Abstract

Non-paraxial corrections for a scalar optical field that follows the Helmotz equation are extracted for the first time, to the best of our knowledge, in the angular spectrum representation by taking into account generic boundary conditions. Those integration constants are compared with closed-form solutions and approximate series expansions usually obtained by other authors. This method is particularized to the direct electron acceleration with a tightly focused TM0,1 laser beam to demonstrate that these constants have a strong effect on the final average energy and quality of the electron beam.

© 2014 Optical Society of America

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  1. S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, Appl. Phys. Lett. 101, 041105 (2012).
    [CrossRef]
  2. C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
    [CrossRef]
  3. A. April, J. Opt. Soc. Am. A 28, 2100 (2011).
    [CrossRef]
  4. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  5. A. Wünsche, J. Opt. Soc. Am. A 9, 765 (1992).
    [CrossRef]
  6. G. P. Agrawal and M. Lax, Phys. Rev. A 27, 1693 (1983).
    [CrossRef]
  7. O. E. Gawhary and S. Severini, J. Opt. Soc. Am. A 27, 458 (2010).
    [CrossRef]
  8. V. Marceau, C. Varin, and M. Piché, Opt. Lett. 38, 821 (2013).
    [CrossRef]
  9. L. W. Davis, Phys. Rev. A 19, 1177 (1979).
    [CrossRef]
  10. A. G. van Nie, Philips Res. Rep. 19, 378 (1964).
  11. G. P. Agrawal and D. N. Pattanayak, J. Opt. Soc. Am. 69, 575 (1979).
    [CrossRef]
  12. M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
    [CrossRef]
  13. C. J. R. Sheppard and S. Saghafi, Opt. Lett. 24, 1543 (1999).
    [CrossRef]
  14. Y. I. Salamin, Opt. Lett. 31, 2619 (2006).
    [CrossRef]
  15. A. April, Opt. Lett. 33, 1563 (2008).
    [CrossRef]
  16. L. W. Davis and G. Patsakos, Opt. Lett. 6, 22 (1981).
    [CrossRef]
  17. L. J. Wong and F. X. Kärtner, Opt. Express 18, 25035 (2010).
    [CrossRef]
  18. A. April and M. Piché, Opt. Express 18, 22128 (2010).
    [CrossRef]
  19. L. Dai, J.-X. Li, W.-P. Zang, and J.-G. Tian, Opt. Express 19, 9303 (2011).
    [CrossRef]
  20. R. Borghi and M. Santarsiero, Opt. Lett. 28, 774 (2003).
    [CrossRef]

2013

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

V. Marceau, C. Varin, and M. Piché, Opt. Lett. 38, 821 (2013).
[CrossRef]

2012

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, Appl. Phys. Lett. 101, 041105 (2012).
[CrossRef]

2011

2010

2008

2006

2003

1999

1992

1983

G. P. Agrawal and M. Lax, Phys. Rev. A 27, 1693 (1983).
[CrossRef]

1981

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

L. W. Davis and G. Patsakos, Opt. Lett. 6, 22 (1981).
[CrossRef]

1979

1975

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

1964

A. G. van Nie, Philips Res. Rep. 19, 378 (1964).

Agrawal, G. P.

April, A.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

A. April, J. Opt. Soc. Am. A 28, 2100 (2011).
[CrossRef]

A. April and M. Piché, Opt. Express 18, 22128 (2010).
[CrossRef]

A. April, Opt. Lett. 33, 1563 (2008).
[CrossRef]

Belanger, P.-A.

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Borghi, R.

Brabec, T.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

Couture, M.

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Dai, L.

Davis, L. W.

Fortin, P.-L.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

Fourmaux, S.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, Appl. Phys. Lett. 101, 041105 (2012).
[CrossRef]

Gawhary, O. E.

Kärtner, F. X.

Kieffer, J. C.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, Appl. Phys. Lett. 101, 041105 (2012).
[CrossRef]

Kieffer, J.-C.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

Lax, M.

G. P. Agrawal and M. Lax, Phys. Rev. A 27, 1693 (1983).
[CrossRef]

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

Légaré, F.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, Appl. Phys. Lett. 101, 041105 (2012).
[CrossRef]

Li, J.-X.

Louisell, W. H.

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

MacLean, J. P.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, Appl. Phys. Lett. 101, 041105 (2012).
[CrossRef]

Marceau, V.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

V. Marceau, C. Varin, and M. Piché, Opt. Lett. 38, 821 (2013).
[CrossRef]

McKnight, W. B.

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

Patsakos, G.

Pattanayak, D. N.

Payeur, S.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, Appl. Phys. Lett. 101, 041105 (2012).
[CrossRef]

Piché, M.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

V. Marceau, C. Varin, and M. Piché, Opt. Lett. 38, 821 (2013).
[CrossRef]

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, Appl. Phys. Lett. 101, 041105 (2012).
[CrossRef]

A. April and M. Piché, Opt. Express 18, 22128 (2010).
[CrossRef]

Saghafi, S.

Salamin, Y. I.

Santarsiero, M.

Schmidt, B.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

Schmidt, B. E.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, Appl. Phys. Lett. 101, 041105 (2012).
[CrossRef]

Severini, S.

Sheppard, C. J. R.

Tchervenkov, C.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, Appl. Phys. Lett. 101, 041105 (2012).
[CrossRef]

Thiré, N.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

Tian, J.-G.

van Nie, A. G.

A. G. van Nie, Philips Res. Rep. 19, 378 (1964).

Varin, C.

V. Marceau, C. Varin, and M. Piché, Opt. Lett. 38, 821 (2013).
[CrossRef]

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

Wong, L. J.

Wünsche, A.

Zang, W.-P.

Appl. Phys. Lett.

S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, Appl. Phys. Lett. 101, 041105 (2012).
[CrossRef]

Appl. Sci.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, Appl. Sci. 3, 70 (2013).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Philips Res. Rep.

A. G. van Nie, Philips Res. Rep. 19, 378 (1964).

Phys. Rev. A

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

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

G. P. Agrawal and M. Lax, Phys. Rev. A 27, 1693 (1983).
[CrossRef]

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

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Figures (3)

Fig. 1.
Fig. 1.

Energy of the electron after 15 ps as a function of its initial radial position ρ0 in units of the wavelength λ. (a) Dependence on the choice of a set of parameters taken from Table 1 with n=1. (b) Same as (a) for few asymptotic coefficients C2,2 while C2,4=0. (c) Effect of corrections at n=1 and 2 for Solution (A) of Table 1 on a wider ρ0 range.

Fig. 2.
Fig. 2.

Energy of the electron after 15 ps as a function of (a) the modulus and the argument of C2,2 and ρ0=0 and (b) C2,2 and ρ0 where C2,2 is varied on the real axis, both for C2,4=0.

Fig. 3.
Fig. 3.

Energy and radial position of a set of 5000 electrons generated with conditions of Fig. 1(c) with n=1 after 15 ps. The same simulation is performed in the conditions of the paraxial approximation, which corresponds to points indicated by the arrows, with a very small extension.

Tables (1)

Tables Icon

Table 1. Values of the Integration Constants C2n,2(n+i) for Various References

Equations (5)

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

Ψ˜=n=0ϵ2nΨ˜2n,
κ2Ψ˜2n+4iΨ˜2nζ={0,ifn=02Ψ˜2(n1)/ζ2,ifn>0,
4iΨ˜0Qnζ=2Ψ˜2(n1)ζ2,n>0,
Ψ2Ψ0=f(4C2,212)+f2(32C2,4+12)+ρ2(f3(64C2,41)+f2(14C2,2))+ρ4(f4(16C2,4+14)f34),
Ψ4Ψ0=f2(32C4,46C2,2)+f3(6C2,296C2,4+384C4,634)+f4(96C2,4+6144C4,8+34)+ρ2(3f5(128C2,4+8192C4,8+1)+f4(18C2,2+384C2,41152C4,6+154)+f3(18C2,264C4,434))+ρ4(f6(288C2,4+18432C4,8+94)+f5(9C2,2288C2,4+576C4,6278)+f4(9C2,2+16C4,4+98))+ρ6(f7(64C2,44096C4,812)+f6(C2,2+64C2,464C4,6+78)+f5(C2,238))+ρ8(f8(4C2,4+256C4,8+132)+f7(4C2,4116)+f632).

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