A. Wünsche, J. Opt. Soc. Am. A 9, 765 (1992).
    [CrossRef]
  12. Y. A. Brychkov, Handbook of Special Functions (CRC Press, 2008).
  13. S. R. Seshadri, J. Opt. Soc. Am. A 19, 2134 (2002).
    [CrossRef]
  14. In evaluating the integral, use has been made of formula 2.19.3.3 in .
  15. C. M. Bender and S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, 1978).
  16. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon Breach, 1986), Vol.  II.

2010 (2)

C. Brezinski, M. Redivo-Zaglia, and E. J. Weniger, Appl. Numer. Math. 60, 1183 (2010).
[CrossRef]

R. Borghi, Appl. Numer. Math. 60, 1242 (2010).
[CrossRef]

2009 (1)

2007 (1)

E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, Phys. Rep. 446, 1 (2007).
[CrossRef]

2006 (1)

2003 (1)

2002 (1)

1992 (1)

1989 (1)

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

1975 (1)

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

Bender, C. M.

C. M. Bender and S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, 1978).

Borghi, R.

Brezinski, C.

C. Brezinski, M. Redivo-Zaglia, and E. J. Weniger, Appl. Numer. Math. 60, 1183 (2010).
[CrossRef]

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon Breach, 1986), Vol.  II.

Y. A. Brychkov, Handbook of Special Functions (CRC Press, 2008).

Caliceti, E.

E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, Phys. Rep. 446, 1 (2007).
[CrossRef]

Jentschura, U. D.

E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, Phys. Rep. 446, 1 (2007).
[CrossRef]

Lax, M.

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

Li, J.

Louisell, W. H.

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

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon Breach, 1986), Vol.  II.

McKnight, W. B.

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

Meyer-Hermann, M.

E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, Phys. Rep. 446, 1 (2007).
[CrossRef]

Orzag, S. A.

C. M. Bender and S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, 1978).

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon Breach, 1986), Vol.  II.

Redivo-Zaglia, M.

C. Brezinski, M. Redivo-Zaglia, and E. J. Weniger, Appl. Numer. Math. 60, 1183 (2010).
[CrossRef]

Ribeca, P.

E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, Phys. Rep. 446, 1 (2007).
[CrossRef]

Salamin, Y.

Santarsiero, M.

Seshadri, S. R.

Surzhykov, A.

E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, Phys. Rep. 446, 1 (2007).
[CrossRef]

Tian, J.

Weniger, E. J.

C. Brezinski, M. Redivo-Zaglia, and E. J. Weniger, Appl. Numer. Math. 60, 1183 (2010).
[CrossRef]

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

Wünsche, A.

Zang, W.

Appl. Numer. Math. (2)

C. Brezinski, M. Redivo-Zaglia, and E. J. Weniger, Appl. Numer. Math. 60, 1183 (2010).
[CrossRef]

R. Borghi, Appl. Numer. Math. 60, 1242 (2010).
[CrossRef]

Comput. Phys. Rep. (1)

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

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (2)

Phys. Rep. (1)

E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, Phys. Rep. 446, 1 (2007).
[CrossRef]

Phys. Rev. A (1)

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

Other (6)

There is a nice quotation of a letter from Euler to Goldbach (1745) in that, translated from Latin, reads, “The sum of any given series is the value of the specific finite expression whose expansion gave rise to that same series.”

National Institute of Standards and Technology, Digital Library of Mathematical Functions (National Institute of Standards and Technology, 2010), http://dlmf.nist.gov/.

Y. A. Brychkov, Handbook of Special Functions (CRC Press, 2008).

In evaluating the integral, use has been made of formula 2.19.3.3 in .

C. M. Bender and S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, 1978).

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon Breach, 1986), Vol.  II.

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Equations (18)

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s n s a n + 1 = m = 0 c m ( n + 1 ) m ,
δ n ( m ) = Δ m { ( n + 1 ) m + 1 s n / a n + 1 } Δ m { ( n + 1 ) m + 1 / a n + 1 } ,
{ × × E = k 2 E , · E = 0.
F ( 0 ) ( r , z ) = i k 2 π z d 2 ρ E ( ρ , 0 ) exp [ i k 2 z ( r ρ ) 2 ] ,
F z ( 1 ) ( r , z ) = i k · F ( 0 ) ( r , z ) .
F ( 2 m ) ( r , z ) = ( i 2 k ) m z m ! z 2 m [ z m 1 F ( 0 ) ( r , z ) ] .
F z ( 2 m + 1 ) ( r , z ) = i k · F ( 2 m ) ( r , z ) + i k z F z ( 2 m 1 ) ( r , z ) ,
F ( 2 m ) ( r , z ) = i k 2 π ( i 2 k ) m z m ! d 2 ρ E ( ρ , 0 ) z 2 m { z m 2 exp [ i k 2 z ( r ρ ) 2 ] } .
F ( 2 m ) ( r , z ) = i k 2 π z ( 2 m ) ! m ! ( i 2 k z ) m × d 2 ρ E ( ρ , 0 ) L 2 m 1 m [ i k 2 z ( r ρ ) 2 ] ,
L n α ( t ) = exp ( t ) L n α ( t ) ,
L 2 m 1 m ( t ) = ( t ) m 1 ( m + 1 ) ! ( 2 m ) ! L m + 1 m 1 ( t ) ,
F ( 2 m ) ( r , z ) = m + 1 π ( 1 4 z 2 ) m d 2 ρ E ( ρ , 0 ) ( r ρ ) 2 ( m 1 ) L m + 1 m 1 [ i k 2 z ( r ρ ) 2 ] .
F ( 2 m ) ( 0 , z ) = u ^ ( 2 m ) ! m ! ( z R 2 k z 2 ) m ( 1 i z R z ) 2 m + 1 ,
L m + 1 m 1 ( t ) ( 2 m m + 1 ) exp ( t ) , for     m 1 ,
F ( 2 m ) ( r , z ) 1 π ( 2 m ) ! m ! ( m 1 ) ! ( 1 4 z 2 ) m × d 2 ρ E ( ρ , 0 ) ( r ρ ) 2 ( m 1 ) exp [ i k z ( r ρ ) 2 ] .
d 2 ξ E ( r ξ , 0 ) exp [ ϕ ( ξ ) ] ,
exp [ ϕ ( ξ ¯ ) ] = [ i ( m 1 ) β ] m 1 exp [ ( m 1 ) ] ( i z k ) m 1 ( m 1 ) ! 2 π ( m 1 ) ,
( 2 m ) ! m ! m 1 ,

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