Abstract

Extremely intense laser beams penetrating into a plasma are focused because of the dependence of the index of refraction on the relativistic electron mass. Under the assumptions of a collision-free plasma and circularly polarized laser radiation, we calculate the focus length by using the WKB approximation. The resulting analytical formulas are applied to the case of a neodymium:glass laser.

© 1988 Optical Society of America

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References

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  1. H. Hora, Physics of Laser Driven Plasmas (Wiley, New York, 1981).
  2. H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
    [CrossRef]
  3. S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
    [CrossRef]
  4. O. Svelto, in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974), p. 1.
    [CrossRef]
  5. M. S. Sodha, A. K. Ghatak, V. K. Tripathi, in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976), p. 169.
    [CrossRef]
  6. P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
    [CrossRef]
  7. S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. JETP 23, 1025 (1966).
  8. H. Hora, J. Opt Soc. Am. 65, 882 (1975).
    [CrossRef]
  9. K. H. Spatscheck, J. Plasma Phys. 18, 293 (1977).
    [CrossRef]
  10. E. L. Kane, H. Hora, Aust. J. Phys. 34, 385 (1981).
    [CrossRef]
  11. D. A. Jones, E. L. Kane, P. Lalousis, H. Hora, Phys. Fluids 25, 2295 (1982);F. J. Limpouch, G. Lončar, I. G. Lebo, V. B. Rozanov, Laser Part. Beams 6, 295 (1988);N. G. Basov, K. Götz, A. M. Maksimchovk, Ya. A. Mikhailov, A. V. Rode, G. V. Sklizkov, S. I. Fedotov, E. Förster, H. Hora, Zh. Eks. Teor. Fiz. 92, 1299 (1987) [Sov. Phys. JETP 65, 727 (1987)].
    [CrossRef]
  12. H. Hora, Phys. Fluids 28, 3705 (1985).
    [CrossRef]
  13. M. Lax, W. H. Louisell, W. B. McKnight, Phys. Rev. A 11, 1395 (1975).
    [CrossRef]
  14. B. Boreham, H. Hora, Phys. Rev. Lett. 42, 776 (1979).
    [CrossRef]
  15. L. Cicchitelli, Ph.D. dissertation (University of New South Wales, New South Wales, Australia, 1988).
  16. H. Hora, Nucl. Instrum. Methods A271, 117 (1988);L. Cicchitelli, S. Eliezer, M. P. Goldsworthy, F. Green, H. Hora, P. S. Ray, R. J. Stening, H. Szichman, Laser Part. Beams 6, 163 (1988).
    [CrossRef]
  17. S. Eliezer, A. Ludmirski, Laser Part. Beams 1, 251 (1983).
    [CrossRef]
  18. P. Lalousis, H. Hora, Laser Part. Beams 1, 151 (1983);H. Hora, Laser Part. Beams 3, 359 (1985);M. P. Goldsworthy, F. Green, H. Hora, Laser Part. Beams 5, 269 (1987).
    [CrossRef]

1988 (1)

H. Hora, Nucl. Instrum. Methods A271, 117 (1988);L. Cicchitelli, S. Eliezer, M. P. Goldsworthy, F. Green, H. Hora, P. S. Ray, R. J. Stening, H. Szichman, Laser Part. Beams 6, 163 (1988).
[CrossRef]

1985 (1)

H. Hora, Phys. Fluids 28, 3705 (1985).
[CrossRef]

1983 (2)

S. Eliezer, A. Ludmirski, Laser Part. Beams 1, 251 (1983).
[CrossRef]

P. Lalousis, H. Hora, Laser Part. Beams 1, 151 (1983);H. Hora, Laser Part. Beams 3, 359 (1985);M. P. Goldsworthy, F. Green, H. Hora, Laser Part. Beams 5, 269 (1987).
[CrossRef]

1982 (1)

D. A. Jones, E. L. Kane, P. Lalousis, H. Hora, Phys. Fluids 25, 2295 (1982);F. J. Limpouch, G. Lončar, I. G. Lebo, V. B. Rozanov, Laser Part. Beams 6, 295 (1988);N. G. Basov, K. Götz, A. M. Maksimchovk, Ya. A. Mikhailov, A. V. Rode, G. V. Sklizkov, S. I. Fedotov, E. Förster, H. Hora, Zh. Eks. Teor. Fiz. 92, 1299 (1987) [Sov. Phys. JETP 65, 727 (1987)].
[CrossRef]

1981 (1)

E. L. Kane, H. Hora, Aust. J. Phys. 34, 385 (1981).
[CrossRef]

1979 (1)

B. Boreham, H. Hora, Phys. Rev. Lett. 42, 776 (1979).
[CrossRef]

1977 (1)

K. H. Spatscheck, J. Plasma Phys. 18, 293 (1977).
[CrossRef]

1975 (2)

M. Lax, W. H. Louisell, W. B. McKnight, Phys. Rev. A 11, 1395 (1975).
[CrossRef]

H. Hora, J. Opt Soc. Am. 65, 882 (1975).
[CrossRef]

1968 (1)

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

1966 (1)

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. JETP 23, 1025 (1966).

1965 (1)

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. JETP 23, 1025 (1966).

Boreham, B.

B. Boreham, H. Hora, Phys. Rev. Lett. 42, 776 (1979).
[CrossRef]

Cicchitelli, L.

L. Cicchitelli, Ph.D. dissertation (University of New South Wales, New South Wales, Australia, 1988).

Eliezer, S.

S. Eliezer, A. Ludmirski, Laser Part. Beams 1, 251 (1983).
[CrossRef]

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

Farley, F. J. M.

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

Ghatak, A. K.

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

M. S. Sodha, A. K. Ghatak, V. K. Tripathi, in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976), p. 169.
[CrossRef]

Goldsworthy, M. P.

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

Green, F.

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

Hora, H.

H. Hora, Nucl. Instrum. Methods A271, 117 (1988);L. Cicchitelli, S. Eliezer, M. P. Goldsworthy, F. Green, H. Hora, P. S. Ray, R. J. Stening, H. Szichman, Laser Part. Beams 6, 163 (1988).
[CrossRef]

H. Hora, Phys. Fluids 28, 3705 (1985).
[CrossRef]

P. Lalousis, H. Hora, Laser Part. Beams 1, 151 (1983);H. Hora, Laser Part. Beams 3, 359 (1985);M. P. Goldsworthy, F. Green, H. Hora, Laser Part. Beams 5, 269 (1987).
[CrossRef]

D. A. Jones, E. L. Kane, P. Lalousis, H. Hora, Phys. Fluids 25, 2295 (1982);F. J. Limpouch, G. Lončar, I. G. Lebo, V. B. Rozanov, Laser Part. Beams 6, 295 (1988);N. G. Basov, K. Götz, A. M. Maksimchovk, Ya. A. Mikhailov, A. V. Rode, G. V. Sklizkov, S. I. Fedotov, E. Förster, H. Hora, Zh. Eks. Teor. Fiz. 92, 1299 (1987) [Sov. Phys. JETP 65, 727 (1987)].
[CrossRef]

E. L. Kane, H. Hora, Aust. J. Phys. 34, 385 (1981).
[CrossRef]

B. Boreham, H. Hora, Phys. Rev. Lett. 42, 776 (1979).
[CrossRef]

H. Hora, J. Opt Soc. Am. 65, 882 (1975).
[CrossRef]

H. Hora, Physics of Laser Driven Plasmas (Wiley, New York, 1981).

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

Jones, D. A.

D. A. Jones, E. L. Kane, P. Lalousis, H. Hora, Phys. Fluids 25, 2295 (1982);F. J. Limpouch, G. Lončar, I. G. Lebo, V. B. Rozanov, Laser Part. Beams 6, 295 (1988);N. G. Basov, K. Götz, A. M. Maksimchovk, Ya. A. Mikhailov, A. V. Rode, G. V. Sklizkov, S. I. Fedotov, E. Förster, H. Hora, Zh. Eks. Teor. Fiz. 92, 1299 (1987) [Sov. Phys. JETP 65, 727 (1987)].
[CrossRef]

Kane, E. L.

D. A. Jones, E. L. Kane, P. Lalousis, H. Hora, Phys. Fluids 25, 2295 (1982);F. J. Limpouch, G. Lončar, I. G. Lebo, V. B. Rozanov, Laser Part. Beams 6, 295 (1988);N. G. Basov, K. Götz, A. M. Maksimchovk, Ya. A. Mikhailov, A. V. Rode, G. V. Sklizkov, S. I. Fedotov, E. Förster, H. Hora, Zh. Eks. Teor. Fiz. 92, 1299 (1987) [Sov. Phys. JETP 65, 727 (1987)].
[CrossRef]

E. L. Kane, H. Hora, Aust. J. Phys. 34, 385 (1981).
[CrossRef]

Kelley, P. L.

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Kentwell, G. W.

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. JETP 23, 1025 (1966).

Lalousis, P.

P. Lalousis, H. Hora, Laser Part. Beams 1, 151 (1983);H. Hora, Laser Part. Beams 3, 359 (1985);M. P. Goldsworthy, F. Green, H. Hora, Laser Part. Beams 5, 269 (1987).
[CrossRef]

D. A. Jones, E. L. Kane, P. Lalousis, H. Hora, Phys. Fluids 25, 2295 (1982);F. J. Limpouch, G. Lončar, I. G. Lebo, V. B. Rozanov, Laser Part. Beams 6, 295 (1988);N. G. Basov, K. Götz, A. M. Maksimchovk, Ya. A. Mikhailov, A. V. Rode, G. V. Sklizkov, S. I. Fedotov, E. Förster, H. Hora, Zh. Eks. Teor. Fiz. 92, 1299 (1987) [Sov. Phys. JETP 65, 727 (1987)].
[CrossRef]

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, Phys. Rev. A 11, 1395 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, Phys. Rev. A 11, 1395 (1975).
[CrossRef]

Ludmirski, A.

S. Eliezer, A. Ludmirski, Laser Part. Beams 1, 251 (1983).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, Phys. Rev. A 11, 1395 (1975).
[CrossRef]

Scheid, W.

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

Sodha, M. S.

M. S. Sodha, A. K. Ghatak, V. K. Tripathi, in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976), p. 169.
[CrossRef]

Spatscheck, K. H.

K. H. Spatscheck, J. Plasma Phys. 18, 293 (1977).
[CrossRef]

Stening, R.

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. JETP 23, 1025 (1966).

Svelto, O.

O. Svelto, in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974), p. 1.
[CrossRef]

Tapalaga, S.

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

Tripathi, V. K.

M. S. Sodha, A. K. Ghatak, V. K. Tripathi, in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976), p. 169.
[CrossRef]

Aust. J. Phys. (1)

E. L. Kane, H. Hora, Aust. J. Phys. 34, 385 (1981).
[CrossRef]

J. Opt Soc. Am. (1)

H. Hora, J. Opt Soc. Am. 65, 882 (1975).
[CrossRef]

J. Plasma Phys. (1)

K. H. Spatscheck, J. Plasma Phys. 18, 293 (1977).
[CrossRef]

Laser Part. Beams (2)

S. Eliezer, A. Ludmirski, Laser Part. Beams 1, 251 (1983).
[CrossRef]

P. Lalousis, H. Hora, Laser Part. Beams 1, 151 (1983);H. Hora, Laser Part. Beams 3, 359 (1985);M. P. Goldsworthy, F. Green, H. Hora, Laser Part. Beams 5, 269 (1987).
[CrossRef]

Nucl. Instrum. Methods (1)

H. Hora, Nucl. Instrum. Methods A271, 117 (1988);L. Cicchitelli, S. Eliezer, M. P. Goldsworthy, F. Green, H. Hora, P. S. Ray, R. J. Stening, H. Szichman, Laser Part. Beams 6, 163 (1988).
[CrossRef]

Phys. Fluids (2)

D. A. Jones, E. L. Kane, P. Lalousis, H. Hora, Phys. Fluids 25, 2295 (1982);F. J. Limpouch, G. Lončar, I. G. Lebo, V. B. Rozanov, Laser Part. Beams 6, 295 (1988);N. G. Basov, K. Götz, A. M. Maksimchovk, Ya. A. Mikhailov, A. V. Rode, G. V. Sklizkov, S. I. Fedotov, E. Förster, H. Hora, Zh. Eks. Teor. Fiz. 92, 1299 (1987) [Sov. Phys. JETP 65, 727 (1987)].
[CrossRef]

H. Hora, Phys. Fluids 28, 3705 (1985).
[CrossRef]

Phys. Rev. A (1)

M. Lax, W. H. Louisell, W. B. McKnight, Phys. Rev. A 11, 1395 (1975).
[CrossRef]

Phys. Rev. Lett. (2)

B. Boreham, H. Hora, Phys. Rev. Lett. 42, 776 (1979).
[CrossRef]

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Sov. Phys. JETP (1)

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. JETP 23, 1025 (1966).

Sov. Phys. Usp. (1)

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Other (5)

O. Svelto, in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974), p. 1.
[CrossRef]

M. S. Sodha, A. K. Ghatak, V. K. Tripathi, in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976), p. 169.
[CrossRef]

H. Hora, Physics of Laser Driven Plasmas (Wiley, New York, 1981).

H. Hora, S. Eliezer, F. J. M. Farley, A. K. Ghatak, M. P. Goldsworthy, F. Green, G. W. Kentwell, P. Lalousis, W. Scheid, R. Stening, S. Tapalaga, in Laser Interaction and Related Plasma Phenomena, H. Hora, G. H. Miley, eds. (Plenum, New York, 1986), Vol. 7, p. 347.
[CrossRef]

L. Cicchitelli, Ph.D. dissertation (University of New South Wales, New South Wales, Australia, 1988).

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Figures (2)

Fig. 1
Fig. 1

Focal length for relativistic self-focusing as a function of the ratio β of the maximum intensity to the relativistic intensity Irel = 2.44 × 1018 W cm−2 for a Nd:glass laser with a wavelength of λ = 1.06 μm. The solid curves are calculated from Eq. (50) for contant electron densities (f = 0) 0.1, 0.01, and 0.001 × 1021 cm−3. The dashed curves are obtained with by using the arc-approximation formula [see Eq. (53)] for the same electron densities.

Fig. 2
Fig. 2

The ratio of the intensity along the beam axis to the relativistic intensity Irel as a function of the ratio x/xf for an initial intensity I = Irel = 2.44 × 1018 W cm−2 for a Nd:glass laser with a wavelength of λ = 1.06 μm. The density of electrons is ne = 1020 cm−3 (f = 0).

Equations (62)

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

n 2 = 1 ω p 2 / ω 2 , ω p 2 = 4 π n e e 2 / γ m 0 .
γ 2 = 1 + ( e m 0 c ) 2 ( E d t ) 2 .
× E = 1 c H t ,
× H = 4 π c j + 1 c E t
j t = ω p 2 4 π E .
Δ E ( E ) 1 c 2 2 E d t 2 ω p 2 c 2 E = 0 .
Δ E ( E ) + k 2 E = 0 ,
( k 2 E ) = 0
E = 1 n 2 n 2 E ( ln γ ) .
E = E 0 Re [ ψ exp ( i ω t ) ( e y i e z ) ] .
Δ ψ + k 2 ψ = 0 .
ψ = exp ( if ) .
( f ) 2 + i λ 2 f + k 2 = 0 .
f = f 0 ( x ) + f 2 ( x ) r 2 + i λ [ g 0 ( x ) + g 2 ( x ) r 2 ] ,
k 2 ( x , r ) = k 0 2 + k 2 r 2 | r = 0 r 2 ,
d f 0 d x = k 0 ,
2 k 0 d f 2 d x + 4 f 2 2 = k 2 r 2 | r = 0 ,
2 k 0 d g 0 d x 4 f 2 = d k 0 d x .
exp ( 2 g 0 ) ( k 0 g 2 + 1 2 g 2 2 d f 2 d x ) = 4 A ,
P = c 4 π 0 ( E × H ) x 2 π r d r = c 2 E 0 2 2 ω A ,
A = 1 2 i 0 ( ψ * ψ x ψ ψ * x ) r d r .
u ( x ) = exp [ g 0 ( x ) ] .
E ( x , r = 0 , t ) = E 0 u ( x ) { cos [ f 0 ( x ) ω t ] e y + sin [ f 0 ( x ) ω t ] e z } .
k 2 r 2 | r = 0 = 2 g 2 u k 0 d k 0 d u .
f 2 = 1 2 ( k 0 u + 1 2 d k 0 d u ) d u d x .
1 2 ( 1 + 1 2 u k 0 d k 0 d u ) d υ d u 2 [ 1 u 2 4 k 0 d 2 k 0 d u 2 + u 2 8 k 0 2 ( d k 0 d u ) 2 ] υ u = 2 g 2 u 2 k 0 d k 0 d u ,
( 1 u 2 k 0 d k 0 d u ) k 0 g 2 k 0 4 u 2 ( 1 + u 2 k 0 d k 0 d u ) 2 υ = 4 A g 2 2 / u 2 .
| d k 0 d u | k 0 u , | d 2 k 0 d u 2 | k 0 u 2 .
γ 2 = 1 + ( e E 0 m 0 c ω ) 2 u 2 exp ( 2 g 2 r 2 ) .
1 n 2 n 2 γ 2 1 γ 2 2 .
g 2 = k 0 u 2 8 A [ 1 + ( 1 4 υ A k 0 u 4 ) 1 / 2 ] .
υ > k 0 u 4 4 A
1 u | d u d x | > 1 2 R 0 ( x ) ,
R 0 ( x ) = [ 2 A k 0 ( x ) ] 1 / 2 / u ( x ) = 2 c E 0 [ ω P k 0 ( x ) ] 1 / 2 / u ( x ) . .
d υ d u 4 υ u = ( υ 4 A υ k 0 ) d k 0 d u .
υ = u 4 2 A [ k 0 ( u ) k 0 2 ( u 0 ) k 0 ( u ) ] .
u 0 u [ k 0 ( u ) ] 1 / 2 [ k 0 2 ( u ) k 0 2 ( u 0 ) ] 1 / 2 d u u 2 = 1 ( 2 A ) 1 / 2 x .
1 / u f 0 if υ ( u f ) / u f 4 < k 0 ( u f ) / 4 A ,
k 0 ( u f ) = 2 k 0 ( u 0 ) if υ ( u f ) / u f 4 = k 0 ( u f ) / 4 A .
n 0 2 = 1 α ( 1 + β u 2 u 0 2 ) 1 / 2 ,
α = 4 π n e e 2 / m 0 ω 2 = n e / n e c ,
β = e 2 E 0 2 u 0 2 ( m 0 ω c ) 2 = I max / [ I rel n 0 ( x = 0 ) ] .
n e = n e o exp [ f ( γ 1 ) ] .
n 0 2 = 1 α ( 1 + β u 2 u 0 2 ) 1 / 2 exp { f [ ( 1 + β u 2 u 0 2 ) 1 / 2 1 ] } .
( d s d x ) 2 = 1 A [ a ( s s 0 ) + b ( s s 0 ) 2 ] ,
a = u 0 2 d k 0 ( u ) d u | u = u 0 < 0 .
s = s 0 + 1 2 b { cosh [ ( b A ) 1 / 2 x ] 1 } .
x f = ( A b ) 1 / 2 arccosh ( 1 2 b u 0 a )
x f 2 = 4 A | α | u 0 = 2 n 0 u 0 d n 0 d u | u = u 0 R 0 2 .
b a u 0 = 1 2 ( 3 + β β / n 0 2 ) / ( 1 + β ) ,
x f 2 = 4 α β ( 1 + β ) 3 / 2 × 1 α ( 1 + β ) 1 / 2 exp { f [ ( 1 + β ) 1 / 2 1 ] } [ 1 + f ( 1 + β ) 1 / 2 ] exp { f [ ( 1 + β ) 1 / 2 1 ] } R 0 2 .
x f 2 R 0 2 = 4 α β ( 1 + β ) 3 / 2 [ 1 α ( 1 + β ) 1 / 2 ] ,
x f 2 R 0 2 = 4 β 1 / 2 / α = 4 n e c n e ( I max / I rel ) 1 / 2 .
β = 2 2 α ( 1 + β ) 1 / 2 .
( d u d x ) 2 = u 4 ω α β ( 1 + f ) 4 A c ( 1 α ) 1 / 2 ( u 2 u 0 2 1 ) .
u 2 = u 0 2 / ( 1 x 2 / x f 2 ) ,
x f 2 R 0 2 = 2 ( 1 α ) α β ( 1 + f ) .
x f 2 R 0 2 = F ( β ) 2 α β ( 1 + β ) 3 / 2 × 1 α ( 1 + β ) 1 / 2 exp { f [ ( 1 + β ) 1 / 2 1 ] } [ 1 + f ( 1 + β ) 1 / 2 ] exp { f [ ( 1 + β ) 1 / 2 1 ] } ,
F ( β ) = ( 1 + 4 β ) / ( 1 + 2 β ) .
l SF d 0 = 0.5 [ | n ( I max ) | + | n ( I max / 2 ) | | n ( I max ) | | n ( I max / 2 ) | ] 1 / 2 ,
x f 0 R 0 2 = n 0 ( α , β ) + n 0 ( α , β / 2 ) n 0 ( α , β ) n 0 ( α , β / 2 ) ,
n 0 ( α , β ) = [ 1 α ( 1 + β ) 1 / 2 ] 1 / 2 .

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