Abstract

The wave equation describing an ultrashort, tightly focused laser pulse in vacuum is solved analytically. Plasma dispersive effects are also included. Based on exact short-pulse solutions, analytical expressions are obtained for the pulse-length evolution, the pulse centroid motion, and the group velocity. Approximate short-pulse solutions are obtained to arbitrary order in the parameter λ/2πL < 1, where λ is the pulse wavelength and L is the length of the pulse envelope. Comparisons are made to the solutions of the paraxial wave equation and to numerical solutions of the full wave equation. The exact analytical expression for the pulse group velocity vg, which is correctly determined from the motion of the pulse centroid, is in excellent agreement with the numerical solution. In vacuum, 1 − vg/c ≅ (λ/2πr0)2, where r0 is the laser spot size at focus. Estimates for the quantity 1 − vg/c, based on the paraxial wave equation, are found to be in error by a factor of 2.

© 1995 Optical Society of America

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  1. See, e.g., J. Wurtele, ed., Advanced Accelerator Concepts, AIP Conf. Proc. 279 (American Institute of Physics, New York, 1993).
  2. C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, "Ultra-gradient acceleration of injected electrons by laser-excited relativistic electron plasma waves," Phys. Rev. Lett. 70, 37 (1993); C. E. Clayton, M. J. Everett, A. Lal, D. Gordon, K. A. Marsh, and C. Joshi, "Acceleration and scattering of injected electrons in plasma beat wave accelerator experiments," Phys, Plasmas 1, 1753 (1994).
    [CrossRef] [PubMed]
  3. P. Sprangle, E. Esarey, J. Krall, and G. Joyce, "Propagation and guiding of intense laser pulses in plasmas," Phys. Rev. Lett. 69, 2200 (1992); E. Esarey, P. Sprangle, J. Krall, A. Ting, and G. Joyce, "Optically guided laser wakefield acceleration," Phys. Fluids B 5, 2690 (1993).
    [CrossRef] [PubMed]
  4. N. H. Burnett and P. B. Corkum, "Cold-plasma production for recombination extreme-ultraviolet lasers by optical-field-induced ionization," J. Opt. Soc. Am. 6, 1195 (1989).
    [CrossRef]
  5. D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
    [CrossRef]
  6. P. Sprangle, A. Ting, E. Esarey, and A. Fisher, "Tunable, short pulse hard x-rays from a compact laser synchrotron source," J. Appl. Phys. 72, 5032 (1992); E. Esarey, S. K. Ride, and P. Sprangle, "Nonlinear Thomson scattering of intense laser pulses from beams and plasmas," Phys. Rev. E 48, 3003 (1993).
    [CrossRef]
  7. K. J. Kim, S. Chattopadhyay, and C. V. Shank, "Generation of femtosecond x-rays by 90° Thomson scattering," Nucl. Instrum. Meth. A 341, 351 (1994).
    [CrossRef]
  8. M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
    [CrossRef]
  9. See, e.g., J. L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail, eds., Ultrafast Phenomena VIII (Springer-Verlag, Berlin, 1993).
    [CrossRef]
  10. D. Strickland and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 56, 216 (1985); G. Mourou and D. Umstadter, "Development and applications of compact high-intensity lasers," Phys. Fluids B 4, 2315 (1992); M. D. Perry and G. Mourou, "Terawatt to petawatt subpicosecond lasers," Science 264, 917 (1994).
    [CrossRef] [PubMed]
  11. P. W. Milonni and J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 14.
  12. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.
  13. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 5.
  14. I. P. Christov, "Propagation of femtosecond laser pulses," Opt. Commun. 53, 364 (1985).
    [CrossRef]
  15. Z. L. Horvath and Zs. Bor, "Behaviour of femtosecond pulses on the optical axis of a lens: analytical description," Opt. Commun. 108, 333 (1994); "Focusing of femtosecond pulses having Gaussian spatial distribution," Opt. Commun. 100, 6 (1993); Zs. Bor and Z. L. Horvath, "Distortion of femtosecond pulses in lenses: wave optical description," Opt. Commun. 94, 249 (1992).
    [CrossRef]
  16. C. D. Decker, "Group velocity of large amplitude electromagnetic waves in plasma," Phys. Rev. Lett. 72, 490 (1994).
    [CrossRef] [PubMed]
  17. W. B. Mori, "Group velocity of large amplitude electromagnetic waves in plasma," Phys. Rev. E 51, 1364 (1995).
    [CrossRef]

1995 (1)

W. B. Mori, "Group velocity of large amplitude electromagnetic waves in plasma," Phys. Rev. E 51, 1364 (1995).
[CrossRef]

1994 (5)

Z. L. Horvath and Zs. Bor, "Behaviour of femtosecond pulses on the optical axis of a lens: analytical description," Opt. Commun. 108, 333 (1994); "Focusing of femtosecond pulses having Gaussian spatial distribution," Opt. Commun. 100, 6 (1993); Zs. Bor and Z. L. Horvath, "Distortion of femtosecond pulses in lenses: wave optical description," Opt. Commun. 94, 249 (1992).
[CrossRef]

C. D. Decker, "Group velocity of large amplitude electromagnetic waves in plasma," Phys. Rev. Lett. 72, 490 (1994).
[CrossRef] [PubMed]

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

K. J. Kim, S. Chattopadhyay, and C. V. Shank, "Generation of femtosecond x-rays by 90° Thomson scattering," Nucl. Instrum. Meth. A 341, 351 (1994).
[CrossRef]

M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
[CrossRef]

1993 (1)

C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, "Ultra-gradient acceleration of injected electrons by laser-excited relativistic electron plasma waves," Phys. Rev. Lett. 70, 37 (1993); C. E. Clayton, M. J. Everett, A. Lal, D. Gordon, K. A. Marsh, and C. Joshi, "Acceleration and scattering of injected electrons in plasma beat wave accelerator experiments," Phys, Plasmas 1, 1753 (1994).
[CrossRef] [PubMed]

1992 (2)

P. Sprangle, E. Esarey, J. Krall, and G. Joyce, "Propagation and guiding of intense laser pulses in plasmas," Phys. Rev. Lett. 69, 2200 (1992); E. Esarey, P. Sprangle, J. Krall, A. Ting, and G. Joyce, "Optically guided laser wakefield acceleration," Phys. Fluids B 5, 2690 (1993).
[CrossRef] [PubMed]

P. Sprangle, A. Ting, E. Esarey, and A. Fisher, "Tunable, short pulse hard x-rays from a compact laser synchrotron source," J. Appl. Phys. 72, 5032 (1992); E. Esarey, S. K. Ride, and P. Sprangle, "Nonlinear Thomson scattering of intense laser pulses from beams and plasmas," Phys. Rev. E 48, 3003 (1993).
[CrossRef]

1989 (1)

N. H. Burnett and P. B. Corkum, "Cold-plasma production for recombination extreme-ultraviolet lasers by optical-field-induced ionization," J. Opt. Soc. Am. 6, 1195 (1989).
[CrossRef]

1985 (2)

D. Strickland and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 56, 216 (1985); G. Mourou and D. Umstadter, "Development and applications of compact high-intensity lasers," Phys. Fluids B 4, 2315 (1992); M. D. Perry and G. Mourou, "Terawatt to petawatt subpicosecond lasers," Science 264, 917 (1994).
[CrossRef] [PubMed]

I. P. Christov, "Propagation of femtosecond laser pulses," Opt. Commun. 53, 364 (1985).
[CrossRef]

Amendt, P.

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Bor, Zs.

Z. L. Horvath and Zs. Bor, "Behaviour of femtosecond pulses on the optical axis of a lens: analytical description," Opt. Commun. 108, 333 (1994); "Focusing of femtosecond pulses having Gaussian spatial distribution," Opt. Commun. 100, 6 (1993); Zs. Bor and Z. L. Horvath, "Distortion of femtosecond pulses in lenses: wave optical description," Opt. Commun. 94, 249 (1992).
[CrossRef]

Burnett, N. H.

N. H. Burnett and P. B. Corkum, "Cold-plasma production for recombination extreme-ultraviolet lasers by optical-field-induced ionization," J. Opt. Soc. Am. 6, 1195 (1989).
[CrossRef]

Campbell, E. M.

M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
[CrossRef]

Chattopadhyay, S.

K. J. Kim, S. Chattopadhyay, and C. V. Shank, "Generation of femtosecond x-rays by 90° Thomson scattering," Nucl. Instrum. Meth. A 341, 351 (1994).
[CrossRef]

Christov, I. P.

I. P. Christov, "Propagation of femtosecond laser pulses," Opt. Commun. 53, 364 (1985).
[CrossRef]

Clayton, C. E.

C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, "Ultra-gradient acceleration of injected electrons by laser-excited relativistic electron plasma waves," Phys. Rev. Lett. 70, 37 (1993); C. E. Clayton, M. J. Everett, A. Lal, D. Gordon, K. A. Marsh, and C. Joshi, "Acceleration and scattering of injected electrons in plasma beat wave accelerator experiments," Phys, Plasmas 1, 1753 (1994).
[CrossRef] [PubMed]

Corkum, P. B.

N. H. Burnett and P. B. Corkum, "Cold-plasma production for recombination extreme-ultraviolet lasers by optical-field-induced ionization," J. Opt. Soc. Am. 6, 1195 (1989).
[CrossRef]

DaSilva, L. B.

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Decker, C. D.

C. D. Decker, "Group velocity of large amplitude electromagnetic waves in plasma," Phys. Rev. Lett. 72, 490 (1994).
[CrossRef] [PubMed]

Donnelly, T. D.

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Dyson, A.

C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, "Ultra-gradient acceleration of injected electrons by laser-excited relativistic electron plasma waves," Phys. Rev. Lett. 70, 37 (1993); C. E. Clayton, M. J. Everett, A. Lal, D. Gordon, K. A. Marsh, and C. Joshi, "Acceleration and scattering of injected electrons in plasma beat wave accelerator experiments," Phys, Plasmas 1, 1753 (1994).
[CrossRef] [PubMed]

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 14.

Eder, D. C.

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Esarey, E.

P. Sprangle, E. Esarey, J. Krall, and G. Joyce, "Propagation and guiding of intense laser pulses in plasmas," Phys. Rev. Lett. 69, 2200 (1992); E. Esarey, P. Sprangle, J. Krall, A. Ting, and G. Joyce, "Optically guided laser wakefield acceleration," Phys. Fluids B 5, 2690 (1993).
[CrossRef] [PubMed]

P. Sprangle, A. Ting, E. Esarey, and A. Fisher, "Tunable, short pulse hard x-rays from a compact laser synchrotron source," J. Appl. Phys. 72, 5032 (1992); E. Esarey, S. K. Ride, and P. Sprangle, "Nonlinear Thomson scattering of intense laser pulses from beams and plasmas," Phys. Rev. E 48, 3003 (1993).
[CrossRef]

Everett, M.

C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, "Ultra-gradient acceleration of injected electrons by laser-excited relativistic electron plasma waves," Phys. Rev. Lett. 70, 37 (1993); C. E. Clayton, M. J. Everett, A. Lal, D. Gordon, K. A. Marsh, and C. Joshi, "Acceleration and scattering of injected electrons in plasma beat wave accelerator experiments," Phys, Plasmas 1, 1753 (1994).
[CrossRef] [PubMed]

Falcone, R. W.

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Fisher, A.

P. Sprangle, A. Ting, E. Esarey, and A. Fisher, "Tunable, short pulse hard x-rays from a compact laser synchrotron source," J. Appl. Phys. 72, 5032 (1992); E. Esarey, S. K. Ride, and P. Sprangle, "Nonlinear Thomson scattering of intense laser pulses from beams and plasmas," Phys. Rev. E 48, 3003 (1993).
[CrossRef]

Glinsky, M. E.

M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
[CrossRef]

Hammer, J.

M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
[CrossRef]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 5.

Horvath, Z. L.

Z. L. Horvath and Zs. Bor, "Behaviour of femtosecond pulses on the optical axis of a lens: analytical description," Opt. Commun. 108, 333 (1994); "Focusing of femtosecond pulses having Gaussian spatial distribution," Opt. Commun. 100, 6 (1993); Zs. Bor and Z. L. Horvath, "Distortion of femtosecond pulses in lenses: wave optical description," Opt. Commun. 94, 249 (1992).
[CrossRef]

Joshi, C.

C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, "Ultra-gradient acceleration of injected electrons by laser-excited relativistic electron plasma waves," Phys. Rev. Lett. 70, 37 (1993); C. E. Clayton, M. J. Everett, A. Lal, D. Gordon, K. A. Marsh, and C. Joshi, "Acceleration and scattering of injected electrons in plasma beat wave accelerator experiments," Phys, Plasmas 1, 1753 (1994).
[CrossRef] [PubMed]

Joyce, G.

P. Sprangle, E. Esarey, J. Krall, and G. Joyce, "Propagation and guiding of intense laser pulses in plasmas," Phys. Rev. Lett. 69, 2200 (1992); E. Esarey, P. Sprangle, J. Krall, A. Ting, and G. Joyce, "Optically guided laser wakefield acceleration," Phys. Fluids B 5, 2690 (1993).
[CrossRef] [PubMed]

Kim, K. J.

K. J. Kim, S. Chattopadhyay, and C. V. Shank, "Generation of femtosecond x-rays by 90° Thomson scattering," Nucl. Instrum. Meth. A 341, 351 (1994).
[CrossRef]

Krall, J.

P. Sprangle, E. Esarey, J. Krall, and G. Joyce, "Propagation and guiding of intense laser pulses in plasmas," Phys. Rev. Lett. 69, 2200 (1992); E. Esarey, P. Sprangle, J. Krall, A. Ting, and G. Joyce, "Optically guided laser wakefield acceleration," Phys. Fluids B 5, 2690 (1993).
[CrossRef] [PubMed]

Kruer, W. L.

M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
[CrossRef]

Lal, A.

C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, "Ultra-gradient acceleration of injected electrons by laser-excited relativistic electron plasma waves," Phys. Rev. Lett. 70, 37 (1993); C. E. Clayton, M. J. Everett, A. Lal, D. Gordon, K. A. Marsh, and C. Joshi, "Acceleration and scattering of injected electrons in plasma beat wave accelerator experiments," Phys, Plasmas 1, 1753 (1994).
[CrossRef] [PubMed]

Leemans, W. P.

C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, "Ultra-gradient acceleration of injected electrons by laser-excited relativistic electron plasma waves," Phys. Rev. Lett. 70, 37 (1993); C. E. Clayton, M. J. Everett, A. Lal, D. Gordon, K. A. Marsh, and C. Joshi, "Acceleration and scattering of injected electrons in plasma beat wave accelerator experiments," Phys, Plasmas 1, 1753 (1994).
[CrossRef] [PubMed]

London, R. A.

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

MacGowan, B. J.

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Marsh, K. A.

C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, "Ultra-gradient acceleration of injected electrons by laser-excited relativistic electron plasma waves," Phys. Rev. Lett. 70, 37 (1993); C. E. Clayton, M. J. Everett, A. Lal, D. Gordon, K. A. Marsh, and C. Joshi, "Acceleration and scattering of injected electrons in plasma beat wave accelerator experiments," Phys, Plasmas 1, 1753 (1994).
[CrossRef] [PubMed]

Mason, R. J.

M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
[CrossRef]

Matthews, D. L.

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 14.

Mori, W. B.

W. B. Mori, "Group velocity of large amplitude electromagnetic waves in plasma," Phys. Rev. E 51, 1364 (1995).
[CrossRef]

Mourou, G.

D. Strickland and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 56, 216 (1985); G. Mourou and D. Umstadter, "Development and applications of compact high-intensity lasers," Phys. Fluids B 4, 2315 (1992); M. D. Perry and G. Mourou, "Terawatt to petawatt subpicosecond lasers," Science 264, 917 (1994).
[CrossRef] [PubMed]

Penetrante, B. M.

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Perry, M. D.

M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
[CrossRef]

Rosen, M. D.

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Shank, C. V.

K. J. Kim, S. Chattopadhyay, and C. V. Shank, "Generation of femtosecond x-rays by 90° Thomson scattering," Nucl. Instrum. Meth. A 341, 351 (1994).
[CrossRef]

Sprangle, P.

P. Sprangle, E. Esarey, J. Krall, and G. Joyce, "Propagation and guiding of intense laser pulses in plasmas," Phys. Rev. Lett. 69, 2200 (1992); E. Esarey, P. Sprangle, J. Krall, A. Ting, and G. Joyce, "Optically guided laser wakefield acceleration," Phys. Fluids B 5, 2690 (1993).
[CrossRef] [PubMed]

P. Sprangle, A. Ting, E. Esarey, and A. Fisher, "Tunable, short pulse hard x-rays from a compact laser synchrotron source," J. Appl. Phys. 72, 5032 (1992); E. Esarey, S. K. Ride, and P. Sprangle, "Nonlinear Thomson scattering of intense laser pulses from beams and plasmas," Phys. Rev. E 48, 3003 (1993).
[CrossRef]

Strickland, D.

D. Strickland and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 56, 216 (1985); G. Mourou and D. Umstadter, "Development and applications of compact high-intensity lasers," Phys. Fluids B 4, 2315 (1992); M. D. Perry and G. Mourou, "Terawatt to petawatt subpicosecond lasers," Science 264, 917 (1994).
[CrossRef] [PubMed]

Strobel, G. L.

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Tabak, M.

M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
[CrossRef]

Ting, A.

P. Sprangle, A. Ting, E. Esarey, and A. Fisher, "Tunable, short pulse hard x-rays from a compact laser synchrotron source," J. Appl. Phys. 72, 5032 (1992); E. Esarey, S. K. Ride, and P. Sprangle, "Nonlinear Thomson scattering of intense laser pulses from beams and plasmas," Phys. Rev. E 48, 3003 (1993).
[CrossRef]

Wilks, S. C.

M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
[CrossRef]

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Williams, R.

C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, "Ultra-gradient acceleration of injected electrons by laser-excited relativistic electron plasma waves," Phys. Rev. Lett. 70, 37 (1993); C. E. Clayton, M. J. Everett, A. Lal, D. Gordon, K. A. Marsh, and C. Joshi, "Acceleration and scattering of injected electrons in plasma beat wave accelerator experiments," Phys, Plasmas 1, 1753 (1994).
[CrossRef] [PubMed]

Woodworth, J.

M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.

J. Appl. Phys. (1)

P. Sprangle, A. Ting, E. Esarey, and A. Fisher, "Tunable, short pulse hard x-rays from a compact laser synchrotron source," J. Appl. Phys. 72, 5032 (1992); E. Esarey, S. K. Ride, and P. Sprangle, "Nonlinear Thomson scattering of intense laser pulses from beams and plasmas," Phys. Rev. E 48, 3003 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

N. H. Burnett and P. B. Corkum, "Cold-plasma production for recombination extreme-ultraviolet lasers by optical-field-induced ionization," J. Opt. Soc. Am. 6, 1195 (1989).
[CrossRef]

Nucl. Instrum. Meth. A (1)

K. J. Kim, S. Chattopadhyay, and C. V. Shank, "Generation of femtosecond x-rays by 90° Thomson scattering," Nucl. Instrum. Meth. A 341, 351 (1994).
[CrossRef]

Opt. Commun. (3)

D. Strickland and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 56, 216 (1985); G. Mourou and D. Umstadter, "Development and applications of compact high-intensity lasers," Phys. Fluids B 4, 2315 (1992); M. D. Perry and G. Mourou, "Terawatt to petawatt subpicosecond lasers," Science 264, 917 (1994).
[CrossRef] [PubMed]

I. P. Christov, "Propagation of femtosecond laser pulses," Opt. Commun. 53, 364 (1985).
[CrossRef]

Z. L. Horvath and Zs. Bor, "Behaviour of femtosecond pulses on the optical axis of a lens: analytical description," Opt. Commun. 108, 333 (1994); "Focusing of femtosecond pulses having Gaussian spatial distribution," Opt. Commun. 100, 6 (1993); Zs. Bor and Z. L. Horvath, "Distortion of femtosecond pulses in lenses: wave optical description," Opt. Commun. 94, 249 (1992).
[CrossRef]

Phys. Plasmas (2)

M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, "Ignition and high gain with ultrapowerful lasers," Phys. Plasmas 1, 1626 (1994).
[CrossRef]

D. C. Eder, P. Amendt, L. B. DaSilva, R. A. London, B. J. MacGowan, D. L. Matthews, B. M. Penetrante, M. D. Rosen, S. C. Wilks, T. D. Donnelly, R. W. Falcone, and G. L. Strobel, "Tabletop x-ray lasers," Phys. Plasmas 1, 1744 (1994).
[CrossRef]

Phys. Rev. E (1)

W. B. Mori, "Group velocity of large amplitude electromagnetic waves in plasma," Phys. Rev. E 51, 1364 (1995).
[CrossRef]

Phys. Rev. Lett. (3)

C. D. Decker, "Group velocity of large amplitude electromagnetic waves in plasma," Phys. Rev. Lett. 72, 490 (1994).
[CrossRef] [PubMed]

C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, "Ultra-gradient acceleration of injected electrons by laser-excited relativistic electron plasma waves," Phys. Rev. Lett. 70, 37 (1993); C. E. Clayton, M. J. Everett, A. Lal, D. Gordon, K. A. Marsh, and C. Joshi, "Acceleration and scattering of injected electrons in plasma beat wave accelerator experiments," Phys, Plasmas 1, 1753 (1994).
[CrossRef] [PubMed]

P. Sprangle, E. Esarey, J. Krall, and G. Joyce, "Propagation and guiding of intense laser pulses in plasmas," Phys. Rev. Lett. 69, 2200 (1992); E. Esarey, P. Sprangle, J. Krall, A. Ting, and G. Joyce, "Optically guided laser wakefield acceleration," Phys. Fluids B 5, 2690 (1993).
[CrossRef] [PubMed]

Other (5)

See, e.g., J. L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail, eds., Ultrafast Phenomena VIII (Springer-Verlag, Berlin, 1993).
[CrossRef]

See, e.g., J. Wurtele, ed., Advanced Accelerator Concepts, AIP Conf. Proc. 279 (American Institute of Physics, New York, 1993).

P. W. Milonni and J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 14.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 5.

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Figures (3)

Fig. 1
Fig. 1

Second-order solution: Real part of a = âexp(ik0ζ) plotted versus r and ζ = zct at (a) t = 0 and (b) t = 2ZR/c for L/λ = r0/λ = 2 and λ = 2.5 μm. This shows the evolution of the normalized vector potential of the laser field for a pulse of 15-fs duration with a central wavelength of 2.5 μm. (a) The initial laser field at t = 0 at the focal point where the minimum spot size is r0 = 5 μm. (b) The laser field after propagating a distance of two Rayleigh lengths from the focal point, ct = 2ZR ≃ 63 μm. Notice in (b) the curvature of the wave fronts as the pulse diffracts.

Fig. 2
Fig. 2

Percentage difference 100[a(n)a(num)]/Max[a(n)] between the numerical solution to the wave equation a(num) and the nth-order analytical solution a(n) plotted versus r and ζ = zct for (a) the first-order solution a(1) and (b) the second-order solution a(2). The parameters are L/λ = 2 and λ = 2.5 μm, r0/λ = and the comparisons are done after the wave has propagated a distance η = (z + ct)/2 = ZR from the focal point.

Fig. 3
Fig. 3

Quantity 1 − v/c versus propagation distance η = (z + ct)/2 as given by analysis (solid curve) and by numerical solution of the wave equation (dashed curve). Comparisons are done for two cases: (a) v = vg, where vg is the velocity of the pulse centroid [see Eq. (19)], and (b) v = ve, where ve = ve(r = 0, η) is the local envelope velocity [see relation (32)] evaluated along the axis r = 0. Here L/λ = r0/λ = 2, λ = 2.5 μm, and η = 0 is the focal point.

Equations (58)

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( 2 - 1 c 2 2 t 2 ) a = k p 2 a ,
( 2 + 2 i k 0 z + 2 z 2 - k p 2 ) a ^ = 0.
( 2 + 2 i k 0 z - k p 2 ) a ^ = 0.
ψ p ( r , k 0 , z ) = - 1 2 ln ( 1 + α z 2 ) - r 2 / r 0 1 + i α z - i tan - 1 α z - i 4 k p 2 r 0 2 α z .
a p ( r , z , t ) = ( a 0 r 0 / r s ) exp ( - r 2 / r s 2 + i ϕ p ) ,
ϕ p ( r , z , t ) = k 0 ( z - c t ) + α z r 2 / r s 2 - tan - 1 α z - k p 2 r 0 2 α z / 4 ,
k z = k 0 - k p 2 2 k 0 - 2 k 0 r s 2 [ 1 - r 2 r s 2 ( 1 - α z ) ] .
a ( r , z , t ) = f ( z - v e t ) a p ( r , z , t ) .
v e ( r , z ) c 1 - k p 2 2 k 0 2 - 2 k 0 2 r 0 2 × [ ( 1 - α z 2 ) ( 1 + α z 2 ) 1 - r 2 r 0 2 ( 1 - 6 α z 2 + α z 4 ) ( 1 + α z 2 ) 3 ] .
( 2 + 2 2 η ζ - k p 2 ) a = 0.
[ 2 + 2 ( i k 0 + ζ ) η - k p 2 ] a ^ ( r , ζ , η ) = 0.
[ 2 + 2 i ( k 0 + k ) η - k p 2 ] a ^ k = 0 ,
a ^ k ( r , k , η ) = 1 2 π - d ζ exp ( - i k ζ ) a ^ ( r , ζ , η )
a ^ k ( r , k , η ) = f k a ^ p k ( r , k 0 + k , η ) = a 0 f k exp ψ k ( r , k 0 + k , η ) ,
ψ k ( r , k 0 + k , η ) = - 1 2 ln ( 1 + α k 2 ) - r 2 / r 0 2 1 + i α k - i tan - 1 α k - i 4 k p 2 r 0 2 α k ,
α k = η / Z R k ,
Z R k = ( k 0 + k ) r 0 2 / 2.
f = ( 1 + 2 i ζ k 0 L 2 ) exp ( - ζ 2 L 2 ) ,
f k = ( 1 + k k 0 ) L 2 exp ( - k 2 L 2 4 ) .
I = I 0 [ 1 + 4 ( z - c t ) 2 k 0 2 L 4 ] exp [ - 2 ( z - c t ) 2 L 2 - 2 r 2 r 0 2 ] ,
W 0 d r r - d ζ | ( i k 0 + ζ ) a ^ | 2
E 2 + B 2 ~ a / ζ 2 + a / η 2 / 4 a / ζ 2 ( i k 0 + / ζ ) a ^ 2 ,
W = a 0 2 r 0 2 4 - d k ( k 0 + k ) 2 f k 2 ,
W = 2 π 8 L a 0 2 r 0 2 k 0 2 ( 1 + 6 k 0 2 L 2 + 3 k 0 4 L 4 ) ,
ζ ¯ ( η ) ζ = 0 d r r - d ζ ζ b 2 / W .
ζ ¯ = - 1 W 0 d r r - d k ( k 0 + k ) 2 a ^ k 2 k Im ( ψ k ) = - η a 0 2 4 W ( 1 + k p 2 r 0 2 2 ) - d k f k 2 .
ζ ¯ = - n ( 1 + k p 2 r 0 2 / 2 ) ( 1 + 1 / k 0 2 L 2 ) k 0 2 r 0 2 ( 1 + 6 / k 0 2 L 2 + 3 / k 0 4 L 4 ) - η .
ζ ¯ = - η ( 1 + k p 2 r 0 2 / 2 ) k 0 2 r 0 2 ( 1 + 2 / k 0 2 L 2 ) .
v g / c = ( 1 - / 2 ) / ( 1 + / 2 ) ,
v g / c 1 - 1 - 1 k 0 2 r 0 2 ( 1 + k p 2 r 0 2 2 - 5 k 0 2 L 2 ) .
ζ 2 = 0 d r r - d ζ ζ 2 b 2 / W .
ζ 2 = 0 d r r - d k ( k 0 + k ) 2 a p k 2 W [ ( f k k ) 2 + f k 2 | ψ k k | 2 + ( f k 2 k ) Re ( ψ k k ) ] = a 0 2 r 0 2 4 W - d k [ ( k 0 + k ) 2 ( f k k ) 2 + 2 f k 2 η 2 ( k 0 + k ) 2 r 0 × ( 1 + k p 2 r 0 2 2 + k p 4 r 0 4 8 ) ] .
ζ 2 = L 2 4 [ 1 + 10 k 0 2 L 2 + 7 k 0 4 L 4 + 2 η 2 / Z R 2 k 0 2 L 2 × ( 1 + k p 2 r 0 2 2 + k p 4 r 0 4 8 ) ] × ( 1 + 6 k 0 2 L 2 + 3 k 0 4 L 4 ) - 1 .
L p 2 L 2 4 [ 1 + 4 k 0 2 L 2 + η 2 / Z R 2 k 0 2 L 2 ( 1 - 2 k 0 2 L 2 ) ] .
a ^ ( r , ζ , η ) = 1 2 π - d k exp ( i k ζ ) f k a 0 exp ( ψ k ) .
ψ k ( r , k 0 + k , η ) = ψ ( k 0 ) + ψ k 0 k + 1 2 2 ψ k 0 2 k 2 + ,
ψ ( k 0 ) = - 1 2 ln ( 1 + α ) - r 2 / r 0 2 1 + i α - i tan - 1 α - i 4 k p 2 r 0 2 α ,
ψ k 0 = i α k 0 [ 1 1 + i α - r 2 / r 0 2 ( 1 + i α ) + k p 2 r 0 2 4 ] ,
2 ψ k 0 2 = - i α k 0 2 [ 2 + i α ( 1 + i α ) 2 - 2 r 2 / r 0 2 ( 1 + i α ) 3 + k p 2 r 0 2 2 ] ,
a ^ ( 0 ) ( r , ζ , η ) = a 0 f ( ζ ) exp ( ψ ) .
a ( 0 ) = a 0 r 0 r s ( 1 + 2 i ζ k 0 L 2 ) exp [ - r 2 r s 2 - ζ 2 L 2 + i ϕ ( 0 ) ] ,
ϕ ( 0 ) = k 0 ζ + α r 2 / r s 2 - tan - 1 α - k p 2 r 0 2 α / 4 ,
ω ( 0 ) = c k 0 { 1 + k p 2 4 k 0 2 + 1 k 0 2 r s 2 [ 1 - r 2 r s 2 ( 1 - α 2 ) ] } ,
k z ( 0 ) = k 0 { 1 - k p 2 4 k 0 2 - 1 k 0 2 r s 2 [ 1 - r 2 r s 2 ( 1 - α 2 ) ] } .
[ ω ( 0 ) ] 2 - c 2 [ k z ( 0 ) ] 2 = c 2 k p 2 + 4 r s 2 [ 1 - r 2 r s 2 ( 1 - α 2 ) ] .
a ^ ( 1 ) ( r , ζ , η ) = 1 2 π - d k exp [ i k ( ζ - i ψ ) ] f k a 0 exp ( ψ ) = a 0 f ( ζ - i ψ ) exp ( ψ ) .
v e ( r , η ) c 1 - 1 = 1 - 2 k 0 2 r 0 2 × [ ( 1 - α 2 ) ( 1 + α 2 ) - r 2 ( 1 - 6 α 2 + α 4 ) r 0 2 ( 1 + α 2 ) 3 + k p 2 r 0 2 4 ] ,
ζ ζ = d ζ ζ f 2 d ζ f 2 = - ψ i ( r , α ) ,
ζ ¯ = d r r d ζ ζ f 2 exp ( 2 ψ r ) d r r d ζ f 2 exp ( 2 ψ r ) = - η k 0 2 r 0 2 ( 1 + k p 2 r 0 2 2 ) = - 2 η .
v g / c = ( 1 - 2 / 2 ) ( 1 + 2 / 2 ) 1 - 1 k 0 2 r 0 2 ( 1 + k p 2 r 0 2 2 ) ,
δ ω ( 1 ) c k 0 = 2 k 0 2 L 2 { α 2 ( 1 + α 2 ) ( 1 - 2 r 2 r s 2 ) - ζ α Z R ( 1 + α 2 ) 2 [ 1 - 2 r 2 r s 2 ( 1 - α 2 ) ] } .
a ^ ( 2 ) ( r , ζ , η ) = a 0 1 - 2 ψ / L 2 [ 1 + 2 i ( ζ - i ψ ) k 0 L 2 ( 1 - 2 ψ / L 2 ) ] × exp [ ψ - ( ζ - i ψ ) 2 / L 2 ( 1 - 2 ψ / L 2 ) ] .
a ^ ( n ) ( r , ζ , η ) = 1 2 π - d k ^ ( 1 + S + 1 2 S 2 + + 1 n ! S n ) × exp [ ψ + i ( ζ - i ψ ) k ] ,
S = 1 2 2 ψ k 0 2 k 2 + + 1 n ! n ψ k 0 n k n .
a ^ ( 4 ) ( r , ζ , η ) = a 0 e ψ { 1 - ψ 2 2 ζ 2 + i ψ 6 3 ζ 3 + [ ψ ( i v ) 24 - ( ψ ) 2 8 ] 4 ζ 4 } f ( ζ - i ψ ) .
a ^ ( r , ζ , 0 ) = a 0 ( 1 + 2 i ζ k 0 L 2 ) exp [ - ( r 2 r 0 2 + ζ 2 L 2 ) ] .
1 - v g c 1 k 0 2 r 0 2 ( 1 + k p 2 r 0 2 2 - 5 k 0 2 L 2 ) .
γ g 2 ( π r 0 / λ ) ( 1 + k p 2 r 0 2 / 2 ) - 1 / 2 .

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