Abstract

A purely time-domain approach is proposed for the propagation of vectorial ultrafast beams in free space beyond the paraxial and the slowly varying envelope approximations. As an example of application of this method, we describe in detail the vectorial properties of an ultrafast tightly focused transverse-magnetic (TM01) beam, where special attention is given to the longitudinal electric field component. We show that for spot sizes at the waist comparable to the wavelength, the beam diverges more rapidly than expected from paraxial theory. A consequence of this phenomenon is a faster decrease of the amplitude of the longitudinal field away from the waist and a faster evolution of the axial Gouy phase shift in the vicinity of the focus. It has been observed that the phase of the beam has an overall variation of 2π from z= to , independent of the beam spot size at the waist and pulse duration.

© 2006 Optical Society of America

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  1. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
    [Crossref]
  2. R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
    [Crossref] [PubMed]
  3. A. E. Siegman, Lasers (University Science, 1986).
  4. C. Varin, M. Piché, and M. A. Porras, "Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space," Phys. Rev. E 71, 026603 (2005).
    [Crossref]
  5. T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
    [Crossref]
  6. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  7. T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
    [Crossref]
  8. M. A. Porras, "Pulse correction to monochromatic light-beam propagation," Opt. Lett. 26, 44-46 (2001).
    [Crossref]
  9. G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
    [Crossref]
  10. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
    [Crossref]
  11. E. H. Haselhoff, "Free-electron-laser model without the slowly-varying-envelope approximation," Phys. Rev. E 49, R47-R50 (1994).
    [Crossref]
  12. L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
    [Crossref]
  13. D. Lu, W. Hu, Y. Zheng, and Z. Yang, "Propagation of pulsed beam beyond the paraxial approximation in free space," Opt. Commun. 228, 217-223 (2003).
    [Crossref]
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    [Crossref] [PubMed]
  15. K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000).
    [Crossref] [PubMed]
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    [Crossref]
  17. M. A. Bandres, "Elegant Ince-Gaussian beams," Opt. Lett. 29, 1724-1726 (2004).
    [Crossref] [PubMed]
  18. U. T. Schwarz, M. A. Bandres, and J. C. Gutierrez-Vega, "Observation of Ince-Gaussian modes in stable resonators," Opt. Lett. 29, 1870-1872 (2004).
    [Crossref] [PubMed]
  19. L. W. Davis, "Vector electromagnetic modes of an optical resonator," Phys. Rev. A 30, 3092-3096 (1984).
    [Crossref]
  20. W. L. Erikson and S. Singh, "Polarization properties of Maxwell-Gaussian laser beams," Phys. Rev. E 49, 5778-5786 (1994).
    [Crossref]
  21. D. Lu, W. Hu, Z. Yang, and Y. Zheng, "Vectorial nature of nonparaxial ultrashort pulsed beam," J. Opt. A, Pure Appl. Opt. 5, 263-267 (2003).
    [Crossref]
  22. W. H. Carter, "Anomalies in the field of a gaussian beam near focus," Opt. Commun. 7, 211-218 (1973).
    [Crossref]
  23. F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Gouy phase shift for few-cycle laser pulses," Phys. Rev. Lett. 92, 113001 (2004).
    [Crossref] [PubMed]
  24. S. Feng and H. G. Winful, "Physical origin of the Gouy phase shift," Opt. Lett. 26, 485-487 (2001).
    [Crossref]
  25. H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Principle of high accuracy for the nonlinear theory of the acceleration of electrons in a vacuum by lasers at relativistic intensities," Laser Part. Beams 18, 135-144 (2000).
    [Crossref]
  26. N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun. 204, 7-15 (2002).
    [Crossref]
  27. M. O. Scully and M. S. Zubairy, "Simple laser accelerator: optics and particle dynamics," Phys. Rev. A 44, 2656-2663 (1991).
    [Crossref] [PubMed]

2005 (1)

C. Varin, M. Piché, and M. A. Porras, "Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space," Phys. Rev. E 71, 026603 (2005).
[Crossref]

2004 (4)

2003 (3)

D. Lu, W. Hu, Z. Yang, and Y. Zheng, "Vectorial nature of nonparaxial ultrashort pulsed beam," J. Opt. A, Pure Appl. Opt. 5, 263-267 (2003).
[Crossref]

D. Lu, W. Hu, Y. Zheng, and Z. Yang, "Propagation of pulsed beam beyond the paraxial approximation in free space," Opt. Commun. 228, 217-223 (2003).
[Crossref]

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

2002 (1)

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun. 204, 7-15 (2002).
[Crossref]

2001 (2)

2000 (4)

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[Crossref]

T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
[Crossref]

K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000).
[Crossref] [PubMed]

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Principle of high accuracy for the nonlinear theory of the acceleration of electrons in a vacuum by lasers at relativistic intensities," Laser Part. Beams 18, 135-144 (2000).
[Crossref]

1997 (1)

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[Crossref]

1994 (2)

E. H. Haselhoff, "Free-electron-laser model without the slowly-varying-envelope approximation," Phys. Rev. E 49, R47-R50 (1994).
[Crossref]

W. L. Erikson and S. Singh, "Polarization properties of Maxwell-Gaussian laser beams," Phys. Rev. E 49, 5778-5786 (1994).
[Crossref]

1991 (1)

M. O. Scully and M. S. Zubairy, "Simple laser accelerator: optics and particle dynamics," Phys. Rev. A 44, 2656-2663 (1991).
[Crossref] [PubMed]

1990 (1)

1984 (1)

L. W. Davis, "Vector electromagnetic modes of an optical resonator," Phys. Rev. A 30, 3092-3096 (1984).
[Crossref]

1983 (1)

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

1979 (1)

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

1975 (1)

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

1973 (1)

W. H. Carter, "Anomalies in the field of a gaussian beam near focus," Opt. Commun. 7, 211-218 (1973).
[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]

Baltuska, A.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Gouy phase shift for few-cycle laser pulses," Phys. Rev. Lett. 92, 113001 (2004).
[Crossref] [PubMed]

Bandres, M. A.

Brabec, T.

T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
[Crossref]

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[Crossref]

Brown, T. G.

Cao, N.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun. 204, 7-15 (2002).
[Crossref]

Carter, W. H.

W. H. Carter, "Anomalies in the field of a gaussian beam near focus," Opt. Commun. 7, 211-218 (1973).
[Crossref]

Castillo, R.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Principle of high accuracy for the nonlinear theory of the acceleration of electrons in a vacuum by lasers at relativistic intensities," Laser Part. Beams 18, 135-144 (2000).
[Crossref]

Davis, L. W.

L. W. Davis, "Vector electromagnetic modes of an optical resonator," Phys. Rev. A 30, 3092-3096 (1984).
[Crossref]

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

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[Crossref]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[Crossref]

Erikson, W. L.

W. L. Erikson and S. Singh, "Polarization properties of Maxwell-Gaussian laser beams," Phys. Rev. E 49, 5778-5786 (1994).
[Crossref]

Feng, S.

Ford, D. H.

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[Crossref]

Goulielmakis, E.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Gouy phase shift for few-cycle laser pulses," Phys. Rev. Lett. 92, 113001 (2004).
[Crossref] [PubMed]

Gutierrez-Vega, J. C.

Haselhoff, E. H.

E. H. Haselhoff, "Free-electron-laser model without the slowly-varying-envelope approximation," Phys. Rev. E 49, R47-R50 (1994).
[Crossref]

Ho, Y. K.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun. 204, 7-15 (2002).
[Crossref]

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Principle of high accuracy for the nonlinear theory of the acceleration of electrons in a vacuum by lasers at relativistic intensities," Laser Part. Beams 18, 135-144 (2000).
[Crossref]

Hoelss, M.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Principle of high accuracy for the nonlinear theory of the acceleration of electrons in a vacuum by lasers at relativistic intensities," Laser Part. Beams 18, 135-144 (2000).
[Crossref]

Hora, H.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Principle of high accuracy for the nonlinear theory of the acceleration of electrons in a vacuum by lasers at relativistic intensities," Laser Part. Beams 18, 135-144 (2000).
[Crossref]

Hu, W.

D. Lu, W. Hu, Z. Yang, and Y. Zheng, "Vectorial nature of nonparaxial ultrashort pulsed beam," J. Opt. A, Pure Appl. Opt. 5, 263-267 (2003).
[Crossref]

D. Lu, W. Hu, Y. Zheng, and Z. Yang, "Propagation of pulsed beam beyond the paraxial approximation in free space," Opt. Commun. 228, 217-223 (2003).
[Crossref]

Ito, H.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun. 204, 7-15 (2002).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Kimura, D.

Kong, Q.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun. 204, 7-15 (2002).
[Crossref]

Krausz, F.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Gouy phase shift for few-cycle laser pulses," Phys. Rev. Lett. 92, 113001 (2004).
[Crossref] [PubMed]

T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
[Crossref]

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[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, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[Crossref]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[Crossref]

Lezius, M.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Gouy phase shift for few-cycle laser pulses," Phys. Rev. Lett. 92, 113001 (2004).
[Crossref] [PubMed]

Lindner, F.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Gouy phase shift for few-cycle laser pulses," Phys. Rev. Lett. 92, 113001 (2004).
[Crossref] [PubMed]

Louisell, W. H.

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

Lu, D.

D. Lu, W. Hu, Y. Zheng, and Z. Yang, "Propagation of pulsed beam beyond the paraxial approximation in free space," Opt. Commun. 228, 217-223 (2003).
[Crossref]

D. Lu, W. Hu, Z. Yang, and Y. Zheng, "Vectorial nature of nonparaxial ultrashort pulsed beam," J. Opt. A, Pure Appl. Opt. 5, 263-267 (2003).
[Crossref]

McKnight, W. B.

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

Nishida, Y.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun. 204, 7-15 (2002).
[Crossref]

Osman, F.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Principle of high accuracy for the nonlinear theory of the acceleration of electrons in a vacuum by lasers at relativistic intensities," Laser Part. Beams 18, 135-144 (2000).
[Crossref]

Paulus, G. G.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Gouy phase shift for few-cycle laser pulses," Phys. Rev. Lett. 92, 113001 (2004).
[Crossref] [PubMed]

Piché, M.

C. Varin, M. Piché, and M. A. Porras, "Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space," Phys. Rev. E 71, 026603 (2005).
[Crossref]

Porras, M. A.

C. Varin, M. Piché, and M. A. Porras, "Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space," Phys. Rev. E 71, 026603 (2005).
[Crossref]

M. A. Porras, "Pulse correction to monochromatic light-beam propagation," Opt. Lett. 26, 44-46 (2001).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[Crossref]

Scheid, W.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Principle of high accuracy for the nonlinear theory of the acceleration of electrons in a vacuum by lasers at relativistic intensities," Laser Part. Beams 18, 135-144 (2000).
[Crossref]

Schwarz, U. T.

Scully, M. O.

M. O. Scully and M. S. Zubairy, "Simple laser accelerator: optics and particle dynamics," Phys. Rev. A 44, 2656-2663 (1991).
[Crossref] [PubMed]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Singh, S.

W. L. Erikson and S. Singh, "Polarization properties of Maxwell-Gaussian laser beams," Phys. Rev. E 49, 5778-5786 (1994).
[Crossref]

Tidwell, S.

Varin, C.

C. Varin, M. Piché, and M. A. Porras, "Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space," Phys. Rev. E 71, 026603 (2005).
[Crossref]

Walther, H.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Gouy phase shift for few-cycle laser pulses," Phys. Rev. Lett. 92, 113001 (2004).
[Crossref] [PubMed]

Wang, J. W.

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Principle of high accuracy for the nonlinear theory of the acceleration of electrons in a vacuum by lasers at relativistic intensities," Laser Part. Beams 18, 135-144 (2000).
[Crossref]

Wang, P. X.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun. 204, 7-15 (2002).
[Crossref]

Winful, H. G.

Yang, Z.

D. Lu, W. Hu, Z. Yang, and Y. Zheng, "Vectorial nature of nonparaxial ultrashort pulsed beam," J. Opt. A, Pure Appl. Opt. 5, 263-267 (2003).
[Crossref]

D. Lu, W. Hu, Y. Zheng, and Z. Yang, "Propagation of pulsed beam beyond the paraxial approximation in free space," Opt. Commun. 228, 217-223 (2003).
[Crossref]

Youngworth, K. S.

Yuan, X. Q.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun. 204, 7-15 (2002).
[Crossref]

Yugami, N.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun. 204, 7-15 (2002).
[Crossref]

Zheng, Y.

D. Lu, W. Hu, Z. Yang, and Y. Zheng, "Vectorial nature of nonparaxial ultrashort pulsed beam," J. Opt. A, Pure Appl. Opt. 5, 263-267 (2003).
[Crossref]

D. Lu, W. Hu, Y. Zheng, and Z. Yang, "Propagation of pulsed beam beyond the paraxial approximation in free space," Opt. Commun. 228, 217-223 (2003).
[Crossref]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, "Simple laser accelerator: optics and particle dynamics," Phys. Rev. A 44, 2656-2663 (1991).
[Crossref] [PubMed]

Appl. Opt. (1)

J. Opt. A, Pure Appl. Opt. (1)

D. Lu, W. Hu, Z. Yang, and Y. Zheng, "Vectorial nature of nonparaxial ultrashort pulsed beam," J. Opt. A, Pure Appl. Opt. 5, 263-267 (2003).
[Crossref]

J. Opt. Soc. Am. A (1)

Laser Part. Beams (1)

H. Hora, M. Hoelss, W. Scheid, J. W. Wang, Y. K. Ho, F. Osman, and R. Castillo, "Principle of high accuracy for the nonlinear theory of the acceleration of electrons in a vacuum by lasers at relativistic intensities," Laser Part. Beams 18, 135-144 (2000).
[Crossref]

Opt. Commun. (4)

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun. 204, 7-15 (2002).
[Crossref]

W. H. Carter, "Anomalies in the field of a gaussian beam near focus," Opt. Commun. 7, 211-218 (1973).
[Crossref]

D. Lu, W. Hu, Y. Zheng, and Z. Yang, "Propagation of pulsed beam beyond the paraxial approximation in free space," Opt. Commun. 228, 217-223 (2003).
[Crossref]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (5)

M. O. Scully and M. S. Zubairy, "Simple laser accelerator: optics and particle dynamics," Phys. Rev. A 44, 2656-2663 (1991).
[Crossref] [PubMed]

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

L. W. Davis, "Vector electromagnetic modes of an optical resonator," Phys. Rev. A 30, 3092-3096 (1984).
[Crossref]

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

Phys. Rev. E (3)

W. L. Erikson and S. Singh, "Polarization properties of Maxwell-Gaussian laser beams," Phys. Rev. E 49, 5778-5786 (1994).
[Crossref]

E. H. Haselhoff, "Free-electron-laser model without the slowly-varying-envelope approximation," Phys. Rev. E 49, R47-R50 (1994).
[Crossref]

C. Varin, M. Piché, and M. A. Porras, "Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space," Phys. Rev. E 71, 026603 (2005).
[Crossref]

Phys. Rev. Lett. (3)

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[Crossref]

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Gouy phase shift for few-cycle laser pulses," Phys. Rev. Lett. 92, 113001 (2004).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
[Crossref]

Other (2)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

A. E. Siegman, Lasers (University Science, 1986).

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of the focusing of a pulsed TM 01 beam. The waist of the entrance beam E i is located at z = 0 , the focusing optics with a given focal length f is placed at z = z l , and the output beam E f has its waist at z = z f .

Fig. 2
Fig. 2

Radial distribution of the zeroth-order (superscript 0) and first-order envelopes of the field components of a TM 01 beam at different positions along the z axis for t = 0 , as defined in Eqs. (15, 18). (a) w 0 f = λ 0 and (b) w 0 f = 3 λ 0 . Fields are not to scale; they have been normalized so that the maximum value of their respective zeroth-order expression is equal to 1 at the waist ( Δ z = 0 ) . It should be noticed that at t = 0 , the first-order expressions for the field components do not depend upon T f .

Fig. 3
Fig. 3

Fraction of the total beam power stored in the components of a continuous TM 01 beam at the waist ( z = z f ) .

Fig. 4
Fig. 4

Zeroth-, first-, and second-order intensities of a continuous TM 01 beam at z = z f for various beam-spot sizes (given in each graph). The intensity of each curve is normalized so that the total power is conserved and is given in units of I 0 = E 0 i 2 ( π 2 ) 3 2 exp ( 1 ) 2 η 0 .

Fig. 5
Fig. 5

Temporal distribution of the longitudinal electric field of ultrafast paraxial TM 01 beams ( w 0 f = 10 λ 0 ) for various positions along the z axis (at the beam center r = 0 ). (a) T f = 6 π ω 0 and (b) T f = 2 π ω 0 . The rectified fields have been normalized so that their envelope is equal to 1 at the center of the pulse ( t = 0 ) and at the waist ( Δ z = 0 ) . The complex envelope of the second-order field is given in Eq. (26).

Fig. 6
Fig. 6

Temporal distribution of the longitudinal electric field of ultrafast ( T f = 2 π ω 0 ) tightly focused TM 01 beams for various positions along the z axis (at the beam center r = 0 ). (a) w 0 f = λ 0 and (b) w 0 f = 0.5 λ 0 . See also the caption of Fig. 5.

Fig. 7
Fig. 7

Axial frequency shift of an ultrafast tightly focused TM 01 beam ( T f = 2 π ω 0 and w 0 f = λ 0 ). (a) At the waist ( Z = 0 ) , the solid curve is Eq. (35) and the dashed curve is relation (36). (b) Far from the waist ( Z 1 ) , the solid curve is Eq. (37a) and the dashed curve is relation (37b). It should be noticed that in the far field ( Δ z z R f ) the frequency shift does not depend upon w 0 f , the beam-spot size at the waist.

Fig. 8
Fig. 8

(a) Normalized amplitude and (b) axial Gouy phase of the longitudinal electric field of an ultrafast tightly focused TM 01 beam ( t = 0 ) .

Equations (75)

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2 E 1 c 2 t 2 E = 0 ,
E = Re [ E ̃ exp [ j ( ω 0 t k 0 z ) ] ] ,
2 E ̃ 2 j k 0 z E ̃ 2 j ω 0 c 2 t E ̃ + z 2 E ̃ 1 c 2 t 2 E ̃ = 0 ,
2 E ̃ 2 j k 0 z E ̃ + z 2 E ̃ 2 c z t E ̃ = 0 .
2 E ̃ 2 j k 0 z [ 1 Θ ] E ̃ = 0 ,
E ̃ = n = 0 Θ n Ψ ̃ ( n ) .
2 Ψ ̃ ( 0 ) 2 j k 0 z Ψ ̃ ( 0 ) = 0 ,
2 Ψ ̃ ( n ) 2 j k 0 z Ψ ̃ ( n ) + 2 j k 0 z Ψ ̃ ( n 1 ) = 0 .
Ψ ̃ ( n ) = z n 1 ( z n n ! z Ψ ̃ ( 0 ) ) ,
E ̃ = n = 0 j n ( 1 ω 0 t 1 2 k 0 z ) n z n 1 ( z n n ! z Ψ ̃ ( 0 ) ) .
E ̃ z = m = 0 n = 0 j m + n ( 1 ω 0 t 1 k 0 z ) m ( 1 ω 0 t 1 2 k 0 z ) n z n 1 ( z n n ! z E ̃ z ( 0 ) ) ,
B r = 0 ,
B z = 0 ,
z B θ = c 2 t E r ,
r 1 r ( r B θ ) = c 2 t E z .
B ̃ θ = p = 0 n = 0 j p + n ( 1 ω 0 t 1 k 0 z ) p ( 1 j ω 0 t ) ( 1 ω 0 t 1 2 k 0 z ) n z n 1 ( z n n ! z B ̃ θ ( 0 ) ) ,
E ̃ r = n = 0 ( j ω 0 ) n t n z n 1 ( z n n ! z E ̃ r ( 0 ) ) ,
E ̃ z = m = 0 n = 0 ( j ω 0 ) m + n t m + n z n 1 ( z n n ! z E ̃ z ( 0 ) ) ,
B ̃ θ = E ̃ r c ,
E ̃ r = n = 0 ( j 2 k 0 ) n z 2 n 1 ( z n n ! z E ̃ r ( 0 ) ) ,
E ̃ z = m = 0 n = 0 ( j k 0 ) m + n 2 n z 2 n 1 + m ( z n n ! z E ̃ z ( 0 ) ) ,
B ̃ θ = c 1 p = 0 n = 0 ( j k 0 ) p + n 2 n z 2 n 1 + p ( z n n ! z E ̃ r ( 0 ) ) .
E ̃ r i ( 0 ) = E 0 i exp ( 1 2 ) ( j z R i q ̃ i ) 2 2 r w 0 i exp ( j k 0 r 2 2 q ̃ i ) g i ( t ) ,
E ̃ z i ( 0 ) = 2 j E 0 i exp ( 1 2 ) ( j z R i q ̃ i ) 2 2 k 0 w 0 i ( 1 j k 0 r 2 2 q ̃ i ) exp ( j k 0 r 2 2 q ̃ i ) g i ( t ) ,
B ̃ θ i ( 0 ) = E ̃ r i ( 0 ) c ,
E ̃ r f ( 0 ) = E 0 f exp ( 1 2 ) ( j z R f q ̃ f ) 2 2 r w 0 f exp ( j k 0 r 2 2 q ̃ f ) g f ( t ) ,
E ̃ z f ( 0 ) = 2 j E 0 f exp ( 1 2 ) ( j z R f q ̃ f ) 2 2 k 0 w 0 f ( 1 j k 0 r 2 2 q ̃ f ) exp ( j k 0 r 2 2 q ̃ f ) g f ( t ) ,
B ̃ θ f ( 0 ) = E ̃ r f ( 0 ) c ,
w 0 i [ 1 + ( z l z R i ) 2 ] 1 2 = w 0 f [ 1 + ( z l z f z R f ) 2 ] 1 2 .
w 0 f w 0 i ( f z l ) ,
z R f z R i ( f z l ) 2 .
E ̃ r i E ̃ r i ( 0 ) + j ω 0 1 z t z E ̃ r i ( 0 ) ,
E ̃ r i ( 0 ) [ 1 + 2 j z t ω 0 T i 2 ( 2 q ̃ i j k 0 r 2 2 q ̃ i 2 ) ] ,
E ̃ z i E ̃ z i ( 0 ) + j ω 0 1 t E ̃ z i ( 0 ) + j ω 0 1 z t z E ̃ z i ( 0 ) ,
E ̃ z i ( 0 ) [ 1 2 j t ω 0 T i 2 + 2 j z t ω 0 T i 2 ( 1 j k 0 r 2 2 q ̃ i ) 1 ( 2 q ̃ i 2 j k 0 r 2 q ̃ i 2 k 0 2 r 4 4 q ̃ i 3 ) ] ,
B ̃ θ i E ̃ r i c ,
B ̃ θ i ( 0 ) [ 1 + 2 j z t ω 0 T i 2 ( 2 q ̃ i j k 0 r 2 2 q ̃ i 2 ) ] ,
E ̃ r f E ̃ r f ( 0 ) + j ω 0 1 Δ z t z E ̃ r f ( 0 ) j ( 2 k 0 ) 1 ( z E ̃ r f ( 0 ) + Δ z z 2 E ̃ r f ( 0 ) ) ,
E ̃ r f ( 0 ) [ 1 + ( j 2 k 0 + 2 j Δ z t ω 0 T f 2 ) ( 2 q ̃ f j k 0 r 2 2 q ̃ f 2 ) j Δ z 2 k 0 ( 6 q ̃ f 2 3 j k 0 r 2 q ̃ f 3 k 0 2 r 4 4 q ̃ f 4 ) ] ,
E ̃ z f E ̃ z f ( 0 ) + j ω 0 1 t E ̃ z f ( 0 ) j k 0 1 z E ̃ z f ( 0 ) + j ω 0 1 Δ z t z E ̃ z f ( 0 ) j ( 2 k 0 ) 1 ( z E ̃ z f ( 0 ) + Δ z z 2 E ̃ z f ( 0 ) ) ,
E ̃ z f ( 0 ) [ 1 2 j t ω 0 T f 2 + ( 1 j k 0 r 2 2 q ̃ f ) 1 ( 3 j 2 k 0 + 2 j Δ z t ω 0 T f 2 ) ( 2 q ̃ f 2 j k 0 r 2 q ̃ f 2 k 0 2 r 4 4 q ̃ f 3 ) j Δ z 2 k 0 ( 1 j k 0 r 2 2 q ̃ f ) 1 ( 6 q ̃ f 2 9 j k 0 r 2 q ̃ f 3 9 k 0 2 r 4 4 q ̃ f 4 + j k 0 3 r 6 8 q ̃ f 5 ) ] ,
B ̃ θ f B ̃ θ f ( 0 ) j k 0 1 z B ̃ θ f ( 0 ) + j ω 0 1 Δ z t z B ̃ θ f ( 0 ) j ( 2 k 0 ) 1 ( z B ̃ θ f ( 0 ) + Δ z z 2 B ̃ θ f ( 0 ) ) ,
B ̃ θ f ( 0 ) [ 1 + ( 3 j 2 k 0 + 2 j Δ z t ω 0 T f 2 ) ( 2 q ̃ f j k 0 r 2 2 q ̃ f 2 ) j Δ z 2 k 0 ( 6 q ̃ f 2 3 j k 0 r 2 q ̃ f 3 k 0 2 r 4 4 q ̃ f 4 ) ] .
E ̃ r f ( z = z f ) [ E ̃ r f ( 0 ) ] z = z f [ 1 + ( k 0 w 0 f ) 2 ( 2 r 2 w 0 f 2 ) + ( k 0 w 0 f ) 4 ( 18 32 r 2 w 0 f 2 + 3 r 4 w 0 f 4 ) ] ,
E ̃ z f ( z = z f ) [ E ̃ z f ( 0 ) ] z = z f [ 1 + 6 ( k 0 w 0 f ) 2 ( 1 r 2 w 0 f 2 ) + 33 2 ( k 0 w 0 f ) 4 ( 1 r 2 w 0 f 2 ) 1 ( 1 12 r 2 w 0 f 2 + 6 r 4 w 0 f 4 2 r 6 3 w 0 f 6 ) ] ,
B ̃ θ f ( z = z f ) c 1 [ E ̃ r f ( 0 ) ] z = z f [ 1 + 3 ( k 0 w 0 f ) 2 ( 2 r 2 w 0 f 2 ) + 11 3 ( k 0 w 0 f ) 4 ( 18 32 r 2 w 0 f 2 + 3 r 4 w 0 f 4 ) ] ,
[ E ̃ r f ( 0 ) ] z = z f = E 0 f exp ( 1 2 ) 2 w 0 f r exp ( r 2 w 0 f 2 ) g f ( t ) ,
[ E ̃ z f ( 0 ) ] z = z f = 2 j E 0 f exp ( 1 2 ) 2 k 0 w 0 f ( 1 r 2 w 0 f 2 ) exp ( r 2 w 0 f 2 ) g f ( t ) .
u i ϵ 0 4 ( E ̃ r i 2 + c 2 B ̃ θ i 2 ) = ϵ 0 2 E ̃ r i ( 0 ) 2 ,
u f = ϵ 0 4 ( E ̃ r f 2 + E ̃ z f 2 + c 2 B ̃ θ f 2 ) .
W i = E 0 i 2 2 η 0 ( π 2 ) 3 2 w 0 i 2 T i exp ( 1 ) ,
W f = E 0 f 2 2 η 0 ( π 2 ) 3 2 w 0 f 2 T f exp ( 1 ) [ 1 + 6 ( k 0 w 0 f ) 2 149 6 ( k 0 w 0 f ) 4 + 30 ( k 0 w 0 f ) 6 + 13 289 6 ( k 0 w 0 f ) 8 + 11 979 ( k 0 w 0 f ) 10 ] ,
E 0 f = E 0 i ( w 0 i w 0 f ) ( T i T f ) 1 2 [ 1 + 6 ( k 0 w 0 f ) 2 149 6 ( k 0 w 0 f ) 4 + 30 ( k 0 w 0 f ) 6 + 13 289 6 ( k 0 w 0 f ) 8 + 11 979 ( k 0 w 0 f ) 10 ] 1 2 ,
u e f = ϵ 0 4 E ̃ f 2 = ϵ 0 4 ( E ̃ r f 2 + E ̃ z f 2 ) .
I f = c u e f ,
= E ̃ r f 2 + E ̃ z f 2 4 η 0 .
E ̃ z f = E ̃ z f ( 0 ) [ 1 2 j t ω 0 T f 2 ( 1 2 Δ z q ̃ f ) + 3 j k 0 q ̃ f ( 1 Δ z q ̃ f ) 2 ω 0 2 T f 2 ( 1 2 t 2 T f 2 ) ( 4 Δ z q ̃ f 3 Δ z 2 q ̃ f 2 ) + 2 t k 0 ω 0 T f 2 q ̃ f ( 5 12 Δ z q ̃ f + 12 Δ z 2 q ̃ f 2 ) 3 2 k 0 2 q ̃ f 2 ( 7 20 Δ z q ̃ f + 10 Δ z 2 q ̃ f 2 ) ] ,
E ̃ z f ( 0 ) = 2 j E 0 f exp ( 1 2 ) 2 k 0 w 0 f ( j z R f q ̃ f ) 2 exp ( t 2 T f 2 ) ,
E z f = Re [ E ̃ z f exp ( j ω 0 t ) ] .
E ̃ z f = E ̃ z f ( 0 ) Ω ̃
= E ̃ z f ( 0 ) Ω ̃ exp ( 2 j Φ G + j Φ C j π 2 ) ,
E ̃ z f ( 0 ) = 2 E 0 f exp ( 1 2 ) 2 k 0 w 0 f ( w 0 f w f ( z ) ) 2 exp ( t 2 T f 2 ) ,
Φ G = tan 1 ( Δ z z R f ) ,
Re [ Ω ̃ ] = 1 + 4 t Z ω 0 T f 2 ( 1 + Z 2 ) + 3 ( 1 Z 2 ) k 0 z R f ( 1 + Z 2 ) 2 2 ( ω 0 T f ) 2 ( 1 2 t 2 T f 2 ) ( 7 Z 2 + Z 4 ) ( 1 + Z 2 ) 2 + 2 t k 0 z R f ω 0 T f 2 Z ( 17 26 Z 2 + 5 Z 4 ) ( 1 + Z 2 ) 3 + 3 2 ( k 0 z R f ) 2 ( 7 63 Z 2 + 13 Z 4 + 3 Z 6 ) ( 1 + Z 2 ) 4 ,
Im [ Ω ̃ ] = 2 t ω 0 T f 2 ( 1 Z 2 1 + Z 2 ) + 6 Z k 0 z R f ( 1 + Z 2 ) 2 + 2 ( ω 0 T f ) 2 ( 1 2 t 2 T f 2 ) ( 4 Z 2 Z 3 ) ( 1 + Z 2 ) 2 2 t k 0 z R f ω 0 T f 2 ( 5 + 26 Z 2 + 17 Z 4 ) ( 1 + Z 2 ) 3 + 3 ( k 0 z R f ) 2 Z ( 17 26 Z 2 + 3 Z 4 ) ( 1 + Z 2 ) 4 ,
Φ ( Z , t ) = ω 0 t + 2 Φ G ( Z ) + Φ C ( Z , t ) π 2 .
Φ ( 0 , t ) = ω 0 t + Φ C ( 0 , t ) π 2 ,
Φ ( 0 , t ) = ω 0 t π 2 ,
( ω 0 ω 0 ω 0 ) Z = 0 = 1 ω 0 t tan 1 ( 2 K t ω 0 T f 2 ) ,
( ω 0 ω 0 ω 0 ) Z = 0 2 K ( ω 0 T f ) 2 [ 1 1 3 ( 2 K t ω 0 T f 2 ) 2 ] .
( ω 0 ω 0 ω 0 ) Z = ± = 1 ω 0 t tan 1 ( 2 t ω 0 T f 2 1 2 ( ω 0 T f ) 2 + ( 2 t ω 0 T f 2 ) 2 ) ,
2 ( ω 0 T f ) 2 1 2 ( ω 0 T f ) 2 + ( 2 t ω 0 T f 2 ) 2 ,
Φ G C ( Z ) = 2 Φ G ( Z ) + Φ C ( Z , 0 ) .
Φ G C ( Z = 0 ) = 0 ,
Φ G C ( Z = ± ) = ± π .

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