Abstract

Propagation into a homogeneous plasma of a laser beam at irradiances higher than 1/500 of the relativistic threshold can result in self-focusing due to the highly sensitive relativistic dependence of the optical constants on laser irradiance. Electron densities slightly less than the relativistic-cutoff densities are required. Simultaneously with the self-focusing, it is also possible to achieve a dielectric increase (swelling) of laser energy density in the plasma that could reach 1/3 of its maximum value. In prepulsed plasmas, generated by Nd-glass-laser pulses of 3 × 1016 W/cm2, relativistic diffraction-limited self-focusing can generate relativistic electron-oscillation energies and hence pair production.

© 1975 Optical Society of America

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  1. A. G. Litvak, Zh. Eksp. Teor. Fiz. 57, 629 (1969) [Sov. Phys. JETP 30, 344 (1970)].
  2. M. S. Sodha, Repts. Prog. Phys. 37, 621 (1974).
  3. H. Hora, Z. Physik 226, 156 (1969); a derivation based on electron motion was given by Claire Ellen Max, J. Arons, and A. B. Langdon, Phys. Rev. Lett. 33, 209 (1974).
    [CrossRef]
  4. The nonlinear force is the electrodynamic part of the gradient of the momentum-flux-density of a more general form than used by Landau and Lifshitz (Ref. 5). The generalization with respect to dispersion was confirmed by consistency with the derivation from other models, e.g., the two-fluid model (Ref. 6). Because Landau and Lifshitz (Ref. 5) used the expression “ponderomotive force” for gas dynamic (thermokinetic) and other forces, the expression “nonlinear force” for its electrodynamic part may be preferred with respect to its quadratic terms. Shearer and Eddleman (Ref. 7) specialized the general derivation to reproduce the electric term (without the magnetic term) of the nonlinear force, as used in microwave theory (Ref. 8), from which I needed further specializations to derive the general expression (Ref. 6).
  5. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1966), p. 242.
  6. H. Hora, Phys. Fluids 12, 182 (1969).
    [CrossRef]
  7. J. W. Shearer and J. L. Eddleman, Phys. Fluids 16, 1522 (1973).
    [CrossRef]
  8. H. Motz and C. J. H. Watson, in Advances in Electronics, Vol. 23, edited by L. Marton (Academic, New York, 1967), p. 153.
  9. A. Schlüter, Plasma Phys. 10, 471 (1968).
  10. H. Hora, D. Pfirsch, and A. Schlüter, Z. Naturforschung 22a, 278 (1967).
  11. F. F. Chen, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3A, p. 291.
    [CrossRef]
  12. V. V. Korobkin and A. J. Alcock, Phys. Rev. Lett. 21, 1433 (1968).
    [CrossRef]
  13. P. Kaw, Appl. Phys. Lett. 15, 16 (1969).
    [CrossRef]
  14. A. J. Palmer, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1972), Vol. 2, p. 367.
  15. R. L. Dewar, Phys. Fluids 16, 431 (1973).
    [CrossRef]
  16. M. S. Sodha, A. K. Chakravarti, W. P. Phadke, G. D. Gautama, and I. Rattan, Appl. Phys. Lett. 22, 121 (1972).
    [CrossRef]
  17. T. W. B. Kibble, Phys. Rev. Lett.16, 1054;Phys. Res. 150, 1060 (1966).
  18. A. Schlüter, Z. Naturforschung 5a, 72 (1950).
  19. R. Lüst, Z. Astrophysik 57, 67 (1953).
  20. H. Hora, Laser Plasmas and Nuclear Energy (Plenum, New York, 1975), p. 47.
  21. H. Hora, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974) Vol. 3B, p. 803.
  22. L. Spitzer, Physics of Fully Ionized Gases (Wiley Interscience, New York, 1962).
  23. H. Hora, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1970), Vol. 1, p. 427.
  24. N. G. Basov and O. N. Krokhin, in Third International Quantum Electronic Conference, edited by N. Bloembergen (Academic, New York, 1963).
  25. T. W. Johnston and J. M. Dawson, Phys. Fluids 16, 722 (1973).
    [CrossRef]
  26. J. M. Dawson and C. Oberman, Phys. Fluids 5, 517 (1962).
    [CrossRef]
  27. H. Hora and H. Wilhelm, Nucl. Fusion 10, 111 (1970).
    [CrossRef]
  28. F. Lindl and P. Kaw, Phys. Fluids 14, 371 (1972).
    [CrossRef]
  29. R. B. White and F. F. Chen, Plasma Phys. 16, 587 (1974).
    [CrossRef]
  30. H. Hora, Atomkernenergie 24, 187 (1974).
  31. H. Hora, Opto-Electronics 5, 491 (1973).
    [CrossRef]
  32. J. L. Hughes, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 849.
    [CrossRef]
  33. J. W. Shearer, J. Garrison, J. Wong, and J. E. Swain, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 803.
    [CrossRef]
  34. J. H. Nuckolls, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 397.
  35. K. Brueckner, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 427.
    [CrossRef]

1974 (3)

M. S. Sodha, Repts. Prog. Phys. 37, 621 (1974).

R. B. White and F. F. Chen, Plasma Phys. 16, 587 (1974).
[CrossRef]

H. Hora, Atomkernenergie 24, 187 (1974).

1973 (4)

H. Hora, Opto-Electronics 5, 491 (1973).
[CrossRef]

T. W. Johnston and J. M. Dawson, Phys. Fluids 16, 722 (1973).
[CrossRef]

J. W. Shearer and J. L. Eddleman, Phys. Fluids 16, 1522 (1973).
[CrossRef]

R. L. Dewar, Phys. Fluids 16, 431 (1973).
[CrossRef]

1972 (2)

M. S. Sodha, A. K. Chakravarti, W. P. Phadke, G. D. Gautama, and I. Rattan, Appl. Phys. Lett. 22, 121 (1972).
[CrossRef]

F. Lindl and P. Kaw, Phys. Fluids 14, 371 (1972).
[CrossRef]

1970 (1)

H. Hora and H. Wilhelm, Nucl. Fusion 10, 111 (1970).
[CrossRef]

1969 (4)

P. Kaw, Appl. Phys. Lett. 15, 16 (1969).
[CrossRef]

A. G. Litvak, Zh. Eksp. Teor. Fiz. 57, 629 (1969) [Sov. Phys. JETP 30, 344 (1970)].

H. Hora, Z. Physik 226, 156 (1969); a derivation based on electron motion was given by Claire Ellen Max, J. Arons, and A. B. Langdon, Phys. Rev. Lett. 33, 209 (1974).
[CrossRef]

H. Hora, Phys. Fluids 12, 182 (1969).
[CrossRef]

1968 (2)

A. Schlüter, Plasma Phys. 10, 471 (1968).

V. V. Korobkin and A. J. Alcock, Phys. Rev. Lett. 21, 1433 (1968).
[CrossRef]

1967 (1)

H. Hora, D. Pfirsch, and A. Schlüter, Z. Naturforschung 22a, 278 (1967).

1962 (1)

J. M. Dawson and C. Oberman, Phys. Fluids 5, 517 (1962).
[CrossRef]

1953 (1)

R. Lüst, Z. Astrophysik 57, 67 (1953).

1950 (1)

A. Schlüter, Z. Naturforschung 5a, 72 (1950).

Alcock, A. J.

V. V. Korobkin and A. J. Alcock, Phys. Rev. Lett. 21, 1433 (1968).
[CrossRef]

Basov, N. G.

N. G. Basov and O. N. Krokhin, in Third International Quantum Electronic Conference, edited by N. Bloembergen (Academic, New York, 1963).

Brueckner, K.

K. Brueckner, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 427.
[CrossRef]

Chakravarti, A. K.

M. S. Sodha, A. K. Chakravarti, W. P. Phadke, G. D. Gautama, and I. Rattan, Appl. Phys. Lett. 22, 121 (1972).
[CrossRef]

Chen, F. F.

R. B. White and F. F. Chen, Plasma Phys. 16, 587 (1974).
[CrossRef]

F. F. Chen, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3A, p. 291.
[CrossRef]

Dawson, J. M.

T. W. Johnston and J. M. Dawson, Phys. Fluids 16, 722 (1973).
[CrossRef]

J. M. Dawson and C. Oberman, Phys. Fluids 5, 517 (1962).
[CrossRef]

Dewar, R. L.

R. L. Dewar, Phys. Fluids 16, 431 (1973).
[CrossRef]

Eddleman, J. L.

J. W. Shearer and J. L. Eddleman, Phys. Fluids 16, 1522 (1973).
[CrossRef]

Garrison, J.

J. W. Shearer, J. Garrison, J. Wong, and J. E. Swain, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 803.
[CrossRef]

Gautama, G. D.

M. S. Sodha, A. K. Chakravarti, W. P. Phadke, G. D. Gautama, and I. Rattan, Appl. Phys. Lett. 22, 121 (1972).
[CrossRef]

Hora, H.

H. Hora, Atomkernenergie 24, 187 (1974).

H. Hora, Opto-Electronics 5, 491 (1973).
[CrossRef]

H. Hora and H. Wilhelm, Nucl. Fusion 10, 111 (1970).
[CrossRef]

H. Hora, Phys. Fluids 12, 182 (1969).
[CrossRef]

H. Hora, Z. Physik 226, 156 (1969); a derivation based on electron motion was given by Claire Ellen Max, J. Arons, and A. B. Langdon, Phys. Rev. Lett. 33, 209 (1974).
[CrossRef]

H. Hora, D. Pfirsch, and A. Schlüter, Z. Naturforschung 22a, 278 (1967).

H. Hora, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1970), Vol. 1, p. 427.

H. Hora, Laser Plasmas and Nuclear Energy (Plenum, New York, 1975), p. 47.

H. Hora, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974) Vol. 3B, p. 803.

Hughes, J. L.

J. L. Hughes, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 849.
[CrossRef]

Johnston, T. W.

T. W. Johnston and J. M. Dawson, Phys. Fluids 16, 722 (1973).
[CrossRef]

Kaw, P.

F. Lindl and P. Kaw, Phys. Fluids 14, 371 (1972).
[CrossRef]

P. Kaw, Appl. Phys. Lett. 15, 16 (1969).
[CrossRef]

Kibble, T. W. B.

T. W. B. Kibble, Phys. Rev. Lett.16, 1054;Phys. Res. 150, 1060 (1966).

Korobkin, V. V.

V. V. Korobkin and A. J. Alcock, Phys. Rev. Lett. 21, 1433 (1968).
[CrossRef]

Krokhin, O. N.

N. G. Basov and O. N. Krokhin, in Third International Quantum Electronic Conference, edited by N. Bloembergen (Academic, New York, 1963).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1966), p. 242.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1966), p. 242.

Lindl, F.

F. Lindl and P. Kaw, Phys. Fluids 14, 371 (1972).
[CrossRef]

Litvak, A. G.

A. G. Litvak, Zh. Eksp. Teor. Fiz. 57, 629 (1969) [Sov. Phys. JETP 30, 344 (1970)].

Lüst, R.

R. Lüst, Z. Astrophysik 57, 67 (1953).

Motz, H.

H. Motz and C. J. H. Watson, in Advances in Electronics, Vol. 23, edited by L. Marton (Academic, New York, 1967), p. 153.

Nuckolls, J. H.

J. H. Nuckolls, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 397.

Oberman, C.

J. M. Dawson and C. Oberman, Phys. Fluids 5, 517 (1962).
[CrossRef]

Palmer, A. J.

A. J. Palmer, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1972), Vol. 2, p. 367.

Pfirsch, D.

H. Hora, D. Pfirsch, and A. Schlüter, Z. Naturforschung 22a, 278 (1967).

Phadke, W. P.

M. S. Sodha, A. K. Chakravarti, W. P. Phadke, G. D. Gautama, and I. Rattan, Appl. Phys. Lett. 22, 121 (1972).
[CrossRef]

Rattan, I.

M. S. Sodha, A. K. Chakravarti, W. P. Phadke, G. D. Gautama, and I. Rattan, Appl. Phys. Lett. 22, 121 (1972).
[CrossRef]

Schlüter, A.

A. Schlüter, Plasma Phys. 10, 471 (1968).

H. Hora, D. Pfirsch, and A. Schlüter, Z. Naturforschung 22a, 278 (1967).

A. Schlüter, Z. Naturforschung 5a, 72 (1950).

Shearer, J. W.

J. W. Shearer and J. L. Eddleman, Phys. Fluids 16, 1522 (1973).
[CrossRef]

J. W. Shearer, J. Garrison, J. Wong, and J. E. Swain, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 803.
[CrossRef]

Sodha, M. S.

M. S. Sodha, Repts. Prog. Phys. 37, 621 (1974).

M. S. Sodha, A. K. Chakravarti, W. P. Phadke, G. D. Gautama, and I. Rattan, Appl. Phys. Lett. 22, 121 (1972).
[CrossRef]

Spitzer, L.

L. Spitzer, Physics of Fully Ionized Gases (Wiley Interscience, New York, 1962).

Swain, J. E.

J. W. Shearer, J. Garrison, J. Wong, and J. E. Swain, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 803.
[CrossRef]

Watson, C. J. H.

H. Motz and C. J. H. Watson, in Advances in Electronics, Vol. 23, edited by L. Marton (Academic, New York, 1967), p. 153.

White, R. B.

R. B. White and F. F. Chen, Plasma Phys. 16, 587 (1974).
[CrossRef]

Wilhelm, H.

H. Hora and H. Wilhelm, Nucl. Fusion 10, 111 (1970).
[CrossRef]

Wong, J.

J. W. Shearer, J. Garrison, J. Wong, and J. E. Swain, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 803.
[CrossRef]

Appl. Phys. Lett. (2)

P. Kaw, Appl. Phys. Lett. 15, 16 (1969).
[CrossRef]

M. S. Sodha, A. K. Chakravarti, W. P. Phadke, G. D. Gautama, and I. Rattan, Appl. Phys. Lett. 22, 121 (1972).
[CrossRef]

Atomkernenergie (1)

H. Hora, Atomkernenergie 24, 187 (1974).

Nucl. Fusion (1)

H. Hora and H. Wilhelm, Nucl. Fusion 10, 111 (1970).
[CrossRef]

Opto-Electronics (1)

H. Hora, Opto-Electronics 5, 491 (1973).
[CrossRef]

Phys. Fluids (6)

F. Lindl and P. Kaw, Phys. Fluids 14, 371 (1972).
[CrossRef]

T. W. Johnston and J. M. Dawson, Phys. Fluids 16, 722 (1973).
[CrossRef]

J. M. Dawson and C. Oberman, Phys. Fluids 5, 517 (1962).
[CrossRef]

R. L. Dewar, Phys. Fluids 16, 431 (1973).
[CrossRef]

H. Hora, Phys. Fluids 12, 182 (1969).
[CrossRef]

J. W. Shearer and J. L. Eddleman, Phys. Fluids 16, 1522 (1973).
[CrossRef]

Phys. Rev. Lett. (1)

V. V. Korobkin and A. J. Alcock, Phys. Rev. Lett. 21, 1433 (1968).
[CrossRef]

Plasma Phys. (2)

A. Schlüter, Plasma Phys. 10, 471 (1968).

R. B. White and F. F. Chen, Plasma Phys. 16, 587 (1974).
[CrossRef]

Repts. Prog. Phys. (1)

M. S. Sodha, Repts. Prog. Phys. 37, 621 (1974).

Z. Astrophysik (1)

R. Lüst, Z. Astrophysik 57, 67 (1953).

Z. Naturforschung (2)

H. Hora, D. Pfirsch, and A. Schlüter, Z. Naturforschung 22a, 278 (1967).

A. Schlüter, Z. Naturforschung 5a, 72 (1950).

Z. Physik (1)

H. Hora, Z. Physik 226, 156 (1969); a derivation based on electron motion was given by Claire Ellen Max, J. Arons, and A. B. Langdon, Phys. Rev. Lett. 33, 209 (1974).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

A. G. Litvak, Zh. Eksp. Teor. Fiz. 57, 629 (1969) [Sov. Phys. JETP 30, 344 (1970)].

Other (15)

A. J. Palmer, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1972), Vol. 2, p. 367.

T. W. B. Kibble, Phys. Rev. Lett.16, 1054;Phys. Res. 150, 1060 (1966).

The nonlinear force is the electrodynamic part of the gradient of the momentum-flux-density of a more general form than used by Landau and Lifshitz (Ref. 5). The generalization with respect to dispersion was confirmed by consistency with the derivation from other models, e.g., the two-fluid model (Ref. 6). Because Landau and Lifshitz (Ref. 5) used the expression “ponderomotive force” for gas dynamic (thermokinetic) and other forces, the expression “nonlinear force” for its electrodynamic part may be preferred with respect to its quadratic terms. Shearer and Eddleman (Ref. 7) specialized the general derivation to reproduce the electric term (without the magnetic term) of the nonlinear force, as used in microwave theory (Ref. 8), from which I needed further specializations to derive the general expression (Ref. 6).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1966), p. 242.

F. F. Chen, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3A, p. 291.
[CrossRef]

H. Motz and C. J. H. Watson, in Advances in Electronics, Vol. 23, edited by L. Marton (Academic, New York, 1967), p. 153.

H. Hora, Laser Plasmas and Nuclear Energy (Plenum, New York, 1975), p. 47.

H. Hora, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974) Vol. 3B, p. 803.

L. Spitzer, Physics of Fully Ionized Gases (Wiley Interscience, New York, 1962).

H. Hora, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1970), Vol. 1, p. 427.

N. G. Basov and O. N. Krokhin, in Third International Quantum Electronic Conference, edited by N. Bloembergen (Academic, New York, 1963).

J. L. Hughes, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 849.
[CrossRef]

J. W. Shearer, J. Garrison, J. Wong, and J. E. Swain, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 803.
[CrossRef]

J. H. Nuckolls, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 397.

K. Brueckner, in Laser Interaction and Related Plasma Phenomena, edited by H. Schwarz and H. Hora (Plenum, New York, 1974), Vol. 3B, p. 427.
[CrossRef]

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Figures (3)

FIG. 1
FIG. 1

Evaluation of the self-focusing length lsf for a laser beam with gaussian irradiance profile (pointed line) and diameter d0 (half-irradiance maximum). Its initial plane wave front becomes convex (dashed line) because the effective wavelength λ is less at higher irradiance than at lower irradiance.

FIG. 2
FIG. 2

Calculated self-focusing lengths over the laser-beam diameter in dependence on the irradiance of neodymium-glass-laser radiation for varying electron densities ne.

FIG. 3
FIG. 3

Maximum dielectric increase (swelling) g = 1/|ñ|min and swelling at relativistic-self-focusing conditions gsf in dependence on the laser irradiance.

Equations (30)

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

P = { 1.46 × 10 6 / T 5 / 4 , if ω p < ω ,             T > 30 eV 11.0 × 10 3 T , if ω p ω ,
ω p 2 = 4 π e 2 n e / m e
n ˜ = 1 - ω p 2 ω 2 1 1 - i ν / ω ,
kin = m 0 c 2 [ ( 1 + e 2 E 2 m e 2 ω 2 c 2 ) 1 / 2 - 1 ) ] ,
kin { m e c 2 I I re 1 for kin m 0 c 2 m e c 2 ( I I re 1 ) 1 / 2 for kin m 0 c 2 ,
I re 1 = c E re 1 2 8 π = 3 c 8 π m e 2 c 2 ω 2 e 2 = { 3.66 × 10 18 W / cm 2 for Nd glass lasers 3.66 × 10 16 W / cm 2 for CO 2 lasers .
T 5 × 10 8 K 10 4 eV ;
ω p 2 = 4 π e 2 n e m e m e c 2 m e c 2 + kin { 4 π e 2 n e m e ( 1 + I I re 1 ) - 1 , if I I re 1 4 π e 2 n e m e ( 1 + ( I I re 1 ) 1 / 2 ) - 1 , if I I re 1
ν = π 3 / 2 Z e 4 n e ln Λ ( m e c 2 + kin ) 1 / 4 4 2 m e 1 / 2 ( k T + kin ) 3 / 2 γ E ( Z ) e 1 / 2 ,
γ ( Z ) = { 1 at kin k T γ E S ( Z ) at kin k T ;
ln Λ = ln [ 3 2 Z 2 e 2 ( k 3 T 3 + kin 3 π n e ) 1 / 2 ] .
{ n κ } = [ 1 2 { [ 1 - ω p 2 ω 2 + ν 2 + ( ν ω ω p 2 ω 2 + ν 2 ) 2 ] 1 / 2 ± ( 1 + ω p 2 ω 2 + ν 2 ) } ] 1 / 2 .
K = 2 ω c κ = 2 π 3 / 2 Z n e 2 e 6 ln Λ γ E S c ω 2 m e 3 / 2 ( k T + kin ) 3 / 2
K D 0 = 8 π 1 / 2 2 1 / 2 Z n e 6 ln Λ 3 c ω 2 m e 3 / 2 ( k T ) 3 / 2 ,
K K D 0 = 1.01211 ,
λ = λ 0 n ˜ ( I ) ,
n ˜ ( I max ) > n ˜ ( I max / 2 ) .
l sf = [ 2 ( d 0 / 2 ) ( d 0 / 2 + ρ 0 ) - d 0 / 2 ] 1 / 2 .
n ˜ ( I max / 2 ) - 1 / ( d 0 / 2 + ρ 0 ) = n ˜ ( I max ) - 1 / ρ 0
l sf d 0 = 0.5 ( n ˜ ( I max ) + n ˜ ( I max / 2 ) n ˜ ( I max ) - n ˜ ( I max / 2 ) ) 1 / 2 .
n ˜ = [ ( 1 - ω p 2 ω 2 + ν 2 ) 2 + ( ν ω ω p 2 ω 2 + ν 2 ) 2 ] 1 / 4 ,
n ˜ = [ ( 1 - n e n e co NR m e c 2 m e c 2 + kin ) 2 + C * m e c 2 + kin ( k T + kin ) 3 ( n e n e co NR m e c 2 m e c 2 + kin ) 2 ] 1 / 4 ,
C * = π ω 2 e 4 ln ( 2 Λ ) / ( 512 c 2 ) .
n ˜ ( n e = n e * ) kin = 0.
n e n e co NR kin m 0 c 2 .
n e n e co NR 1             for kin m 0 c 2 0.03 ,
l sf d 0 = f ( I ; n e )
I = I v / n ˜ ,             E = E v / n ˜ 1 / 2 ,
g = 1 / n ˜ min = ( ν / ω ) 1 / 2 ,
g sf = 1 10 1 / 2 g = 1 3.16 n ˜ min ,