Abstract

We critically revise the theory of terahertz emission from a plasma filament induced in a gas media by one or two focusd femtosecond laser pulses. We distinguish a radiation pressure force (RPF) from a ponderomotive force (PF), discuss conditions for one of these forces to be the dominating contribution to the terahertz emission, and also show that the angular distribution of the emitted power critically depends on which of the two forces dominates in a particular experiment. We show that the experimentally observed periodic dependence of the emitted terahertz power on the gas pressure reveals the dominating role of the RPF over the PF, whereas the angular diagram of the emission allows us to determine the predominant direction of the force. We also emphasize that the terahertz emission originated by a transient photocurrent exhibits a different dependency from the phase difference between the first and the second harmonics of the optic laser field, which generally enables the experimental detection of the prevailing mechanism of the terahertz emission from the plasma filament.

© 2009 Optical Society of America

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  1. P. Sprangle, E. Esarey, and A. Ting, “Nonlinear theory of intense laser-plasma interactions,” Phys. Rev. Lett. 64, 2011-2014 (1990).
    [CrossRef] [PubMed]
  2. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994-1997 (1993).
    [CrossRef] [PubMed]
  3. H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71, 2725-2728 (1993).
    [CrossRef] [PubMed]
  4. H. Hamster, A. Sullivan, S. Gordon, and R. W. Falcone, “Short-pulse terahertz radiation from high-intensity-laser-produced plasmas,” Phys. Rev. E 49, 671-677 (1994).
    [CrossRef]
  5. P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E 69, 066415 (2004).
    [CrossRef]
  6. L. M. Gorbunov and A. A. Frolov, “Emission of low-frequency electromagnetic waves by a short laser pulse in stratified rarefied plasma,” J. Exp. Theor. Phys. 83, 967-973 (1996).
  7. D. J. Cook and R. M. Hochstrasser, “Intense terahertz pulses by four-wave rectification in air,” Opt. Lett. 25, 1210-1212 (2000).
    [CrossRef]
  8. T. Bartel, P. Gaal, K. Reimann, M. Woerner, and T. Elsaesser, “Generation of single-cycle THz transients with high electric-field amplitudes,” Opt. Lett. 30, 2805-2807 (2005).
    [CrossRef] [PubMed]
  9. M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
    [CrossRef] [PubMed]
  10. N. Bloembergen, “Recent progress in four-wave mixing spectroscopy,” in Laser Spectroscopy, H.Walther and K.W.Rothe, eds. (Springer, 1979), Vol. IV, pp. 340-348.
  11. X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96, 075005 (2006).
    [CrossRef] [PubMed]
  12. J. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases, 2nd ed. (Academic, 1984).
  13. A. Houard, Y. Liu, B. Prade, and A. Mysyrowicz, “Polarization analysis of terahertz radiation generated by four-wave mixing in air,” Opt. Lett. 33, 1195-1197 (2008).
    [CrossRef] [PubMed]
  14. J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97, 103903 (2006).
    [CrossRef] [PubMed]
  15. X. Lu, N. Karpowicz, and X.-C. Zhang, “Broadband terahertz detection with selected gases,” J. Opt. Soc. Am. B 26, A66-A73 (2009).
    [CrossRef]
  16. K. Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15, 4577-4584 (2007).
    [CrossRef] [PubMed]
  17. M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
    [CrossRef]
  18. C.-C. Cheng, E. M. Wright, and J. V. Moloney, “Generation of electromagnetic pulses from plasma channels induced by femtosecond light strings,” Phys. Rev. Lett. 87, 213001 (2001).
    [CrossRef] [PubMed]
  19. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813-822 (1968).
    [CrossRef]
  20. A. Proulx, A. Talebpour, S. Petit, and S. L. Chin, “Fast pulsed electric field created from the self-generated filament of a femtosecond Ti:sapphire laser pulse in air,” Opt. Commun. 174, 305-309 (2000).
    [CrossRef]
  21. J. F. Ready, Effects of High Power Laser Radiation (Academic, 1971).
  22. G. A. Askar'yan, “Cherenkov radiation from optical pulses,” Phys. Rev. Lett. 57, 2470 (1986).
    [CrossRef] [PubMed]
  23. B. B. Kadomtsev, Collective Phenomena in Plasmas (Pergamon, 1982).
  24. L. A. Artsimovich and R. Z. Sagdeev, Plasma Physics for Physicists (Atomizdat, 1979) (in Russian).
  25. R. Fitzpatrick, The Physics of Plasmas (Lulu, 2008).
  26. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. II of Course of Theoretical Physics, 4th ed. (Butterworth & Heinemann, 1998).
  27. Y. Chen, M. Yamaguchi, M. Wang, and X.-C. Zhang, “Terahertz pulse generation from noble gases,” Appl. Phys. Lett. 91, 251116 (2007).
    [CrossRef]
  28. V. S. Popov, “Tunnel and multiphoton ionization of atoms and ions in a strong laser field (Keldysh theory),” Phys. Usp. 47, 855-885 (2004).
    [CrossRef]
  29. V. S. Popov, “Multiphoton ionization of atoms by an ultrashort laser pulse,” JETP Lett. 73, 1-5 (2001).
    [CrossRef]
  30. This condition can be rewritten as cEω2/8π⪢(α8mc3/32πre3)(J/JH)3/(J/ℏω)2, where α is the fine structure constant, re is the classical electron radius, J/JH is the ratio of the potential of ionization to that of the atom of hydrogen, and J/ℏω is the number of photons required for the ionization. Numerically, this condition means that the laser power flux density should exceed 8.8×1015(J/JH)3/(J/ℏω)2 W/cm2.
  31. N. Karpowicz and X.-C. Zhang, “Coherent terahertz echo of tunnel ionization in gases,” Phys. Rev. Lett. 102, 093001 (2009).
    [CrossRef] [PubMed]
  32. C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of krypton and xenon and their relation to those of helium and argon,” Proc. R. Soc. London, Ser. A 84, 2805-2807 (1910).
  33. C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of neon,” Proc. R. Soc. London, Ser. A 83, 149-151 (1910).
    [CrossRef]
  34. S. A. Korff and G. Breit, “Optical dispersion,” Rev. Mod. Phys. 4, 471-503 (1932).
    [CrossRef]
  35. E. R. Peck and D. J. Fisher, “Dispersion of argon,” J. Opt. Soc. Am. A 54, 1362-1364 (1964).
    [CrossRef]

2009 (2)

X. Lu, N. Karpowicz, and X.-C. Zhang, “Broadband terahertz detection with selected gases,” J. Opt. Soc. Am. B 26, A66-A73 (2009).
[CrossRef]

N. Karpowicz and X.-C. Zhang, “Coherent terahertz echo of tunnel ionization in gases,” Phys. Rev. Lett. 102, 093001 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (2)

2006 (3)

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97, 103903 (2006).
[CrossRef] [PubMed]

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96, 075005 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (3)

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E 69, 066415 (2004).
[CrossRef]

V. S. Popov, “Tunnel and multiphoton ionization of atoms and ions in a strong laser field (Keldysh theory),” Phys. Usp. 47, 855-885 (2004).
[CrossRef]

2001 (2)

V. S. Popov, “Multiphoton ionization of atoms by an ultrashort laser pulse,” JETP Lett. 73, 1-5 (2001).
[CrossRef]

C.-C. Cheng, E. M. Wright, and J. V. Moloney, “Generation of electromagnetic pulses from plasma channels induced by femtosecond light strings,” Phys. Rev. Lett. 87, 213001 (2001).
[CrossRef] [PubMed]

2000 (2)

D. J. Cook and R. M. Hochstrasser, “Intense terahertz pulses by four-wave rectification in air,” Opt. Lett. 25, 1210-1212 (2000).
[CrossRef]

A. Proulx, A. Talebpour, S. Petit, and S. L. Chin, “Fast pulsed electric field created from the self-generated filament of a femtosecond Ti:sapphire laser pulse in air,” Opt. Commun. 174, 305-309 (2000).
[CrossRef]

1996 (1)

L. M. Gorbunov and A. A. Frolov, “Emission of low-frequency electromagnetic waves by a short laser pulse in stratified rarefied plasma,” J. Exp. Theor. Phys. 83, 967-973 (1996).

1994 (1)

H. Hamster, A. Sullivan, S. Gordon, and R. W. Falcone, “Short-pulse terahertz radiation from high-intensity-laser-produced plasmas,” Phys. Rev. E 49, 671-677 (1994).
[CrossRef]

1993 (2)

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994-1997 (1993).
[CrossRef] [PubMed]

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71, 2725-2728 (1993).
[CrossRef] [PubMed]

1990 (1)

P. Sprangle, E. Esarey, and A. Ting, “Nonlinear theory of intense laser-plasma interactions,” Phys. Rev. Lett. 64, 2011-2014 (1990).
[CrossRef] [PubMed]

1986 (1)

G. A. Askar'yan, “Cherenkov radiation from optical pulses,” Phys. Rev. Lett. 57, 2470 (1986).
[CrossRef] [PubMed]

1968 (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813-822 (1968).
[CrossRef]

1964 (1)

E. R. Peck and D. J. Fisher, “Dispersion of argon,” J. Opt. Soc. Am. A 54, 1362-1364 (1964).
[CrossRef]

1932 (1)

S. A. Korff and G. Breit, “Optical dispersion,” Rev. Mod. Phys. 4, 471-503 (1932).
[CrossRef]

1910 (2)

C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of krypton and xenon and their relation to those of helium and argon,” Proc. R. Soc. London, Ser. A 84, 2805-2807 (1910).

C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of neon,” Proc. R. Soc. London, Ser. A 83, 149-151 (1910).
[CrossRef]

Artsimovich, L. A.

L. A. Artsimovich and R. Z. Sagdeev, Plasma Physics for Physicists (Atomizdat, 1979) (in Russian).

Askar'yan, G. A.

G. A. Askar'yan, “Cherenkov radiation from optical pulses,” Phys. Rev. Lett. 57, 2470 (1986).
[CrossRef] [PubMed]

Bartel, T.

Bloembergen, N.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813-822 (1968).
[CrossRef]

N. Bloembergen, “Recent progress in four-wave mixing spectroscopy,” in Laser Spectroscopy, H.Walther and K.W.Rothe, eds. (Springer, 1979), Vol. IV, pp. 340-348.

Breit, G.

S. A. Korff and G. Breit, “Optical dispersion,” Rev. Mod. Phys. 4, 471-503 (1932).
[CrossRef]

Chang, R. K.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813-822 (1968).
[CrossRef]

Chen, Y.

Y. Chen, M. Yamaguchi, M. Wang, and X.-C. Zhang, “Terahertz pulse generation from noble gases,” Appl. Phys. Lett. 91, 251116 (2007).
[CrossRef]

Cheng, C. -C.

C.-C. Cheng, E. M. Wright, and J. V. Moloney, “Generation of electromagnetic pulses from plasma channels induced by femtosecond light strings,” Phys. Rev. Lett. 87, 213001 (2001).
[CrossRef] [PubMed]

Chin, S. L.

A. Proulx, A. Talebpour, S. Petit, and S. L. Chin, “Fast pulsed electric field created from the self-generated filament of a femtosecond Ti:sapphire laser pulse in air,” Opt. Commun. 174, 305-309 (2000).
[CrossRef]

Cook, D. J.

Corkum, P. B.

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994-1997 (1993).
[CrossRef] [PubMed]

Cuthbertson, C.

C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of krypton and xenon and their relation to those of helium and argon,” Proc. R. Soc. London, Ser. A 84, 2805-2807 (1910).

C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of neon,” Proc. R. Soc. London, Ser. A 83, 149-151 (1910).
[CrossRef]

Cuthbertson, M.

C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of neon,” Proc. R. Soc. London, Ser. A 83, 149-151 (1910).
[CrossRef]

C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of krypton and xenon and their relation to those of helium and argon,” Proc. R. Soc. London, Ser. A 84, 2805-2807 (1910).

Dai, J.

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96, 075005 (2006).
[CrossRef] [PubMed]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97, 103903 (2006).
[CrossRef] [PubMed]

Dörner, R.

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

Elsaesser, T.

Ergler, T.

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

Esarey, E.

P. Sprangle, E. Esarey, and A. Ting, “Nonlinear theory of intense laser-plasma interactions,” Phys. Rev. Lett. 64, 2011-2014 (1990).
[CrossRef] [PubMed]

Falcone, R. W.

H. Hamster, A. Sullivan, S. Gordon, and R. W. Falcone, “Short-pulse terahertz radiation from high-intensity-laser-produced plasmas,” Phys. Rev. E 49, 671-677 (1994).
[CrossRef]

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71, 2725-2728 (1993).
[CrossRef] [PubMed]

Fisher, D. J.

E. R. Peck and D. J. Fisher, “Dispersion of argon,” J. Opt. Soc. Am. A 54, 1362-1364 (1964).
[CrossRef]

Fitzpatrick, R.

R. Fitzpatrick, The Physics of Plasmas (Lulu, 2008).

Frolov, A. A.

L. M. Gorbunov and A. A. Frolov, “Emission of low-frequency electromagnetic waves by a short laser pulse in stratified rarefied plasma,” J. Exp. Theor. Phys. 83, 967-973 (1996).

Gaal, P.

Gimpel, H.

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

Glownia, J. H.

Gorbunov, L. M.

L. M. Gorbunov and A. A. Frolov, “Emission of low-frequency electromagnetic waves by a short laser pulse in stratified rarefied plasma,” J. Exp. Theor. Phys. 83, 967-973 (1996).

Gordon, S.

H. Hamster, A. Sullivan, S. Gordon, and R. W. Falcone, “Short-pulse terahertz radiation from high-intensity-laser-produced plasmas,” Phys. Rev. E 49, 671-677 (1994).
[CrossRef]

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71, 2725-2728 (1993).
[CrossRef] [PubMed]

Hafizi, B.

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E 69, 066415 (2004).
[CrossRef]

Hamster, H.

H. Hamster, A. Sullivan, S. Gordon, and R. W. Falcone, “Short-pulse terahertz radiation from high-intensity-laser-produced plasmas,” Phys. Rev. E 49, 671-677 (1994).
[CrossRef]

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71, 2725-2728 (1993).
[CrossRef] [PubMed]

Hochstrasser, R. M.

Houard, A.

Jha, S. S.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813-822 (1968).
[CrossRef]

Kadomtsev, B. B.

B. B. Kadomtsev, Collective Phenomena in Plasmas (Pergamon, 1982).

Kapetanakos, C. A.

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E 69, 066415 (2004).
[CrossRef]

Karpowicz, N.

X. Lu, N. Karpowicz, and X.-C. Zhang, “Broadband terahertz detection with selected gases,” J. Opt. Soc. Am. B 26, A66-A73 (2009).
[CrossRef]

N. Karpowicz and X.-C. Zhang, “Coherent terahertz echo of tunnel ionization in gases,” Phys. Rev. Lett. 102, 093001 (2009).
[CrossRef] [PubMed]

Kim, K. Y.

Korff, S. A.

S. A. Korff and G. Breit, “Optical dispersion,” Rev. Mod. Phys. 4, 471-503 (1932).
[CrossRef]

Kress, M.

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. II of Course of Theoretical Physics, 4th ed. (Butterworth & Heinemann, 1998).

Lee, C. H.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813-822 (1968).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. II of Course of Theoretical Physics, 4th ed. (Butterworth & Heinemann, 1998).

Liu, Y.

Löffler, T.

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

Lu, X.

Moloney, J. V.

C.-C. Cheng, E. M. Wright, and J. V. Moloney, “Generation of electromagnetic pulses from plasma channels induced by femtosecond light strings,” Phys. Rev. Lett. 87, 213001 (2001).
[CrossRef] [PubMed]

Morgner, U.

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

Moshammer, R.

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

Mysyrowicz, A.

Peck, E. R.

E. R. Peck and D. J. Fisher, “Dispersion of argon,” J. Opt. Soc. Am. A 54, 1362-1364 (1964).
[CrossRef]

Peñano, J. R.

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E 69, 066415 (2004).
[CrossRef]

Petit, S.

A. Proulx, A. Talebpour, S. Petit, and S. L. Chin, “Fast pulsed electric field created from the self-generated filament of a femtosecond Ti:sapphire laser pulse in air,” Opt. Commun. 174, 305-309 (2000).
[CrossRef]

Popov, V. S.

V. S. Popov, “Tunnel and multiphoton ionization of atoms and ions in a strong laser field (Keldysh theory),” Phys. Usp. 47, 855-885 (2004).
[CrossRef]

V. S. Popov, “Multiphoton ionization of atoms by an ultrashort laser pulse,” JETP Lett. 73, 1-5 (2001).
[CrossRef]

Prade, B.

Proulx, A.

A. Proulx, A. Talebpour, S. Petit, and S. L. Chin, “Fast pulsed electric field created from the self-generated filament of a femtosecond Ti:sapphire laser pulse in air,” Opt. Commun. 174, 305-309 (2000).
[CrossRef]

Ready, J. F.

J. F. Ready, Effects of High Power Laser Radiation (Academic, 1971).

Reimann, K.

Reintjes, J.

J. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases, 2nd ed. (Academic, 1984).

Rodriguez, G.

Roskos, H. G.

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

Sagdeev, R. Z.

L. A. Artsimovich and R. Z. Sagdeev, Plasma Physics for Physicists (Atomizdat, 1979) (in Russian).

Sprangle, P.

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E 69, 066415 (2004).
[CrossRef]

P. Sprangle, E. Esarey, and A. Ting, “Nonlinear theory of intense laser-plasma interactions,” Phys. Rev. Lett. 64, 2011-2014 (1990).
[CrossRef] [PubMed]

Sullivan, A.

H. Hamster, A. Sullivan, S. Gordon, and R. W. Falcone, “Short-pulse terahertz radiation from high-intensity-laser-produced plasmas,” Phys. Rev. E 49, 671-677 (1994).
[CrossRef]

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71, 2725-2728 (1993).
[CrossRef] [PubMed]

Talebpour, A.

A. Proulx, A. Talebpour, S. Petit, and S. L. Chin, “Fast pulsed electric field created from the self-generated filament of a femtosecond Ti:sapphire laser pulse in air,” Opt. Commun. 174, 305-309 (2000).
[CrossRef]

Taylor, A. J.

Thomson, M. D.

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

Ting, A.

P. Sprangle, E. Esarey, and A. Ting, “Nonlinear theory of intense laser-plasma interactions,” Phys. Rev. Lett. 64, 2011-2014 (1990).
[CrossRef] [PubMed]

Ullrich, J.

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

Wang, M.

Y. Chen, M. Yamaguchi, M. Wang, and X.-C. Zhang, “Terahertz pulse generation from noble gases,” Appl. Phys. Lett. 91, 251116 (2007).
[CrossRef]

White, W.

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71, 2725-2728 (1993).
[CrossRef] [PubMed]

Woerner, M.

Wright, E. M.

C.-C. Cheng, E. M. Wright, and J. V. Moloney, “Generation of electromagnetic pulses from plasma channels induced by femtosecond light strings,” Phys. Rev. Lett. 87, 213001 (2001).
[CrossRef] [PubMed]

Xie, X.

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96, 075005 (2006).
[CrossRef] [PubMed]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97, 103903 (2006).
[CrossRef] [PubMed]

Yamaguchi, M.

Y. Chen, M. Yamaguchi, M. Wang, and X.-C. Zhang, “Terahertz pulse generation from noble gases,” Appl. Phys. Lett. 91, 251116 (2007).
[CrossRef]

Zhang, X. -C.

N. Karpowicz and X.-C. Zhang, “Coherent terahertz echo of tunnel ionization in gases,” Phys. Rev. Lett. 102, 093001 (2009).
[CrossRef] [PubMed]

X. Lu, N. Karpowicz, and X.-C. Zhang, “Broadband terahertz detection with selected gases,” J. Opt. Soc. Am. B 26, A66-A73 (2009).
[CrossRef]

Y. Chen, M. Yamaguchi, M. Wang, and X.-C. Zhang, “Terahertz pulse generation from noble gases,” Appl. Phys. Lett. 91, 251116 (2007).
[CrossRef]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97, 103903 (2006).
[CrossRef] [PubMed]

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96, 075005 (2006).
[CrossRef] [PubMed]

Zrost, K.

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Opt. Lett. 29, 1120-1122 (2004).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

Y. Chen, M. Yamaguchi, M. Wang, and X.-C. Zhang, “Terahertz pulse generation from noble gases,” Appl. Phys. Lett. 91, 251116 (2007).
[CrossRef]

J. Exp. Theor. Phys. (1)

L. M. Gorbunov and A. A. Frolov, “Emission of low-frequency electromagnetic waves by a short laser pulse in stratified rarefied plasma,” J. Exp. Theor. Phys. 83, 967-973 (1996).

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

E. R. Peck and D. J. Fisher, “Dispersion of argon,” J. Opt. Soc. Am. A 54, 1362-1364 (1964).
[CrossRef]

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

JETP Lett. (1)

V. S. Popov, “Multiphoton ionization of atoms by an ultrashort laser pulse,” JETP Lett. 73, 1-5 (2001).
[CrossRef]

Nat. Phys. (1)

M. Kress, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327-331 (2006).
[CrossRef]

Opt. Commun. (1)

A. Proulx, A. Talebpour, S. Petit, and S. L. Chin, “Fast pulsed electric field created from the self-generated filament of a femtosecond Ti:sapphire laser pulse in air,” Opt. Commun. 174, 305-309 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813-822 (1968).
[CrossRef]

Phys. Rev. E (2)

H. Hamster, A. Sullivan, S. Gordon, and R. W. Falcone, “Short-pulse terahertz radiation from high-intensity-laser-produced plasmas,” Phys. Rev. E 49, 671-677 (1994).
[CrossRef]

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E 69, 066415 (2004).
[CrossRef]

Phys. Rev. Lett. (8)

P. Sprangle, E. Esarey, and A. Ting, “Nonlinear theory of intense laser-plasma interactions,” Phys. Rev. Lett. 64, 2011-2014 (1990).
[CrossRef] [PubMed]

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994-1997 (1993).
[CrossRef] [PubMed]

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71, 2725-2728 (1993).
[CrossRef] [PubMed]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97, 103903 (2006).
[CrossRef] [PubMed]

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96, 075005 (2006).
[CrossRef] [PubMed]

C.-C. Cheng, E. M. Wright, and J. V. Moloney, “Generation of electromagnetic pulses from plasma channels induced by femtosecond light strings,” Phys. Rev. Lett. 87, 213001 (2001).
[CrossRef] [PubMed]

N. Karpowicz and X.-C. Zhang, “Coherent terahertz echo of tunnel ionization in gases,” Phys. Rev. Lett. 102, 093001 (2009).
[CrossRef] [PubMed]

G. A. Askar'yan, “Cherenkov radiation from optical pulses,” Phys. Rev. Lett. 57, 2470 (1986).
[CrossRef] [PubMed]

Phys. Usp. (1)

V. S. Popov, “Tunnel and multiphoton ionization of atoms and ions in a strong laser field (Keldysh theory),” Phys. Usp. 47, 855-885 (2004).
[CrossRef]

Proc. R. Soc. London, Ser. A (2)

C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of krypton and xenon and their relation to those of helium and argon,” Proc. R. Soc. London, Ser. A 84, 2805-2807 (1910).

C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of neon,” Proc. R. Soc. London, Ser. A 83, 149-151 (1910).
[CrossRef]

Rev. Mod. Phys. (1)

S. A. Korff and G. Breit, “Optical dispersion,” Rev. Mod. Phys. 4, 471-503 (1932).
[CrossRef]

Other (8)

This condition can be rewritten as cEω2/8π⪢(α8mc3/32πre3)(J/JH)3/(J/ℏω)2, where α is the fine structure constant, re is the classical electron radius, J/JH is the ratio of the potential of ionization to that of the atom of hydrogen, and J/ℏω is the number of photons required for the ionization. Numerically, this condition means that the laser power flux density should exceed 8.8×1015(J/JH)3/(J/ℏω)2 W/cm2.

J. F. Ready, Effects of High Power Laser Radiation (Academic, 1971).

B. B. Kadomtsev, Collective Phenomena in Plasmas (Pergamon, 1982).

L. A. Artsimovich and R. Z. Sagdeev, Plasma Physics for Physicists (Atomizdat, 1979) (in Russian).

R. Fitzpatrick, The Physics of Plasmas (Lulu, 2008).

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. II of Course of Theoretical Physics, 4th ed. (Butterworth & Heinemann, 1998).

J. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases, 2nd ed. (Academic, 1984).

N. Bloembergen, “Recent progress in four-wave mixing spectroscopy,” in Laser Spectroscopy, H.Walther and K.W.Rothe, eds. (Springer, 1979), Vol. IV, pp. 340-348.

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Figures (2)

Fig. 1
Fig. 1

Angular distribution of the terahertz radiation computed by Eq. (29) for L / λ = 50 : (a) driving force is directed along the plasma filament; (b) driving force is perpendicular to the plasma filament.

Fig. 2
Fig. 2

Dependence of the terahertz field amplitude generated in the Xe versus the gas pressure. In the figure, open squares correspond to 980 mW and the dark triangles correspond to the 600 mW of the overage power of the ω fundamental field.

Equations (66)

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m v t + m ( v ) v = e E + e c v × B γ m v ,
E L = E L x ̂ ,     B L = E L y ̂ ,
E ω = E 1 ( t z / v g ( ω ) ) exp [ i k ( ω ) z i ω t ] ,
E 2 ω = E 2 ( t z / v g ( 2 ω ) ) exp [ i k ( 2 ω ) z i 2 ω t ] ,
m v x t = e E L e v z c E L γ m v x ,
m v y t = γ m v y ,
m v z t = e v x c E L γ m v z .
v x e 2 m [ E ω ( γ i ω ) + E ω ( γ + i ω ) + E 2 ω ( γ i 2 ω ) + E 2 ω ( γ + i 2 ω ) ] .
G z = e v x c E L = e 2 2 m c γ | E ω | 2 ( γ 2 + ω 2 ) + e 2 2 m c γ | E 2 ω | 2 ( γ 2 + 4 ω 2 ) ,
v z c = ( e 2 m c ) 2 [ E ω ( γ i ω ) E ω γ + E ω ( γ + i ω ) E ω γ + E 2 ω ( γ i 2 ω ) E 2 ω γ + E 2 ω ( γ + i 2 ω ) E 2 ω γ ] + ( e 2 m c ) 2 [ E ω ( γ i ω ) E 2 ω ( γ + i ω ) + E ω ( γ + i ω ) E 2 ω ( γ i ω ) + E 2 ω ( γ i 2 ω ) E ω ( γ i ω ) + E 2 ω ( γ + i 2 ω ) E ω ( γ + i ω ) ] + ( e 2 m c ) 2 [ E ω ( γ i ω ) E ω ( γ i 2 ω ) + E ω ( γ + i ω ) E ω ( γ + i 2 ω ) ] + ( e 2 m c ) 2 [ E ω ( γ i ω ) E 2 ω ( γ i 3 ω ) + E ω ( γ + i ω ) E 2 ω ( γ + i 3 ω ) + E 2 ω ( γ i 2 ω ) E ω ( γ i 3 ω ) + E 2 ω ( γ + i 2 ω ) E ω ( γ i 3 ω ) ] + ( e 2 m c ) 2 [ E 2 ω ( γ i 2 ω ) E 2 ω ( γ i 4 ω ) + E 2 ω ( γ + i 2 ω ) E 2 ω ( γ + i 4 ω ) ] .
G x = e v z c E L = e 3 8 m 2 c 2 3 γ 2 ( γ 2 + ω 2 ) ( γ 2 + 4 ω 2 ) [ E ω 2 E 2 ω + c .c . ] ,
E L = 1 2 E ω x ̂ + 1 2 E 2 ω y ̂ + c .c . ,
B L = 1 2 E 2 ω x ̂ + 1 2 E ω y ̂ + c .c .
m v x t = e 2 ( E ω + E ω ) e 2 v z c ( E ω + E ω ) γ m v x ,
m v y t = e 2 ( E 2 ω + E 2 ω ) e 2 v z c ( E 2 ω + E 2 ω ) γ m v y ,
m v z t = e 2 v x c ( E ω + E ω ) + e 2 v y c ( E 2 ω + E 2 ω ) γ m v z .
G y = e 2 v z c ( E 2 ω + E 2 ω ) = e 3 8 m 2 c 2 [ E ω 2 E 2 ω ( γ i ω ) ( γ i 2 ω ) + c .c . ] e 3 16 m 2 c 2 ω 2 [ E ω 2 E 2 ω + c .c . ] ,
E ω = E 1 ( x , y , t z / v g ( ω ) ) exp [ i k ( ω ) z i ω t ] ,
E L = 1 2 E ω x ̂ + c .c . ,
B L = 1 2 k c ω E ω y ̂ 1 2 i c ω E ω × x ̂ + c .c .
F = 1 4 m ( v ω ) v ω + 1 4 e c v ω × [ i c ω E ω × x ̂ ] + c .c .
F = e 2 4 m ω 2 | E ω | 2 ,
c ( E 2 ω 2 / 8 π ) π a 2 ( c / 2 r e ) m c 2 ,
2 ξ t 2 + γ ξ t = e m E + f m .
div   E = 4 π e δ n e .
δ n e t + div ( n e ξ t ) = 0 ,
E = 4 π e n e ξ .
2 ξ t 2 + γ ξ t + ω p 2 ξ = f m ,
Δ z p = τ L 1 / v g ( 2 ω ) 1 / v g ( ω ) 8 ω 2 3 ω p 2 c τ L ,
Δ k p = 2 k ( ω ) k ( 2 ω ) 3 4 ω p 2 c ω .
Δ k g = 2 ω c [ n ω n 2 ω ]
p = 2 π / | Δ k p | = 8 π c ω / 3 ω p 2 ,
ξ = 1 m ω p 2 γ 2 / 4 0 exp [ γ τ / 2 ] sin [ ω p 2 γ 2 / 4 τ ] f ( r , t z / c τ ) d τ .
ξ f ( r , t z / c ) / m ω p 2 ,
ξ Im { f ω p   exp [ ( γ / 2 + i ω p ) ( t z / c ) ] } / 2 m ω p ,
f Ω ( r ) = f ( r , τ ) e i Ω τ d τ ,
j e n e m 0 e γ τ / 2   cos ( ω p τ ) f ( r , t z / c τ ) d τ .
d E d Ω d o = Ω 2 4 π c 3 | n × j Ω , K | 2 ,
j Ω , K = d t d 3 r e i Ω t i K r j ( r , t )
f ( r , t z / c τ ) = d Ω 2 π f Ω ( r ) e i Ω ( t z / c τ )
j Ω = e 2 m n e f Ω ( r ) e i Ω z / c [ 1 γ / 2 + i ω p i Ω + 1 γ / 2 i ω p i Ω ] .
n e N e δ ( x ) δ ( y ) ,
f = d x d y n e f / N e ,     f Ω = d x d y n e f Ω / N e ,
j Ω , K = e N e 2 m f Ω e i ( Ω / c K z ) z d z [ 1 γ / 2 + i ω p i Ω + 1 γ / 2 i ω p i Ω ] ,
f Ω = f ( r ) τ L   sinc [ Ω τ L / 2 ] ,
f ( r ) = f 0 H ( z + L / 2 ) H ( L / 2 z ) cos [ φ + Δ k p z ] ,
d E d Ω d o = Ω 2 4 π c 3 ( e N e L τ L 2 m ) 2 ( γ 2 / 4 + Ω 2 ) sinc 2 ( Ω τ L / 2 ) ( γ 2 / 4 + ( ω p Ω ) 2 ) ( γ 2 / 4 + ( ω p + Ω ) 2 ) { [ sinc ( Ω L c sin 2 θ 2 + Δ k p L 2 ) + sinc ( Ω L c sin 2 θ 2 Δ k p L 2 ) ] 2 cos 2 φ + [ sinc ( Ω L c sin 2 θ 2 + Δ k p L 2 ) sinc ( Ω L c sin 2 θ 2 Δ k p L 2 ) ] 2 sin 2 φ } | n × f 0 | 2 .
d E d Ω d o = Ω 2 4 π c 3 ( e N e L τ L m ) 2 ( γ 2 / 4 + Ω 2 ) sinc 2 ( Ω τ L / 2 ) ( γ 2 / 4 + ( ω p Ω ) 2 ) ( γ 2 / 4 + ( ω p + Ω ) 2 ) sinc 2 ( Ω L c sin 2 θ 2 ) cos 2 φ | n × f 0 | 2 ,
θ 2 λ / L .
E ( t ) = E ω   cos ( ω t ) + E 2 ω   cos ( 2 ω t + φ ) .
v ( t ) = v ω   sin ( ω t ) + α v ω   sin ( 2 ω t + φ ) + v N ,
v N = v ω   sin ( ω t N ) α v ω   sin ( 2 ω t N + φ ) .
ω t N = π N 2 α   sin   φ   cos   π N ,
v N = 3 2 α v ω   sin   φ ,
E N | E ( t N ) | = | E ω | [ 1 + α   cos   π N ] .
n e t + div ( n e v ) = n ̇ ,
t n e v + ( n e v v ) = e n e m E γ n e v + n e f m + n ̇ v 0 .
t v + ( v ) v = e m E γ v + f m + n ̇ n e [ v 0 v ] .
n = t n ̇ ( t ) d t ,
δ n e t + div ( n e ξ t ) = 0 ,
2 ξ t 2 + γ ξ t = e m E + f m + n ̇ n [ v 0 ξ t ] .
2 ξ t 2 + γ ξ t + ω p 2 ξ = f m + n ̇ n [ v 0 ξ t ] ,
2 ξ t 2 + γ ξ t + ω p 2 ξ = 0 ,
ξ = 0 ,
ξ t = 1 n [ n f / m + n ̇ v 0 ] d t ,
ξ = e γ ( t z / v g ) / 2 ω p 2 γ 2 / 4 sin [ ω p 2 γ 2 4 ( t z v g ) ] | ξ t | t = z / v g + 0 .

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