Abstract

Starting from the vector Maxwell’s equations and applying the multiscale singular perturbation method, the nonparaxial beam propagation is studied in free space. Two new equations have been derived for transverse and longitudinal electric fields of an arbitrary polarized electromagnetic wave even in the case of tightly focused nonparaxial laser beams. By using the analogy of the optical beam in the space domain and the optical pulse in the time domain, the higher-order diffraction term is introduced. For strongly nonparaxial beams that are characterized by large values of the perturbative parameter, our correction solutions yield an accurate description of the field in the near-field region and are consistent with all other correction results obtained by others in the far-field region. For weakly nonparaxial beams, our correction solutions can be expressed in a very simple form that is proved to be exactly consistent with the solutions obtained by Cao and Deng [J. Opt. Soc. Am. A 15, 1144 (1998) ]. In addition, the lowest-order correction to the paraxial approximation can be found to be in good agreement with the results of Lax et al. [Phys. Rev. A 11, 1365 (1975) ] and Seshadri [J. Opt. Soc. Am. A 19, 2134 (2002) ].

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Cicchitelli, H. Hora, and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
    [Crossref] [PubMed]
  2. B. Quesnel and P. Mora, "Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum," Phys. Rev. E 58, 3719-3732 (1998).
    [Crossref]
  3. S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
    [Crossref] [PubMed]
  4. S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
    [Crossref]
  5. S. Sepke and D. Umstadter, "Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes," Opt. Lett. 31, 1447-1449 (2006).
    [Crossref] [PubMed]
  6. S. Sepke and D. Umstadter, "Analytical solutions for the electromagnetic fields of tightly focused laser beams of arbitrary pulse length," Opt. Lett. 31, 2589-2591 (2006).
    [Crossref] [PubMed]
  7. S. Sepke and D. Umstadter, "Analytical solutions for the electromagnetic fields of flattened and annular Gaussian laser modes. I. Small F-number laser focusing," J. Opt. Soc. Am. B 23, 2157-2165 (2006).
    [Crossref]
  8. S. Sepke and D. Umstadter, "Analytical solutions for the electromagnetic fields of flattened and annular Gaussian laser modes. II. Large F-number laser focusing," J. Opt. Soc. Am. B 23, 2166-2173 (2006).
    [Crossref]
  9. M. Lax, W. H. Louisell, W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
    [Crossref]
  10. G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximation," J. Opt. Soc. Am. 69, 575-578 (1979).
    [Crossref]
  11. L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
    [Crossref]
  12. M. Couture and P. A. Belanger, "From Gaussian beam to complex-source-point spherical wave," Phys. Rev. A 24, 355-359 (1981).
    [Crossref]
  13. G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
    [Crossref]
  14. T. Takenaka, M. Yokota, and O. Fukumitsu, "Propagation of light beams beyond the paraxial approximation," J. Opt. Soc. Am. A 2, 826-829 (1985).
    [Crossref]
  15. E. Zauderer, "Complex argument Hermite-Gaussian and Laguerre-Gaussian beams," J. Opt. Soc. Am. A 3, 465-469 (1986).
    [Crossref]
  16. A. Wünsche, "Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams," J. Opt. Soc. Am. A 9, 765-774 (1992).
    [Crossref]
  17. Q. Cao and X. M. Deng, "Corrections to the paraxial approximation of an arbitrary free-propagation beam," J. Opt. Soc. Am. A 15, 1144-1148 (1998).
    [Crossref]
  18. S. R. Seshadri, "Nonparaxial corrections for the fundamental Gaussian beam," J. Opt. Soc. Am. A 19, 2134-2141 (2002).
    [Crossref]
  19. R. Borghi and M. Santarsiero, "Summing Lax series for nonparaxial beam propagation," Opt. Lett. 28, 774-776 (2003).
    [Crossref] [PubMed]
  20. R. Borghi and M. Santarsiero, "Nonparaxial propagation of spirally polarized optical beams," J. Opt. Soc. Am. A 21, 2029-2037 (2004).
    [Crossref]
  21. D. M. Deng, "Nonparaxial propagation of radially polarized light beams," J. Opt. Soc. Am. B 23, 1228-1234 (2006).
    [Crossref]
  22. D. M. Deng, Q. Guo, L. J. Wu, X. B. Yang, "Propagation of radially polarized elegant light beams," J. Opt. Soc. Am. B 24, 636-643 (2007).
    [Crossref]
  23. P. Varga and P. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
    [Crossref]
  24. J. X. Wang, W. Scheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
    [Crossref]
  25. Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5, 101301 (2002).
    [Crossref]
  26. J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
    [Crossref]
  27. A. Ciattoni, P. Di Porto, B. Crosignani, and A. Yariv, "Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation," J. Opt. Soc. Am. B 17, 809-819 (2000).
    [Crossref]
  28. A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 177, 9-13 (2000).
    [Crossref]
  29. E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989).
    [Crossref]
  30. A. H. Nayfeh, Perturbation Methods (Wiley, 1973), Chap. 6 pp. 231-243.
  31. Q. Guo, G. S. Zhou, W. G. Lin, S. H. Liu, C. J. Liao, and H. C. Jin, "Evolution equation of optical solitons in femtosecond regime," Sci. China A34, 1365-1377 (1991).
  32. Q. Guo, C. J. Liao, and S. H. Liu, "Two-evolution equations for nonlinear subpicosecond pulses in optical fibers and their equivalent solitary solutions," Fiber Integr. Opt. 12, 71-81 (1993).
    [Crossref]
  33. S. Chi, and Q. Guo, "Vector theory of self-focusing of an optical beam in Kerr media," Opt. Lett. 20, 1598-1600 (1995).
    [Crossref] [PubMed]
  34. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1989), Chap. 3.
  35. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, 1989), Chap. 4, 5.
  36. S. Mookherjea and A. Yariv, "Analysis of optical pulse propagation with two-by-two ABCD matrices," Phys. Rev. E 64, 016611 (2001).
    [Crossref]
  37. Q. Guo, "Optical beams in media with spatial dispersion," Chin. Phys. Lasers 20, 64-67 (2003).
  38. D. M. Deng, H. Guo, D. A. Han, C. F. Li, "Analysis of beam propagation and pulse propagation with variance matrix treatment," Opt. Commun. 238, 205-213 (2004).
    [Crossref]

2007 (1)

2006 (5)

2005 (1)

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
[Crossref] [PubMed]

2004 (3)

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[Crossref]

R. Borghi and M. Santarsiero, "Nonparaxial propagation of spirally polarized optical beams," J. Opt. Soc. Am. A 21, 2029-2037 (2004).
[Crossref]

D. M. Deng, H. Guo, D. A. Han, C. F. Li, "Analysis of beam propagation and pulse propagation with variance matrix treatment," Opt. Commun. 238, 205-213 (2004).
[Crossref]

2003 (2)

Q. Guo, "Optical beams in media with spatial dispersion," Chin. Phys. Lasers 20, 64-67 (2003).

R. Borghi and M. Santarsiero, "Summing Lax series for nonparaxial beam propagation," Opt. Lett. 28, 774-776 (2003).
[Crossref] [PubMed]

2002 (2)

S. R. Seshadri, "Nonparaxial corrections for the fundamental Gaussian beam," J. Opt. Soc. Am. A 19, 2134-2141 (2002).
[Crossref]

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5, 101301 (2002).
[Crossref]

2001 (2)

S. Mookherjea and A. Yariv, "Analysis of optical pulse propagation with two-by-two ABCD matrices," Phys. Rev. E 64, 016611 (2001).
[Crossref]

J. X. Wang, W. Scheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[Crossref]

2000 (2)

A. Ciattoni, P. Di Porto, B. Crosignani, and A. Yariv, "Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation," J. Opt. Soc. Am. B 17, 809-819 (2000).
[Crossref]

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 177, 9-13 (2000).
[Crossref]

1998 (3)

P. Varga and P. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[Crossref]

B. Quesnel and P. Mora, "Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum," Phys. Rev. E 58, 3719-3732 (1998).
[Crossref]

Q. Cao and X. M. Deng, "Corrections to the paraxial approximation of an arbitrary free-propagation beam," J. Opt. Soc. Am. A 15, 1144-1148 (1998).
[Crossref]

1995 (1)

1993 (1)

Q. Guo, C. J. Liao, and S. H. Liu, "Two-evolution equations for nonlinear subpicosecond pulses in optical fibers and their equivalent solitary solutions," Fiber Integr. Opt. 12, 71-81 (1993).
[Crossref]

1992 (1)

1990 (1)

L. Cicchitelli, H. Hora, and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
[Crossref] [PubMed]

1989 (2)

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989).
[Crossref]

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[Crossref]

1986 (1)

1985 (1)

1983 (1)

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[Crossref]

1981 (1)

M. Couture and P. A. Belanger, "From Gaussian beam to complex-source-point spherical wave," Phys. Rev. A 24, 355-359 (1981).
[Crossref]

1979 (2)

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[Crossref]

G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximation," J. Opt. Soc. Am. 69, 575-578 (1979).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1989), Chap. 3.

Alexander, D. R.

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[Crossref]

Banerjee, S.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
[Crossref] [PubMed]

Barton, J. P.

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[Crossref]

Belanger, P. A.

M. Couture and P. A. Belanger, "From Gaussian beam to complex-source-point spherical wave," Phys. Rev. A 24, 355-359 (1981).
[Crossref]

Bonnaud, G.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[Crossref]

Borghi, R.

Cao, Q.

Chi, S.

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 177, 9-13 (2000).
[Crossref]

A. Ciattoni, P. Di Porto, B. Crosignani, and A. Yariv, "Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation," J. Opt. Soc. Am. B 17, 809-819 (2000).
[Crossref]

Cicchitelli, L.

L. Cicchitelli, H. Hora, and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
[Crossref] [PubMed]

Couture, M.

M. Couture and P. A. Belanger, "From Gaussian beam to complex-source-point spherical wave," Phys. Rev. A 24, 355-359 (1981).
[Crossref]

Crosignani, B.

A. Ciattoni, P. Di Porto, B. Crosignani, and A. Yariv, "Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation," J. Opt. Soc. Am. B 17, 809-819 (2000).
[Crossref]

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 177, 9-13 (2000).
[Crossref]

Davis, L. W.

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[Crossref]

Deng, D. M.

Deng, X. M.

Di Porto, P.

A. Ciattoni, P. Di Porto, B. Crosignani, and A. Yariv, "Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation," J. Opt. Soc. Am. B 17, 809-819 (2000).
[Crossref]

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 177, 9-13 (2000).
[Crossref]

Fukumitsu, O.

Guo, H.

D. M. Deng, H. Guo, D. A. Han, C. F. Li, "Analysis of beam propagation and pulse propagation with variance matrix treatment," Opt. Commun. 238, 205-213 (2004).
[Crossref]

Guo, Q.

D. M. Deng, Q. Guo, L. J. Wu, X. B. Yang, "Propagation of radially polarized elegant light beams," J. Opt. Soc. Am. B 24, 636-643 (2007).
[Crossref]

Q. Guo, "Optical beams in media with spatial dispersion," Chin. Phys. Lasers 20, 64-67 (2003).

S. Chi, and Q. Guo, "Vector theory of self-focusing of an optical beam in Kerr media," Opt. Lett. 20, 1598-1600 (1995).
[Crossref] [PubMed]

Q. Guo, C. J. Liao, and S. H. Liu, "Two-evolution equations for nonlinear subpicosecond pulses in optical fibers and their equivalent solitary solutions," Fiber Integr. Opt. 12, 71-81 (1993).
[Crossref]

Q. Guo, G. S. Zhou, W. G. Lin, S. H. Liu, C. J. Liao, and H. C. Jin, "Evolution equation of optical solitons in femtosecond regime," Sci. China A34, 1365-1377 (1991).

Han, D. A.

D. M. Deng, H. Guo, D. A. Han, C. F. Li, "Analysis of beam propagation and pulse propagation with variance matrix treatment," Opt. Commun. 238, 205-213 (2004).
[Crossref]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, 1989), Chap. 4, 5.

Ho, Y. K.

J. X. Wang, W. Scheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[Crossref]

Hoelss, M.

J. X. Wang, W. Scheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[Crossref]

Hora, H.

L. Cicchitelli, H. Hora, and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
[Crossref] [PubMed]

Jin, H. C.

Q. Guo, G. S. Zhou, W. G. Lin, S. H. Liu, C. J. Liao, and H. C. Jin, "Evolution equation of optical solitons in femtosecond regime," Sci. China A34, 1365-1377 (1991).

Keitel, C. H.

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5, 101301 (2002).
[Crossref]

Lax, M.

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[Crossref]

M. Lax, W. H. Louisell, W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[Crossref]

Li, C. F.

D. M. Deng, H. Guo, D. A. Han, C. F. Li, "Analysis of beam propagation and pulse propagation with variance matrix treatment," Opt. Commun. 238, 205-213 (2004).
[Crossref]

Liao, C. J.

Q. Guo, C. J. Liao, and S. H. Liu, "Two-evolution equations for nonlinear subpicosecond pulses in optical fibers and their equivalent solitary solutions," Fiber Integr. Opt. 12, 71-81 (1993).
[Crossref]

Q. Guo, G. S. Zhou, W. G. Lin, S. H. Liu, C. J. Liao, and H. C. Jin, "Evolution equation of optical solitons in femtosecond regime," Sci. China A34, 1365-1377 (1991).

Lin, W. G.

Q. Guo, G. S. Zhou, W. G. Lin, S. H. Liu, C. J. Liao, and H. C. Jin, "Evolution equation of optical solitons in femtosecond regime," Sci. China A34, 1365-1377 (1991).

Liu, S. H.

Q. Guo, C. J. Liao, and S. H. Liu, "Two-evolution equations for nonlinear subpicosecond pulses in optical fibers and their equivalent solitary solutions," Fiber Integr. Opt. 12, 71-81 (1993).
[Crossref]

Q. Guo, G. S. Zhou, W. G. Lin, S. H. Liu, C. J. Liao, and H. C. Jin, "Evolution equation of optical solitons in femtosecond regime," Sci. China A34, 1365-1377 (1991).

Loubere, R.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[Crossref]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[Crossref]

Maksimchuk, A.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," 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, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[Crossref]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[Crossref]

Michel, P.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[Crossref]

Mocken, G. R.

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5, 101301 (2002).
[Crossref]

Mookherjea, S.

S. Mookherjea and A. Yariv, "Analysis of optical pulse propagation with two-by-two ABCD matrices," Phys. Rev. E 64, 016611 (2001).
[Crossref]

Mora, P.

B. Quesnel and P. Mora, "Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum," Phys. Rev. E 58, 3719-3732 (1998).
[Crossref]

Nayfeh, A. H.

A. H. Nayfeh, Perturbation Methods (Wiley, 1973), Chap. 6 pp. 231-243.

Ovadia, J.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[Crossref]

Pattanayak, D. N.

Postle, R.

L. Cicchitelli, H. Hora, and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
[Crossref] [PubMed]

Quesnel, B.

B. Quesnel and P. Mora, "Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum," Phys. Rev. E 58, 3719-3732 (1998).
[Crossref]

Riazuelo, G.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[Crossref]

Salamin, Y. I.

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5, 101301 (2002).
[Crossref]

Santarsiero, M.

Scheid, W.

J. X. Wang, W. Scheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[Crossref]

Sepke, S.

Seshadri, S. R.

Shah, R.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
[Crossref] [PubMed]

Takenaka, T.

Tikhonchuk, V. T.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[Crossref]

Török, P.

P. Varga and P. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[Crossref]

Umstadter, D.

Valenzuela, A.

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
[Crossref] [PubMed]

Varga, P.

P. Varga and P. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[Crossref]

Walraet, F.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[Crossref]

Wang, J. X.

J. X. Wang, W. Scheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[Crossref]

Weber, S.

S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[Crossref]

Weniger, E. J.

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989).
[Crossref]

Wu, L. J.

Wünsche, A.

Yang, X. B.

Yariv, A.

Yokota, M.

Zauderer, E.

Zhou, G. S.

Q. Guo, G. S. Zhou, W. G. Lin, S. H. Liu, C. J. Liao, and H. C. Jin, "Evolution equation of optical solitons in femtosecond regime," Sci. China A34, 1365-1377 (1991).

Chin. Phys. Lasers (1)

Q. Guo, "Optical beams in media with spatial dispersion," Chin. Phys. Lasers 20, 64-67 (2003).

Comput. Phys. Rep. (1)

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989).
[Crossref]

Fiber Integr. Opt. (1)

Q. Guo, C. J. Liao, and S. H. Liu, "Two-evolution equations for nonlinear subpicosecond pulses in optical fibers and their equivalent solitary solutions," Fiber Integr. Opt. 12, 71-81 (1993).
[Crossref]

J. Appl. Phys. (1)

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Opt. Soc. Am. B (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, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004).
[Crossref]

Opt. Commun. (3)

P. Varga and P. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998).
[Crossref]

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 177, 9-13 (2000).
[Crossref]

D. M. Deng, H. Guo, D. A. Han, C. F. Li, "Analysis of beam propagation and pulse propagation with variance matrix treatment," Opt. Commun. 238, 205-213 (2004).
[Crossref]

Opt. Lett. (4)

Phys. Rev. A (5)

L. Cicchitelli, H. Hora, and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
[Crossref] [PubMed]

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[Crossref]

M. Couture and P. A. Belanger, "From Gaussian beam to complex-source-point spherical wave," Phys. Rev. A 24, 355-359 (1981).
[Crossref]

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[Crossref]

M. Lax, W. H. Louisell, W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[Crossref]

Phys. Rev. E (3)

S. Mookherjea and A. Yariv, "Analysis of optical pulse propagation with two-by-two ABCD matrices," Phys. Rev. E 64, 016611 (2001).
[Crossref]

B. Quesnel and P. Mora, "Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum," Phys. Rev. E 58, 3719-3732 (1998).
[Crossref]

J. X. Wang, W. Scheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001).
[Crossref]

Phys. Rev. Lett. (1)

S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005).
[Crossref] [PubMed]

Phys. Rev. ST Accel. Beams (1)

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5, 101301 (2002).
[Crossref]

Other (4)

A. H. Nayfeh, Perturbation Methods (Wiley, 1973), Chap. 6 pp. 231-243.

Q. Guo, G. S. Zhou, W. G. Lin, S. H. Liu, C. J. Liao, and H. C. Jin, "Evolution equation of optical solitons in femtosecond regime," Sci. China A34, 1365-1377 (1991).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1989), Chap. 3.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, 1989), Chap. 4, 5.

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 (4)

Fig. 1
Fig. 1

Behavior of the normalized amplitude of the transverse electric field components as a function of (a) z λ and (b) x λ for a Gaussian beam with a spot size w 0 λ = 1 2 for transverse coordinates y = 0 . Solid curves, Lax’s second-order solution; dashed curves, Borghi’s second-order solution; dotted curves, our second-order solution; dash-dotted curves, the exact solution. The parameters are chosen as (a) x = λ , (b) z = λ . Included are the solutions of four models corresponding to Lax’s second-order solution, Borghi’s second-order solution, our second-order solution and the exact solution, which are a little bit different and cannot be distinguished by eye when 0.9 < z λ < 1.1 , z λ > 4 in (a) and when 0.5 < x λ < 1 in (b).

Fig. 2
Fig. 2

Behavior of the normalized amplitude of the longitudinal electric field components as a function of (a) z λ and (b) x λ for a Gaussian beam with a spot size w 0 λ = 1 2 for transverse coordinates y = 0 . Sold curves, Lax’s third-order solution; dashed curves, Borghi’s third-order solution; dotted curves, our third-order solution; dash-dotted curves, the exact solution. The parameters are chosen to be the same as those in Fig. 1. Included are the solutions of four models corresponding to Lax’s second-order solution, Borghi’s second-order solution, our second-order solution, and the exact solution, which are a little bit different and cannot be distinguished by eye when z λ > 3 in (a) and when 0.0 < x λ < 0.8 in (b).

Fig. 3
Fig. 3

Behavior of the relative error for the transverse electric field components as a function of (a) z λ and (b) x λ for a Gaussian beam with a spot size w 0 λ = 1 2 for transverse coordinates y = 0 . Solid curves, Lax’s second-order solution; dashed curves, Borghi’s second-order solution; dotted curves, our second-order solution. The parameters are chosen to be the same as those in Fig. 1.

Fig. 4
Fig. 4

Behavior of the relative error for the longitudinal electric field components as a function of (a) z λ and (b) x λ for a Gaussian beam with a spot size w 0 λ = 1 2 for transverse coordinates y = 0 . Solid curves, Lax’s third-order solution; dashed curves, Borghi’s third-order solution; dotted curves, our third-order solution. The parameters are chosen to be the same as those in Fig. 1.

Equations (82)

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

× E = i ω 0 B ,
× B = i ω 0 c 2 E ,
E = 0 ,
B = 0 ,
× × E = ( ω 0 c ) 2 E ,
E = 0 .
( F + F z z + i k F z ) 2 F 2 F z 2 2 i k F z = 0 ,
z ( F ) + i k F 2 F z = k 2 F z ,
ρ = r w 0 , ζ = z L ,
t ( σ t F t + σ 2 F ζ ζ + i F ζ ) σ t 2 F t σ 3 2 F t ζ 2 2 i σ F t ζ = 0 ,
σ 3 ζ ( t F t ) + i σ t F t σ 2 t 2 F ζ = F ζ ,
ζ n = σ n ζ , n = 0 , 1 , 2 , ,
ζ = ζ 0 + σ ζ 1 + σ 2 ζ 2 + .
F t = F t ( 0 ) + σ F t ( 1 ) + σ 2 F t ( 2 ) + ,
F ζ = F ζ ( 0 ) + σ F ζ ( 1 ) + σ 2 F ζ ( 2 ) + .
E t ( 2 N ) ( ρ , ζ ) = 1 4 π 2 A 0 ( ζ = 0 ) exp ( i φ 1 ) d 2 k t ,
E ζ ( 2 N + 1 ) ( ρ , ζ ) = σ 4 π 2 k t A 0 ( ζ = 0 ) m 2 exp ( i φ 1 ) d 2 k t ,
E t ( ρ , ζ ) = 1 4 π 2 A 0 ( ζ = 0 ) exp ( i φ 2 ) d 2 k t ,
E ζ ( ρ , ζ ) = σ 4 π 2 k t A 0 ( ζ = 0 ) 1 k t 2 σ 2 exp ( i φ 2 ) d 2 k t ,
E t ( 2 N ) ( ρ , ζ ) ζ = i 2 t 2 E t ( 2 N ) ( ρ , ζ ) i n = 2 N + 1 ( 1 ) n × ( 2 n 3 ) ! ! ( 2 n ) ! ! σ ( 2 n 2 ) ( t 2 ) n E t ( 2 N ) ( ρ , ζ ) .
E ζ ( 2 N + 1 ) ( ρ , ζ ) = i σ [ 1 + σ 2 2 t 2 + m = 2 N ( 2 m 1 ) ! ! ( 2 m ) ! ! × σ 2 m ( t 2 ) m ] t E t ( 2 N ) ( ρ , ζ ) .
E t ( 2 ) ( ρ , ζ ) ζ = i 2 t 2 E t ( 2 ) ( ρ , ζ ) i σ 2 8 t 4 E t ( 2 ) ( ρ , ζ ) ,
E ζ ( 3 ) ( ρ , ζ ) = i σ ( 1 + σ 2 2 t 2 ) t E t ( 2 ) ( ρ , ζ ) .
E ( 2 N ) ( r , z ) z = i 2 k 2 E ( 2 N ) ( r , z ) i n = 2 N + 1 ( 1 ) n ( 2 n 3 ) ! ! ( 2 n ) ! ! ( 2 ) n k ( 2 n 1 ) E t ( 2 N ) ( r , z ) ,
E z ( 2 N + 1 ) ( r , z ) = i k [ 1 + 1 2 k 2 2 + m = 2 N ( 2 m 1 ) ! ! ( 2 m ) ! ! ( 2 ) m k 2 m ] E ( 2 N ) ( r , z ) .
E x ( 2 N ) a ( ξ , η , ζ ) = 1 4 π 2 + ϕ ̃ 0 exp ( i φ 1 ) d k ξ d k η ,
E z ( 2 N + 1 ) a ( ξ , η , ζ ) = σ 4 π 2 + k ξ ϕ ̃ 0 m 2 exp ( i φ 1 ) d k ξ d k η ,
ϕ ̃ 0 = + ϕ 0 exp [ i ( k ξ ξ + k η η ) ] d ξ d η .
E x a = 1 4 π 2 + ϕ ̃ 0 exp ( i φ 2 ) d k ξ d k η ,
E z a = σ 4 π 2 + k ξ ϕ ̃ 0 exp ( i φ 2 ) 1 k t 2 σ 2 d k ξ d k η .
E x ( 2 N ) = 1 8 π 2 + ϕ ̃ 0 ( 2 m 1 + σ 2 k η 2 m 2 ) exp ( i φ 1 ) d k ξ d k η ,
E y ( 2 N ) = σ 2 8 π 2 + ϕ ̃ 0 k ξ k η m 2 exp ( i φ 1 ) d k ξ d k η ,
E z ( 2 N + 1 ) = σ 8 π 2 + ϕ ̃ 0 k ξ ( 1 + m 2 ) exp ( i φ 1 ) d k ξ d k η ,
B x ( 2 N ) = E y ( 2 N ) c ,
B y ( 2 N ) = 1 8 π 2 c + ϕ ̃ 0 ( 2 m 1 + σ 2 k ξ 2 m 2 ) exp ( i φ 1 ) d k ξ d k η ,
B z ( 2 N + 1 ) = σ 8 π 2 c + ϕ ̃ 0 k η ( 1 + m 2 ) exp ( i φ 1 ) d k ξ d k η .
E x = 1 8 π 2 + ( 1 + m 3 + k η 2 σ 2 m 3 ) ϕ ̃ 0 exp ( i φ 2 ) d k ξ d k η ,
E y = σ 2 8 π 2 + k ξ k η ϕ ̃ 0 m 3 exp ( i φ 2 ) d k ξ d k η ,
E z = σ 8 π 2 + k ξ ( 1 + 1 m 3 ) ϕ ̃ 0 exp ( i φ 2 ) d k ξ d k η ,
B x = E y c ,
B y = 1 8 c π 2 + ( 1 + m 3 + k ξ 2 σ 2 m 3 ) ϕ ̃ 0 exp ( i φ 2 ) d k ξ d k η ,
B z = σ 8 c π 2 + k η ( 1 + 1 m 3 ) ϕ ̃ 0 exp ( i φ 2 ) d k ξ d k η ,
E x = E 0 4 ( I 1 + σ 2 ξ 2 η 2 r 1 3 I 2 + σ 2 η 2 r 1 2 I 3 ) ,
E y = σ 2 E 0 4 ξ η r 1 3 ( r 1 I 3 2 I 2 ) ,
E z = i σ E 0 4 ξ r 1 I 4 ,
B x = E y c ,
B y = E 0 4 c ( I 1 + σ 2 η 2 ξ 2 r 1 3 I 2 + σ 2 ξ 2 r 1 2 I 3 ) ,
B z = i σ E 0 4 c η r 1 I 4 ,
I 1 = 0 1 σ ( 1 + 1 k t 2 σ 2 ) exp ( k t 2 4 i φ 5 ) J 0 ( k t ρ ) k t d k t ,
I 2 = 0 1 σ 1 1 k t 2 σ 2 exp ( k t 2 4 i φ 5 ) J 1 ( k t ρ ) k t 2 d k t ,
I 3 = 0 1 σ 1 1 k t 2 σ 2 exp ( k t 2 4 i φ 5 ) J 0 ( k t ρ ) k t 3 d k t ,
I 4 = 0 1 σ ( 1 + 1 1 k t 2 σ 2 ) exp ( k t 2 4 i φ 5 ) J 1 ( k t ρ ) k t 2 d k t ,
F ζ ( 0 ) = 0 ,
t 2 F t ( 0 ) + 2 i F t ( 0 ) ζ 0 = 0 ,
F ζ ( 1 ) = i t F t ( 0 ) ,
t 2 F t ( 1 ) + 2 i ( F t ( 1 ) ζ 0 + F t ( 0 ) ζ 1 ) = 0 ,
F ζ ( 2 ) = i t F t ( 1 ) ,
t 2 F t ( 2 ) + 2 i ( F t ( 2 ) ζ 0 + F t ( 1 ) ζ 1 + F t ( 0 ) ζ 2 ) = 2 F t ( 0 ) ζ 0 2 ,
F ζ ( 3 ) = i t F t ( 2 ) + i F ζ ( 1 ) ζ 0 ,
t 2 F t ( N ) + 2 i j = 0 N F t ( N j ) ζ j = j = 0 [ ( N 2 ) 2 ] 2 F t ( N 2 2 j ) ζ j 2 2 m = 0 [ ( N 3 ) 2 ] j = 0 N ( 2 m + 3 ) ( 2 F t ( j ) ζ m ζ N 2 m j ) ,
F ζ ( N + 1 ) = i t F t ( N ) + i j = 1 N 1 F ζ ( j ) ζ N 1 j ( N > 2 ) ,
F t ( 0 ) ( ρ , 0 ) = F t ( 0 ) ( ρ ) , F t ( N ) ( ρ , 0 ) = 0 , N 1 ,
F ζ ( 0 ) ( ρ , 0 ) = 0 , F ζ ( N ) ( ρ , 0 ) = 0 , N 1 .
F ̃ t ( 0 ) ( k t , ζ ) = A 0 ( ζ 0 = 0 , ζ 1 , ) exp ( i k t 2 ζ 0 2 ) ,
F ̃ ζ ( 1 ) ( k t , ζ ) = k t F ̃ t ( 0 ) ( k t , ζ ) ,
F ̃ t ( 0 ) ( k t , ζ ) = F t ( 0 ) ( ρ , ζ ) exp ( i k t ρ ) d 2 ρ ,
F t ( 0 ) ( ρ , ζ ) = 1 4 π 2 F ̃ t ( 0 ) ( k t , ζ ) exp ( i k t ρ ) d 2 k t ;
F ̃ ζ ( 1 ) ( k t , ζ ) = F ζ ( 1 ) ( ρ , ζ ) exp ( i k t ρ ) d 2 ρ ,
F ζ ( 1 ) ( ρ , ζ ) = 1 4 π 2 F ̃ ζ ( 1 ) ( k t , ζ ) exp ( i k t ρ ) d 2 k t .
F ̃ t ( 1 ) ( k t , ζ ) = ζ 0 A 0 ζ 1 exp ( i k t 2 ζ 0 2 ) ,
F ̃ ζ ( 2 ) ( k t , ζ ) = k t F ̃ t ( 1 ) ( k t , ζ ) .
A 0 ζ 1 = 0 .
F ̃ t ( 2 ) ( k t , ζ ) = ζ 0 ( A 0 ζ 2 + i k t 4 8 A 0 ) exp ( i k t 2 ζ 0 2 ) ,
F ̃ ζ ( 3 ) ( k t , ζ ) = k t F ̃ t ( 2 ) ( k t , ζ ) + i F ̃ ζ ( 1 ) ( k t , ζ ) ζ 0 .
A 0 ζ 2 + i k t 4 8 A 0 = 0 .
A 0 ( ζ 0 , ζ 1 , ) = A 0 ( ζ 0 , 1 , 2 = 0 , ζ 3 , ) exp ( i k t 4 ζ 2 8 ) .
F ̃ t ( 3 ) ( k t , ζ ) = 0 ,
F ̃ ζ ( 4 ) ( k t , ζ ) = 0 ,
.
A 0 ( ζ 0 , ζ 1 , , ζ 2 N ) = A 0 ( ζ 0 , 1 , , 2 N = 0 ) × exp [ i n = 2 N + 1 ( 2 n 3 ) ! ! ( 2 n ) ! ! k t 2 n ζ 2 n 2 ] ,
E ̃ t ( 2 N ) ( k t , ζ ) = A 0 ( ζ 0 , 1 , , 2 N = 0 ) exp ( i m 1 ζ σ 2 ) ,
E ̃ ζ ( 2 N + 1 ) ( k t , ζ ) = m 2 σ k t E ̃ t ( 2 N ) ( k t , ζ ) ,

Metrics