Abstract

The transverse modulational instability, or filamentation, of two collinear waves is investigated, using a coupled nonlinear Schrödinger-equation model. For infinite media it is shown that the presence of the second laser field increases the growth rate of the instability and decreases the scale length of the most unstable filaments. Systems of two copropagating waves are shown to be convectively unstable and systems of two counterpropagating waves are shown to be absolutely unstable, even when the ratio of backward- to forward-wave intensity is small. For two counterpropagating waves in finite media, the threshold intensities for the absolute instability depend only weakly on the ratio of wave intensities. The general theory is applied to the pondermotive filamentation of two light waves in homogeneous plasma.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. E. Max, “Physics of the coronal plasma in laser fusion targets,” in Interaction Laser-Matière, R. Balian and J. C. Adam, eds. (North-Holland, Amsterdam, 1982), pp. 305–410.
  2. R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
  3. B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985).
    [Crossref]
  4. T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43, 267–270 (1979).
    [Crossref]
  5. C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature (London) 311, 525–529 (1984).
    [Crossref]
  6. A. L. Berkhoer and V. E. Zakharov, “Self-excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).
  7. K. P. Das and S. Sihi, “Modulational instability of two transverse waves in a cold plasma,” J. Plasma Phys. 21, 183–191 (1979).
    [Crossref]
  8. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
    [Crossref]
  9. Y. Inoue, “Nonlinear interaction of dispersive waves with equal group velocity,” J. Phys. Soc. Jpn. 43, 243–249 (1977).
    [Crossref]
  10. B. K. Som, M. R. Gupta, and B. Dasgupta, “Coupled nonlinear Schrödinger equation for Langmuir and dispersive ion acoustic waves,” Phys. Lett. A 72, 111–114 (1979).
    [Crossref]
  11. A. G. Litvak and G. M. Fraiman, “Interactions of beams of oppositely traveling electromagnetic waves in a transparent nonlinear medium,” Radiophys. Quantum Electron. 15, 1024–1029 (1972).
    [Crossref]
  12. M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrödinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
    [Crossref]
  13. G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [Crossref] [PubMed]
  14. B. Ghosh and K. P. Das, “Nonlinear interactions of two compressional hydromagnetic waves,” J. Plasma Phys. 39, 215–228 (1988).
    [Crossref]
  15. C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
    [Crossref]
  16. C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T-30, 31–40 (1990).
    [Crossref]
  17. S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
    [Crossref] [PubMed]
  18. G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
    [Crossref] [PubMed]
  19. S. Trillo, S. Wabnitz, G. I. Stegeman, and E. M. Wright, “Parametric amplification and modulational instabilities in dispersive nonlinear directional couplers with relaxing nonlinearity,” J. Opt. Soc. Am. B 6, 889–900 (1989).
    [Crossref]
  20. C. T. Law and A. E. Kaplan, “Dispersion-related multimode instabilities and self-sustained oscillations in nonlinear counterpropagating waves,” Opt. Lett. 14, 734–736 (1989).
    [Crossref] [PubMed]
  21. S. N. Vlasov and V. I. Talanov, “About some features of scattering of signal wave on counterpropagating pump beams under conditions of degenerate four-photon interaction,” in Optical Phase Conjugation in Nonlinear Media, V. I. Bespalov, ed. (USSR Academy of Science, Gorkii, 1979), pp. 85–91; a convenient account of this study is to be found in Ref. 3, pp. 165–167.
  22. W. J. Firth and C. Paré, “Transverse modulational instabilities for counterpropagating beams in Kerr media,” Opt. Lett. 13, 1096–1098 (1988).
    [Crossref] [PubMed]
  23. F. W. Perkins and E. J. Valeo, “Thermal self-focusing of electromagnetic waves in plasmas,” Phys. Rev. Lett. 32, 1234–1237 (1974).
    [Crossref]
  24. W. L. Kruer, “Ponderomotive and thermal filamentation of laser light,” Comments Plasma Phys. Controlled Fusion 9, 63–72 (1985), and references therein.
  25. M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, “Self-focusing of laser beams in plasmas and semiconductors,” in Progress In Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 3, pp. 169–265.
    [Crossref]
  26. W. L. Kruer, The Physics of Laser Plasma Interactions (Addison-Wesley, Redwood City, Calif., 1988).
  27. R. D. Jones, W. C. Mead, S. V. Coggeshall, C. H. Aldrich, J. L. Norton, G. D. Pollak, and J. M. Wallace, “Self-focusing and filamentation of laser light in high Z plasmas,” Phys. Fluids 31, 1249–1272 (1988).
    [Crossref]
  28. V. E. Zakharov and E. I. Schulman, “On additional motion invariants of classical Hamiltonian wave systems,” Physica 29D, 283–320 (1988).
  29. V. E. Zakharov and E. I. Schulman, “To the integrability of the system of two coupled nonlinear Schrödinger equations,” Physica 4D, 270–274 (1982).
  30. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
  31. See, for example, C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
    [Crossref]
  32. B. R. Suydam, “Self-focusing of very powerful laser beams,” in Laser-Induced Damage in Optical Materials, A. J. Glass and A. H. Guenther, eds., Natl. Bur. Stand. (U.S.) Spec. Publ.387, 42–49 (1973).
  33. R. J. Briggs, Electron-Stream Interaction with Plasmas (MIT Press, Cambridge, Mass., 1964), pp. 8–46.
  34. A. Bers, “Space-time evolution of plasma instabilities—absolute and convective,” in Basic Physics, A. A. Galeev and R. N. Sudan, eds., Vol. 1 of Handbook of Plasma Physics, M. N. Rosenbluth and R. Z. Sagdeev, eds. (North-Holland, New York, 1983), pp. 451–517.
  35. P. Huerre, “Spatio-temporal instabilities in closed and open flows,” in Instabilities and Nonequilibrium Structures, E. Tirapegui and K. Villarroel, eds. (Reidel, New York, 1987), pp. 141–177.
    [Crossref]
  36. L. S. Hall and W. Heckrotte, “Instabilities: convective versus absolute,” Phys. Rev. 166, 120–126 (1968).
    [Crossref]
  37. A. Bers, A. K. Ram, and G. Francis, “Relativistic analysis of absolute and convective instability evolutions in three dimensions,” Phys. Rev. Lett. 53, 1457–1460 (1984).
    [Crossref]
  38. A. L. Gaeta, The Institute of Optics, University of Rochester, Rochester, New York 14627 (personal communication, 1989).
  39. G. G. Luther, C. J. McKinstrie, and R. W. Short, “The filamentation of two counterpropagating waves,” presented at the 16th IEEE International Conference on Plasma Science, Buffalo, New York, May 22–24, 1989.
  40. G. Grynberg and J. Paye, “Spatial instability for a standing wave in a nonlinear medium,” Europhys. Lett. 8, 29–33 (1989).
    [Crossref]
  41. E. M. Epperlein, Laboratory for Laser Energetics, 250 East River Road, Rochester, New York 14623 (personal communication, 1989).
  42. N. Tan-No, T. Hoshimiya, and H. Inaba, “Dispersion-free amplification and oscillation in phase-conjugate four-wave mixing in an atomic vapor doublet, IEEE J. Quantum Electron. QE-16, 147–153 (1980).
    [Crossref]
  43. J. Pender and L. Hesselink, “Sodium phase-conjugate oscillator in a combusting environment,” Opt. Lett. 12, 693–695 (1987).
    [Crossref] [PubMed]
  44. G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
    [Crossref]
  45. J. Pender and L. Hesselink, “Conical emission and phase conjugation in atomic sodium vapor,” IEEE J. Quantum Electron. 25, 395–402 (1989).
    [Crossref]
  46. D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
    [Crossref] [PubMed]
  47. P. E. Young, H. A. Baldis, R. P. Drake, E. M. Campbell, and K. G. Estabrook, “Direct evidence of ponderomotive filamentation in a laser-produced plasma,” Phys. Rev. Lett. 61, 2336–2339 (1988).
    [Crossref] [PubMed]
  48. R. J. Deissler and H. J. Brandt, “Generation of counterpropagating nonlinear interacting traveling waves by localized noise,” Phys. Lett. A 130, 293–298 (1988).
    [Crossref]
  49. N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34–43 (1964).
    [Crossref]
  50. D. L. Bobroff, “Coupled-mode analysis of the phonon-photon backward-wave oscillator,” J. Appl. Phys. 36, 1760–1769 (1965).
    [Crossref]
  51. C. J. McKinstrie and A. Simon, “Nonlinear saturation of the absolute stimulated Raman scattering instability in a finite collisional plasma,” Phys. Fluids 29, 1959–1970 (1986).
    [Crossref]

1990 (2)

C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T-30, 31–40 (1990).
[Crossref]

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[Crossref] [PubMed]

1989 (6)

G. Grynberg and J. Paye, “Spatial instability for a standing wave in a nonlinear medium,” Europhys. Lett. 8, 29–33 (1989).
[Crossref]

C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
[Crossref]

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[Crossref] [PubMed]

S. Trillo, S. Wabnitz, G. I. Stegeman, and E. M. Wright, “Parametric amplification and modulational instabilities in dispersive nonlinear directional couplers with relaxing nonlinearity,” J. Opt. Soc. Am. B 6, 889–900 (1989).
[Crossref]

C. T. Law and A. E. Kaplan, “Dispersion-related multimode instabilities and self-sustained oscillations in nonlinear counterpropagating waves,” Opt. Lett. 14, 734–736 (1989).
[Crossref] [PubMed]

J. Pender and L. Hesselink, “Conical emission and phase conjugation in atomic sodium vapor,” IEEE J. Quantum Electron. 25, 395–402 (1989).
[Crossref]

1988 (9)

W. J. Firth and C. Paré, “Transverse modulational instabilities for counterpropagating beams in Kerr media,” Opt. Lett. 13, 1096–1098 (1988).
[Crossref] [PubMed]

B. Ghosh and K. P. Das, “Nonlinear interactions of two compressional hydromagnetic waves,” J. Plasma Phys. 39, 215–228 (1988).
[Crossref]

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[Crossref]

P. E. Young, H. A. Baldis, R. P. Drake, E. M. Campbell, and K. G. Estabrook, “Direct evidence of ponderomotive filamentation in a laser-produced plasma,” Phys. Rev. Lett. 61, 2336–2339 (1988).
[Crossref] [PubMed]

R. J. Deissler and H. J. Brandt, “Generation of counterpropagating nonlinear interacting traveling waves by localized noise,” Phys. Lett. A 130, 293–298 (1988).
[Crossref]

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[Crossref] [PubMed]

R. D. Jones, W. C. Mead, S. V. Coggeshall, C. H. Aldrich, J. L. Norton, G. D. Pollak, and J. M. Wallace, “Self-focusing and filamentation of laser light in high Z plasmas,” Phys. Fluids 31, 1249–1272 (1988).
[Crossref]

V. E. Zakharov and E. I. Schulman, “On additional motion invariants of classical Hamiltonian wave systems,” Physica 29D, 283–320 (1988).

See, for example, C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[Crossref]

1987 (3)

J. Pender and L. Hesselink, “Sodium phase-conjugate oscillator in a combusting environment,” Opt. Lett. 12, 693–695 (1987).
[Crossref] [PubMed]

G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[Crossref] [PubMed]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
[Crossref]

1986 (1)

C. J. McKinstrie and A. Simon, “Nonlinear saturation of the absolute stimulated Raman scattering instability in a finite collisional plasma,” Phys. Fluids 29, 1959–1970 (1986).
[Crossref]

1985 (1)

W. L. Kruer, “Ponderomotive and thermal filamentation of laser light,” Comments Plasma Phys. Controlled Fusion 9, 63–72 (1985), and references therein.

1984 (2)

A. Bers, A. K. Ram, and G. Francis, “Relativistic analysis of absolute and convective instability evolutions in three dimensions,” Phys. Rev. Lett. 53, 1457–1460 (1984).
[Crossref]

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature (London) 311, 525–529 (1984).
[Crossref]

1982 (1)

V. E. Zakharov and E. I. Schulman, “To the integrability of the system of two coupled nonlinear Schrödinger equations,” Physica 4D, 270–274 (1982).

1981 (1)

M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrödinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
[Crossref]

1980 (1)

N. Tan-No, T. Hoshimiya, and H. Inaba, “Dispersion-free amplification and oscillation in phase-conjugate four-wave mixing in an atomic vapor doublet, IEEE J. Quantum Electron. QE-16, 147–153 (1980).
[Crossref]

1979 (3)

T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43, 267–270 (1979).
[Crossref]

K. P. Das and S. Sihi, “Modulational instability of two transverse waves in a cold plasma,” J. Plasma Phys. 21, 183–191 (1979).
[Crossref]

B. K. Som, M. R. Gupta, and B. Dasgupta, “Coupled nonlinear Schrödinger equation for Langmuir and dispersive ion acoustic waves,” Phys. Lett. A 72, 111–114 (1979).
[Crossref]

1977 (1)

Y. Inoue, “Nonlinear interaction of dispersive waves with equal group velocity,” J. Phys. Soc. Jpn. 43, 243–249 (1977).
[Crossref]

1974 (1)

F. W. Perkins and E. J. Valeo, “Thermal self-focusing of electromagnetic waves in plasmas,” Phys. Rev. Lett. 32, 1234–1237 (1974).
[Crossref]

1972 (1)

A. G. Litvak and G. M. Fraiman, “Interactions of beams of oppositely traveling electromagnetic waves in a transparent nonlinear medium,” Radiophys. Quantum Electron. 15, 1024–1029 (1972).
[Crossref]

1970 (1)

A. L. Berkhoer and V. E. Zakharov, “Self-excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

1968 (1)

L. S. Hall and W. Heckrotte, “Instabilities: convective versus absolute,” Phys. Rev. 166, 120–126 (1968).
[Crossref]

1965 (1)

D. L. Bobroff, “Coupled-mode analysis of the phonon-photon backward-wave oscillator,” J. Appl. Phys. 36, 1760–1769 (1965).
[Crossref]

1964 (1)

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34–43 (1964).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[Crossref] [PubMed]

G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[Crossref] [PubMed]

Aldrich, C. H.

R. D. Jones, W. C. Mead, S. V. Coggeshall, C. H. Aldrich, J. L. Norton, G. D. Pollak, and J. M. Wallace, “Self-focusing and filamentation of laser light in high Z plasmas,” Phys. Fluids 31, 1249–1272 (1988).
[Crossref]

Alfano, R. R.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[Crossref] [PubMed]

Baldeck, P. L.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[Crossref] [PubMed]

Baldis, H. A.

P. E. Young, H. A. Baldis, R. P. Drake, E. M. Campbell, and K. G. Estabrook, “Direct evidence of ponderomotive filamentation in a laser-produced plasma,” Phys. Rev. Lett. 61, 2336–2339 (1988).
[Crossref] [PubMed]

Berkhoer, A. L.

A. L. Berkhoer and V. E. Zakharov, “Self-excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Bers, A.

A. Bers, A. K. Ram, and G. Francis, “Relativistic analysis of absolute and convective instability evolutions in three dimensions,” Phys. Rev. Lett. 53, 1457–1460 (1984).
[Crossref]

A. Bers, “Space-time evolution of plasma instabilities—absolute and convective,” in Basic Physics, A. A. Galeev and R. N. Sudan, eds., Vol. 1 of Handbook of Plasma Physics, M. N. Rosenbluth and R. Z. Sagdeev, eds. (North-Holland, New York, 1983), pp. 451–517.

Bingham, R.

C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
[Crossref]

Bloch, D.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[Crossref]

Bobroff, D. L.

D. L. Bobroff, “Coupled-mode analysis of the phonon-photon backward-wave oscillator,” J. Appl. Phys. 36, 1760–1769 (1965).
[Crossref]

Boyd, R. W.

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[Crossref] [PubMed]

Brandt, H. J.

R. J. Deissler and H. J. Brandt, “Generation of counterpropagating nonlinear interacting traveling waves by localized noise,” Phys. Lett. A 130, 293–298 (1988).
[Crossref]

Briggs, R. J.

R. J. Briggs, Electron-Stream Interaction with Plasmas (MIT Press, Cambridge, Mass., 1964), pp. 8–46.

Campbell, E. M.

P. E. Young, H. A. Baldis, R. P. Drake, E. M. Campbell, and K. G. Estabrook, “Direct evidence of ponderomotive filamentation in a laser-produced plasma,” Phys. Rev. Lett. 61, 2336–2339 (1988).
[Crossref] [PubMed]

Coggeshall, S. V.

R. D. Jones, W. C. Mead, S. V. Coggeshall, C. H. Aldrich, J. L. Norton, G. D. Pollak, and J. M. Wallace, “Self-focusing and filamentation of laser light in high Z plasmas,” Phys. Fluids 31, 1249–1272 (1988).
[Crossref]

Das, K. P.

B. Ghosh and K. P. Das, “Nonlinear interactions of two compressional hydromagnetic waves,” J. Plasma Phys. 39, 215–228 (1988).
[Crossref]

K. P. Das and S. Sihi, “Modulational instability of two transverse waves in a cold plasma,” J. Plasma Phys. 21, 183–191 (1979).
[Crossref]

Dasgupta, B.

M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrödinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
[Crossref]

B. K. Som, M. R. Gupta, and B. Dasgupta, “Coupled nonlinear Schrödinger equation for Langmuir and dispersive ion acoustic waves,” Phys. Lett. A 72, 111–114 (1979).
[Crossref]

Dawson, J. M.

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature (London) 311, 525–529 (1984).
[Crossref]

T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43, 267–270 (1979).
[Crossref]

Deissler, R. J.

R. J. Deissler and H. J. Brandt, “Generation of counterpropagating nonlinear interacting traveling waves by localized noise,” Phys. Lett. A 130, 293–298 (1988).
[Crossref]

Drake, R. P.

P. E. Young, H. A. Baldis, R. P. Drake, E. M. Campbell, and K. G. Estabrook, “Direct evidence of ponderomotive filamentation in a laser-produced plasma,” Phys. Rev. Lett. 61, 2336–2339 (1988).
[Crossref] [PubMed]

Ducloy, M.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[Crossref]

Epperlein, E. M.

E. M. Epperlein, Laboratory for Laser Energetics, 250 East River Road, Rochester, New York 14623 (personal communication, 1989).

Estabrook, K. G.

P. E. Young, H. A. Baldis, R. P. Drake, E. M. Campbell, and K. G. Estabrook, “Direct evidence of ponderomotive filamentation in a laser-produced plasma,” Phys. Rev. Lett. 61, 2336–2339 (1988).
[Crossref] [PubMed]

Firth, W. J.

Forslund, D. W.

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature (London) 311, 525–529 (1984).
[Crossref]

Fraiman, G. M.

A. G. Litvak and G. M. Fraiman, “Interactions of beams of oppositely traveling electromagnetic waves in a transparent nonlinear medium,” Radiophys. Quantum Electron. 15, 1024–1029 (1972).
[Crossref]

Francis, G.

A. Bers, A. K. Ram, and G. Francis, “Relativistic analysis of absolute and convective instability evolutions in three dimensions,” Phys. Rev. Lett. 53, 1457–1460 (1984).
[Crossref]

Gaeta, A. L.

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[Crossref] [PubMed]

A. L. Gaeta, The Institute of Optics, University of Rochester, Rochester, New York 14627 (personal communication, 1989).

Gauthier, D. J.

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[Crossref] [PubMed]

Ghatak, A. K.

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, “Self-focusing of laser beams in plasmas and semiconductors,” in Progress In Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 3, pp. 169–265.
[Crossref]

Ghosh, B.

B. Ghosh and K. P. Das, “Nonlinear interactions of two compressional hydromagnetic waves,” J. Plasma Phys. 39, 215–228 (1988).
[Crossref]

Grynberg, G.

G. Grynberg and J. Paye, “Spatial instability for a standing wave in a nonlinear medium,” Europhys. Lett. 8, 29–33 (1989).
[Crossref]

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[Crossref]

Gupta, M. R.

M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrödinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
[Crossref]

B. K. Som, M. R. Gupta, and B. Dasgupta, “Coupled nonlinear Schrödinger equation for Langmuir and dispersive ion acoustic waves,” Phys. Lett. A 72, 111–114 (1979).
[Crossref]

Hall, L. S.

L. S. Hall and W. Heckrotte, “Instabilities: convective versus absolute,” Phys. Rev. 166, 120–126 (1968).
[Crossref]

Heckrotte, W.

L. S. Hall and W. Heckrotte, “Instabilities: convective versus absolute,” Phys. Rev. 166, 120–126 (1968).
[Crossref]

Hesselink, L.

J. Pender and L. Hesselink, “Conical emission and phase conjugation in atomic sodium vapor,” IEEE J. Quantum Electron. 25, 395–402 (1989).
[Crossref]

J. Pender and L. Hesselink, “Sodium phase-conjugate oscillator in a combusting environment,” Opt. Lett. 12, 693–695 (1987).
[Crossref] [PubMed]

Hoshimiya, T.

N. Tan-No, T. Hoshimiya, and H. Inaba, “Dispersion-free amplification and oscillation in phase-conjugate four-wave mixing in an atomic vapor doublet, IEEE J. Quantum Electron. QE-16, 147–153 (1980).
[Crossref]

Huerre, P.

P. Huerre, “Spatio-temporal instabilities in closed and open flows,” in Instabilities and Nonequilibrium Structures, E. Tirapegui and K. Villarroel, eds. (Reidel, New York, 1987), pp. 141–177.
[Crossref]

Inaba, H.

N. Tan-No, T. Hoshimiya, and H. Inaba, “Dispersion-free amplification and oscillation in phase-conjugate four-wave mixing in an atomic vapor doublet, IEEE J. Quantum Electron. QE-16, 147–153 (1980).
[Crossref]

Inoue, Y.

Y. Inoue, “Nonlinear interaction of dispersive waves with equal group velocity,” J. Phys. Soc. Jpn. 43, 243–249 (1977).
[Crossref]

Jones, R. D.

R. D. Jones, W. C. Mead, S. V. Coggeshall, C. H. Aldrich, J. L. Norton, G. D. Pollak, and J. M. Wallace, “Self-focusing and filamentation of laser light in high Z plasmas,” Phys. Fluids 31, 1249–1272 (1988).
[Crossref]

Joshi, C.

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature (London) 311, 525–529 (1984).
[Crossref]

Kaplan, A. E.

Katsouleas, T.

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature (London) 311, 525–529 (1984).
[Crossref]

Kindel, J. M.

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature (London) 311, 525–529 (1984).
[Crossref]

Kroll, N. M.

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34–43 (1964).
[Crossref]

Kruer, W. L.

W. L. Kruer, “Ponderomotive and thermal filamentation of laser light,” Comments Plasma Phys. Controlled Fusion 9, 63–72 (1985), and references therein.

W. L. Kruer, The Physics of Laser Plasma Interactions (Addison-Wesley, Redwood City, Calif., 1988).

Law, C. T.

Le Bihan, E.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[Crossref]

Le Boiteux, S.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[Crossref]

Leite, J. R. R.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[Crossref]

Litvak, A. G.

A. G. Litvak and G. M. Fraiman, “Interactions of beams of oppositely traveling electromagnetic waves in a transparent nonlinear medium,” Radiophys. Quantum Electron. 15, 1024–1029 (1972).
[Crossref]

Luther, G. G.

C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T-30, 31–40 (1990).
[Crossref]

See, for example, C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[Crossref]

G. G. Luther, C. J. McKinstrie, and R. W. Short, “The filamentation of two counterpropagating waves,” presented at the 16th IEEE International Conference on Plasma Science, Buffalo, New York, May 22–24, 1989.

Malcuit, M. S.

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[Crossref] [PubMed]

Max, C. E.

C. E. Max, “Physics of the coronal plasma in laser fusion targets,” in Interaction Laser-Matière, R. Balian and J. C. Adam, eds. (North-Holland, Amsterdam, 1982), pp. 305–410.

McKinstrie, C. J.

C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T-30, 31–40 (1990).
[Crossref]

C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
[Crossref]

See, for example, C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[Crossref]

C. J. McKinstrie and A. Simon, “Nonlinear saturation of the absolute stimulated Raman scattering instability in a finite collisional plasma,” Phys. Fluids 29, 1959–1970 (1986).
[Crossref]

G. G. Luther, C. J. McKinstrie, and R. W. Short, “The filamentation of two counterpropagating waves,” presented at the 16th IEEE International Conference on Plasma Science, Buffalo, New York, May 22–24, 1989.

Mead, W. C.

R. D. Jones, W. C. Mead, S. V. Coggeshall, C. H. Aldrich, J. L. Norton, G. D. Pollak, and J. M. Wallace, “Self-focusing and filamentation of laser light in high Z plasmas,” Phys. Fluids 31, 1249–1272 (1988).
[Crossref]

Menyuk, C. R.

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
[Crossref]

Mori, W. B.

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature (London) 311, 525–529 (1984).
[Crossref]

Norton, J. L.

R. D. Jones, W. C. Mead, S. V. Coggeshall, C. H. Aldrich, J. L. Norton, G. D. Pollak, and J. M. Wallace, “Self-focusing and filamentation of laser light in high Z plasmas,” Phys. Fluids 31, 1249–1272 (1988).
[Crossref]

Paré, C.

Paye, J.

G. Grynberg and J. Paye, “Spatial instability for a standing wave in a nonlinear medium,” Europhys. Lett. 8, 29–33 (1989).
[Crossref]

Pender, J.

J. Pender and L. Hesselink, “Conical emission and phase conjugation in atomic sodium vapor,” IEEE J. Quantum Electron. 25, 395–402 (1989).
[Crossref]

J. Pender and L. Hesselink, “Sodium phase-conjugate oscillator in a combusting environment,” Opt. Lett. 12, 693–695 (1987).
[Crossref] [PubMed]

Perkins, F. W.

F. W. Perkins and E. J. Valeo, “Thermal self-focusing of electromagnetic waves in plasmas,” Phys. Rev. Lett. 32, 1234–1237 (1974).
[Crossref]

Pilipetsky, N. F.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985).
[Crossref]

Pollak, G. D.

R. D. Jones, W. C. Mead, S. V. Coggeshall, C. H. Aldrich, J. L. Norton, G. D. Pollak, and J. M. Wallace, “Self-focusing and filamentation of laser light in high Z plasmas,” Phys. Fluids 31, 1249–1272 (1988).
[Crossref]

Ram, A. K.

A. Bers, A. K. Ram, and G. Francis, “Relativistic analysis of absolute and convective instability evolutions in three dimensions,” Phys. Rev. Lett. 53, 1457–1460 (1984).
[Crossref]

Schulman, E. I.

V. E. Zakharov and E. I. Schulman, “On additional motion invariants of classical Hamiltonian wave systems,” Physica 29D, 283–320 (1988).

V. E. Zakharov and E. I. Schulman, “To the integrability of the system of two coupled nonlinear Schrödinger equations,” Physica 4D, 270–274 (1982).

Shkunov, V. V.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985).
[Crossref]

Short, R. W.

G. G. Luther, C. J. McKinstrie, and R. W. Short, “The filamentation of two counterpropagating waves,” presented at the 16th IEEE International Conference on Plasma Science, Buffalo, New York, May 22–24, 1989.

Sihi, S.

K. P. Das and S. Sihi, “Modulational instability of two transverse waves in a cold plasma,” J. Plasma Phys. 21, 183–191 (1979).
[Crossref]

Simon, A.

C. J. McKinstrie and A. Simon, “Nonlinear saturation of the absolute stimulated Raman scattering instability in a finite collisional plasma,” Phys. Fluids 29, 1959–1970 (1986).
[Crossref]

Simoneau, P.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[Crossref]

Sodha, M. S.

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, “Self-focusing of laser beams in plasmas and semiconductors,” in Progress In Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 3, pp. 169–265.
[Crossref]

Som, B. K.

M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrödinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
[Crossref]

B. K. Som, M. R. Gupta, and B. Dasgupta, “Coupled nonlinear Schrödinger equation for Langmuir and dispersive ion acoustic waves,” Phys. Lett. A 72, 111–114 (1979).
[Crossref]

Stegeman, G. I.

Suydam, B. R.

B. R. Suydam, “Self-focusing of very powerful laser beams,” in Laser-Induced Damage in Optical Materials, A. J. Glass and A. H. Guenther, eds., Natl. Bur. Stand. (U.S.) Spec. Publ.387, 42–49 (1973).

Tajima, T.

T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43, 267–270 (1979).
[Crossref]

Talanov, V. I.

S. N. Vlasov and V. I. Talanov, “About some features of scattering of signal wave on counterpropagating pump beams under conditions of degenerate four-photon interaction,” in Optical Phase Conjugation in Nonlinear Media, V. I. Bespalov, ed. (USSR Academy of Science, Gorkii, 1979), pp. 85–91; a convenient account of this study is to be found in Ref. 3, pp. 165–167.

Tan-No, N.

N. Tan-No, T. Hoshimiya, and H. Inaba, “Dispersion-free amplification and oscillation in phase-conjugate four-wave mixing in an atomic vapor doublet, IEEE J. Quantum Electron. QE-16, 147–153 (1980).
[Crossref]

Trillo, S.

Tripathi, V. K.

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, “Self-focusing of laser beams in plasmas and semiconductors,” in Progress In Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 3, pp. 169–265.
[Crossref]

Valeo, E. J.

F. W. Perkins and E. J. Valeo, “Thermal self-focusing of electromagnetic waves in plasmas,” Phys. Rev. Lett. 32, 1234–1237 (1974).
[Crossref]

Verkerk, P.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[Crossref]

Vlasov, S. N.

S. N. Vlasov and V. I. Talanov, “About some features of scattering of signal wave on counterpropagating pump beams under conditions of degenerate four-photon interaction,” in Optical Phase Conjugation in Nonlinear Media, V. I. Bespalov, ed. (USSR Academy of Science, Gorkii, 1979), pp. 85–91; a convenient account of this study is to be found in Ref. 3, pp. 165–167.

Wabnitz, S.

Wallace, J. M.

R. D. Jones, W. C. Mead, S. V. Coggeshall, C. H. Aldrich, J. L. Norton, G. D. Pollak, and J. M. Wallace, “Self-focusing and filamentation of laser light in high Z plasmas,” Phys. Fluids 31, 1249–1272 (1988).
[Crossref]

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

Wright, E. M.

Young, P. E.

P. E. Young, H. A. Baldis, R. P. Drake, E. M. Campbell, and K. G. Estabrook, “Direct evidence of ponderomotive filamentation in a laser-produced plasma,” Phys. Rev. Lett. 61, 2336–2339 (1988).
[Crossref] [PubMed]

Zakharov, V. E.

V. E. Zakharov and E. I. Schulman, “On additional motion invariants of classical Hamiltonian wave systems,” Physica 29D, 283–320 (1988).

V. E. Zakharov and E. I. Schulman, “To the integrability of the system of two coupled nonlinear Schrödinger equations,” Physica 4D, 270–274 (1982).

A. L. Berkhoer and V. E. Zakharov, “Self-excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985).
[Crossref]

Comments Plasma Phys. Controlled Fusion (1)

W. L. Kruer, “Ponderomotive and thermal filamentation of laser light,” Comments Plasma Phys. Controlled Fusion 9, 63–72 (1985), and references therein.

Europhys. Lett. (1)

G. Grynberg and J. Paye, “Spatial instability for a standing wave in a nonlinear medium,” Europhys. Lett. 8, 29–33 (1989).
[Crossref]

IEEE J. Quantum Electron. (3)

N. Tan-No, T. Hoshimiya, and H. Inaba, “Dispersion-free amplification and oscillation in phase-conjugate four-wave mixing in an atomic vapor doublet, IEEE J. Quantum Electron. QE-16, 147–153 (1980).
[Crossref]

J. Pender and L. Hesselink, “Conical emission and phase conjugation in atomic sodium vapor,” IEEE J. Quantum Electron. 25, 395–402 (1989).
[Crossref]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
[Crossref]

J. Appl. Phys. (2)

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34–43 (1964).
[Crossref]

D. L. Bobroff, “Coupled-mode analysis of the phonon-photon backward-wave oscillator,” J. Appl. Phys. 36, 1760–1769 (1965).
[Crossref]

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

J. Phys. Soc. Jpn. (1)

Y. Inoue, “Nonlinear interaction of dispersive waves with equal group velocity,” J. Phys. Soc. Jpn. 43, 243–249 (1977).
[Crossref]

J. Plasma Phys. (3)

K. P. Das and S. Sihi, “Modulational instability of two transverse waves in a cold plasma,” J. Plasma Phys. 21, 183–191 (1979).
[Crossref]

M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrödinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
[Crossref]

B. Ghosh and K. P. Das, “Nonlinear interactions of two compressional hydromagnetic waves,” J. Plasma Phys. 39, 215–228 (1988).
[Crossref]

Nature (London) (1)

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature (London) 311, 525–529 (1984).
[Crossref]

Opt. Commun. (1)

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[Crossref]

Opt. Lett. (3)

Phys. Fluids (2)

R. D. Jones, W. C. Mead, S. V. Coggeshall, C. H. Aldrich, J. L. Norton, G. D. Pollak, and J. M. Wallace, “Self-focusing and filamentation of laser light in high Z plasmas,” Phys. Fluids 31, 1249–1272 (1988).
[Crossref]

C. J. McKinstrie and A. Simon, “Nonlinear saturation of the absolute stimulated Raman scattering instability in a finite collisional plasma,” Phys. Fluids 29, 1959–1970 (1986).
[Crossref]

Phys. Fluids B (1)

C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
[Crossref]

Phys. Lett. A (3)

B. K. Som, M. R. Gupta, and B. Dasgupta, “Coupled nonlinear Schrödinger equation for Langmuir and dispersive ion acoustic waves,” Phys. Lett. A 72, 111–114 (1979).
[Crossref]

See, for example, C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[Crossref]

R. J. Deissler and H. J. Brandt, “Generation of counterpropagating nonlinear interacting traveling waves by localized noise,” Phys. Lett. A 130, 293–298 (1988).
[Crossref]

Phys. Rev. (1)

L. S. Hall and W. Heckrotte, “Instabilities: convective versus absolute,” Phys. Rev. 166, 120–126 (1968).
[Crossref]

Phys. Rev. A (2)

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[Crossref] [PubMed]

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (6)

G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[Crossref] [PubMed]

T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43, 267–270 (1979).
[Crossref]

A. Bers, A. K. Ram, and G. Francis, “Relativistic analysis of absolute and convective instability evolutions in three dimensions,” Phys. Rev. Lett. 53, 1457–1460 (1984).
[Crossref]

F. W. Perkins and E. J. Valeo, “Thermal self-focusing of electromagnetic waves in plasmas,” Phys. Rev. Lett. 32, 1234–1237 (1974).
[Crossref]

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[Crossref] [PubMed]

P. E. Young, H. A. Baldis, R. P. Drake, E. M. Campbell, and K. G. Estabrook, “Direct evidence of ponderomotive filamentation in a laser-produced plasma,” Phys. Rev. Lett. 61, 2336–2339 (1988).
[Crossref] [PubMed]

Phys. Scr. (1)

C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T-30, 31–40 (1990).
[Crossref]

Physica (2)

V. E. Zakharov and E. I. Schulman, “On additional motion invariants of classical Hamiltonian wave systems,” Physica 29D, 283–320 (1988).

V. E. Zakharov and E. I. Schulman, “To the integrability of the system of two coupled nonlinear Schrödinger equations,” Physica 4D, 270–274 (1982).

Radiophys. Quantum Electron. (1)

A. G. Litvak and G. M. Fraiman, “Interactions of beams of oppositely traveling electromagnetic waves in a transparent nonlinear medium,” Radiophys. Quantum Electron. 15, 1024–1029 (1972).
[Crossref]

Sov. Phys. JETP (1)

A. L. Berkhoer and V. E. Zakharov, “Self-excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Other (14)

C. E. Max, “Physics of the coronal plasma in laser fusion targets,” in Interaction Laser-Matière, R. Balian and J. C. Adam, eds. (North-Holland, Amsterdam, 1982), pp. 305–410.

R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1985).
[Crossref]

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

S. N. Vlasov and V. I. Talanov, “About some features of scattering of signal wave on counterpropagating pump beams under conditions of degenerate four-photon interaction,” in Optical Phase Conjugation in Nonlinear Media, V. I. Bespalov, ed. (USSR Academy of Science, Gorkii, 1979), pp. 85–91; a convenient account of this study is to be found in Ref. 3, pp. 165–167.

A. L. Gaeta, The Institute of Optics, University of Rochester, Rochester, New York 14627 (personal communication, 1989).

G. G. Luther, C. J. McKinstrie, and R. W. Short, “The filamentation of two counterpropagating waves,” presented at the 16th IEEE International Conference on Plasma Science, Buffalo, New York, May 22–24, 1989.

B. R. Suydam, “Self-focusing of very powerful laser beams,” in Laser-Induced Damage in Optical Materials, A. J. Glass and A. H. Guenther, eds., Natl. Bur. Stand. (U.S.) Spec. Publ.387, 42–49 (1973).

R. J. Briggs, Electron-Stream Interaction with Plasmas (MIT Press, Cambridge, Mass., 1964), pp. 8–46.

A. Bers, “Space-time evolution of plasma instabilities—absolute and convective,” in Basic Physics, A. A. Galeev and R. N. Sudan, eds., Vol. 1 of Handbook of Plasma Physics, M. N. Rosenbluth and R. Z. Sagdeev, eds. (North-Holland, New York, 1983), pp. 451–517.

P. Huerre, “Spatio-temporal instabilities in closed and open flows,” in Instabilities and Nonequilibrium Structures, E. Tirapegui and K. Villarroel, eds. (Reidel, New York, 1987), pp. 141–177.
[Crossref]

E. M. Epperlein, Laboratory for Laser Energetics, 250 East River Road, Rochester, New York 14623 (personal communication, 1989).

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, “Self-focusing of laser beams in plasmas and semiconductors,” in Progress In Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 3, pp. 169–265.
[Crossref]

W. L. Kruer, The Physics of Laser Plasma Interactions (Addison-Wesley, Redwood City, Calif., 1988).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Temporal growth rate of sinusoidal perturbations in wave amplitude plotted as a function of transverse wave number for the case in which μλ11 is positive and λ1211 is equal to 2. The lower curve corresponds to the transverse modulational instability of a single wave, while the upper curve corresponds to the transverse modulational instability of two equal-amplitude copropagating waves. The temporal growth rates are normalized to |λ11A12|, and the transverse wave numbers are normalized to (λ11/μ)1/2 |A1|, their optimal values for the single-wave instability.

Fig. 2
Fig. 2

Filament amplitudes G± plotted as functions of position for three different values of the wave-damping coefficients: (a) ν = 0.01γ±, (b) ν = 0.05γ±, (c) ν = 0.10γ±. The solid curves represent the exact solutions, while the dashed curves represent the idealized solutions that neglect the effects of pump-wave damping.

Fig. 3
Fig. 3

Imaginary part of the dispersion surface ω(k, k) displayed for σ1 equal to 1: (a) R = 1.0, (b) R = 0.1.

Fig. 4
Fig. 4

Impulse response of a linear system plotted as a function of position at several successive times: (a) the linear system is convectively unstable, (b) the linear system is absolutely unstable.

Fig. 5
Fig. 5

Impulse-response curves of the transverse modulational instability of two collinear waves displayed for the case in which σ is equal to 1, is equal to 2, and υd is equal to 10. The observation-frame velocity is measured relative to υs.

Fig. 6
Fig. 6

(a) Growth rate, (b) velocity, and (c) transverse wave number at the peak of the impulse-response curves plotted as functions of the intensity ratio R for the parameters given in Fig. 5. The velocity of the peak of the impulse response is measured relative to υ1.

Fig. 7
Fig. 7

Pole structure in the complex plane of the longitudinal wave number for an intensity ratio R of 0.1 and a reference-frame velocity of υs. Seven fixed values of the real part of the frequency are mapped into this complex wave-number space by the coupled dispersion equation as the imaginary part of the frequency is varied between zero and twice the maximally unstable value. The imaginary parts of the frequency are zero at the annotated ends, and their real parts increase in equally spaced steps starting with the lines labeled 1. The lines labeled 4 contain the pinching pole. The paths for the other pair of roots are not shown.

Fig. 8
Fig. 8

Modulus of the single-wave dispersion function D2(ω, k, k), evaluated at the pinch-point frequencies and wave vectors in an observation frame moving with velocity υs, plotted as functions of R. It follows that the source function [Eq. (36)] tends to zero as R tends to zero, for arbitrary initial conditions.

Fig. 9
Fig. 9

Impulse-response curves of the transverse modulated inability of two collinear waves displayed for the case in which σ is equal to −1, is equal to 2, and υd is equal to 10. The dimensionless transverse wave number k is equal to 5.3 for all values of R, and the observation-frame velocity is measured relative to υs.

Fig. 10
Fig. 10

Peak temporal growth rate of the impulse response plotted as a function of the intensity ratio R for the parameters given in Fig. 9.

Fig. 11
Fig. 11

Static threshold intensity γ1l/υ in a finite medium plotted as a function of the transverse wave number |μ|k2/υ for the case in which σ is equal to 1 and is equal to 2: (a) R = 1.00 (b) R = 0.67, (c) R = 0.33, (d) R = 0.10.

Fig. 12
Fig. 12

Static threshold intensity γ1l/υ in a finite medium plotted as a function of the transverse wave number |μ|k2l/υ for the case in which σ is equal to 1 and is equal to 2: (a) R = 10−1, (b) R = 10−2, (c) R = 10−3, (d) R = 10−4.

Fig. 13
Fig. 13

Static threshold intensity γ1l/υ plotted as a function of the pump-wave intensity ratio R for the case in which σ is equal to 1 and is equal to 2. Curve 1 corresponds to a transverse wave number |μ|k2l/υ of 3.0, while curve 2 corresponds to a transverse wave number |μ|k2l/υ of 9.0. The static threshold intensity is given by the solid curve.

Fig. 14
Fig. 14

Static threshold intensity γ1l/υ in a finite medium plotted as a function of the transverse wave number |μ|k2l/υ for the case in which σ is equal to −1 and is equal to 2: (a) R = 1.00, (b) R = 0.67, (c) R = 0.33, (d) R = 0.10.

Equations (106)

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

ω 2 = ω e 2 + c 2 k 2 ,
ω e 2 = 4 π e 2 n e / m e
t n s + · ( n s V s ) = 0 ,
t V s + ( V s · ) V s = P s n s m s + q s m s ( E + 1 c V s × B ) ,
ρ = s n s q s , J = s n s q s V s ,
V = e m e c A ,
[ t t + ω e 2 c 2 ( 2 · ) ] V = ω e 2 n V ,
( t t c s 2 2 ) n = Z m e 2 m i 2 V 2 ,
V = 1 2 { V 1 ( x , t ) exp [ i ϕ 1 ( x , t ) ] + V 2 ( x , t ) exp [ i ϕ 2 ( x , t ) ] + c . c . } ,
ϕ j = k j ( 0 ) · x ω j ( 0 ) t
( t t c s 2 2 ) n = Z m e 8 m i 2 [ | V 1 | 2 + | | V 2 | 2 + V 1 · V 2 * exp ( ϕ 1 ϕ 2 ) + V 1 * · V 2 exp ( ϕ 2 ϕ 1 ) ] .
[ 2 i ( ω 1 ( 0 ) t + c 2 k 1 ( 0 ) · ) + c 2 2 ] V 1 = ω e 2 4 υ e 2 ( | V 1 | 2 + | V 2 | 2 ) V 1 , [ 2 i ( ω 2 ( 0 ) t + c 2 k 2 ( 0 ) · ) + c 2 2 ] V 2 = ω e 2 4 υ e 2 ( | V 2 | 2 + | V 1 | 2 ) V 2 ,
[ i ( t + υ 1 z ) + μ 1 2 + λ 11 | A 1 | 2 + λ 12 | A 2 | 2 ] A 1 = 0 , [ i ( t + υ 2 z ) + μ 2 2 + λ 21 | A 1 | 2 + λ 22 | A 2 | 2 ] A 2 = 0 ,
A j = V j υ e , υ j = c 2 k j ( 0 ) ω j ( 0 ) , μ j = c 2 2 ω j ( 0 ) , λ j l = ω e 2 8 ω j ( 0 ) ( 1 δ j l ) ,
τ = t z / υ , z ,
( i υ z + μ 2 + λ 11 | A 1 | 2 ) A 1 = 0
A 1 e ( τ , z ) = A 1 ( τ ) exp [ i λ 11 | A 1 ( τ ) | 2 z / υ ] ,
A 1 ( t , x ) = [ A 1 e ( τ ) + A 1 ( 1 ) ( τ , z ) ϕ ( x ) ] exp [ i λ 11 | A 1 ( τ ) | 2 z / υ ]
( 2 + k 2 ) ϕ = 0
( i υ z μ k 2 + λ 11 | A 1 | 2 ) A 1 ( 1 ) + λ 11 | A 1 | 2 A 1 ( 1 ) * = 0.
[ ( υ z ) 2 μ k 2 ( 2 λ 11 | A 1 | 2 μ k 2 ) ] A 1 r ( 1 ) = 0 ,
υ z A 1 r ( 1 ) μ k 2 A 1 i ( 1 ) = 0 ,
A 1 i ( 1 ) ( τ , 0 ) = 0 ,
A 1 r ( 1 ) ( t , z ) = A 1 r ( 1 ) ( τ , 0 ) cosh ( γ z / υ ) ,
γ 2 ( τ ) = μ k 2 ( 2 λ 11 | A 1 ( τ ) | 2 μ k 2 ) .
A 1 e = A 1 ( τ ) exp { i [ λ 11 | A 1 ( τ ) | 2 + λ 12 | A 2 ( τ ) | 2 ] z / υ } , A 2 e = A 2 ( τ ) exp { i [ λ 21 | A 1 ( τ ) | 2 + λ 22 | A 2 ( τ ) | 2 ] z / υ } ,
[ ( υ z 2 μ k 2 ( 2 λ 11 | A 1 | 2 μ k 2 ) ] A 1 r ( 1 ) 2 λ 12 | A 1 A 2 | μ k 2 A 2 r ( 1 ) = 0 ,
υ z A 1 r ( 1 ) μ k 2 A 1 i ( 1 ) = 0
( υ z ) 2 A M A = 0 ,
[ μ k 2 ( 2 λ 11 | A 1 | 2 μ k 2 ) 2 λ 12 | A 1 A 2 | μ k 2 2 λ 21 | A 2 A 1 | μ k 2 μ k 2 ( 2 λ 22 | A 2 | 2 μ k 2 ) ] .
A ( t , z ) = E + ( τ ) F + ( τ ) G + ( τ , z ) + E ( τ ) F ( τ ) G ( τ , z ) ,
[ ( υ z ) 2 μ k 2 ( 2 Δ ± μ k 2 ) ] G ± = 0
G ± ( τ , 0 ) = 0 , υ z G ± ( τ , 0 ) = 0.
2 Δ ± = ( λ 11 | A 1 | 2 + λ 22 | A 2 | 2 ) ± [ ( λ 11 | A 1 | 2 λ 22 | A 2 | 2 ) 2 + ( 2 λ 12 | A 1 A 2 | ) 2 ] 1 / 2 ,
G ± ( τ , z ) = cosh ( γ ± z / υ ) ,
γ ± 2 ( τ ) = μ k 2 [ 2 Δ ± ( τ ) μ k 2 ]
e 2 + e 1 + = Δ + λ 11 | A 1 | 2 λ 12 | A 1 A 2 | .
| λ 12 | > ( λ 11 λ 22 ) 1 / 2 ,
e 2 e 1 = Δ λ 11 | A 1 | 2 λ 12 | A 1 A 2 | .
γ ± > ν .
A 1 e = A 1 ( τ ) exp { ν z / υ + i [ λ 11 | A 1 ( τ ) | 2 + λ 12 | A 2 ( τ ) | 2 ] × 0 z exp ( 2 ν z / υ ) d z / υ } , A 2 e = A 2 ( τ ) exp { ν z / υ + i [ λ 21 | A 1 ( τ ) | 2 + λ 22 | A 2 ( τ ) | 2 ] × 0 z exp ( 2 ν z / υ ) d z / υ } .
{ ( υ z + ν ) 2 μ k 2 [ 2 Δ ± exp ( 2 ν z / υ ) μ k 2 ] } G ± = 0
G ± ( τ , 0 ) = 1 , ( u z + ν ) G ± ( τ , 0 ) = 0.
H ± = G ± exp ( ν z / υ ) , ζ ± = β ± exp ( ν z / υ ) ,
[ ( ζ ± ζ ± ) ( ζ ± ζ ± ) + α 2 ζ ± 2 ] H ± = 0 ,
α = μ k 2 / ν , β ± = ( 2 μ k 2 Δ ± ) 1 / 2 / ν ,
H ± ( τ , β ± ) = 1 , ζ ± H ± ( τ , β ± ) = 0.
H ± ( τ , ζ ± ) = J i α ( i β ± ) J i α ( i ζ ± ) J i α ( i β ± ) J i α ( i ζ ± ) J i α ( i β ± ) J i α ( i β ± ) J i α ( i β ± ) J i α ( i β ± ) .
A 1 e = A 1 exp [ i ( λ 11 | A 1 | 2 + λ 12 | A 2 | 2 ) t ] , A 2 e = A 2 exp [ i ( λ 21 | A 1 | 2 + λ 22 | A 2 | 2 ) t ] ,
[ i ( t + υ 1 z ) μ k 2 + λ 11 | A 1 | 2 ] A 1 ( 1 ) + λ 11 | A 1 | 2 A 1 ( 1 ) * + λ 12 | A 1 A 2 | A 2 ( 1 ) + λ 12 | A 1 A 2 | A 2 ( 1 ) * = 0 , [ i ( t + υ 2 z ) μ k 2 + λ 22 | A 2 | 2 ] A 2 ( 1 ) + λ 22 | A 2 | 2 A 2 ( 1 ) * + λ 21 | A 2 A 1 | A 1 ( 1 ) + λ 21 | A 2 A 1 | A 1 ( 1 ) * = 0
A 1 r ( 1 ) ( 0 , z ) = F 1 r ( z ) , t A 1 r ( 1 ) ( 0 , z ) = G 1 r ( z ) , A 2 r ( 1 ) ( 0 , z ) = F 2 r ( z ) , t A 2 r ( 1 ) ( 0 , z ) = G 2 r ( z ) ,
A 1 r ( 1 ) ( t , z ) = F L S 1 ( ω , k , k ) D c ( ω , k , k ) exp [ i ( k z ω t ) ] d ω d k ,
S 1 ( ω , k , k ) = 1 2 π { D 2 ( ω , k , k ) [ i ( ω 2 υ 1 k ) F 1 r ( k ) G 1 r ( k ) ] 2 λ 12 | A 1 A 2 | μ k 2 × [ i ( ω 2 υ 1 k ) F 2 r ( k ) G 2 r ( k ) ] }
D 2 ( ω , k , k ) = [ ( ω υ 2 k ) 2 + μ k 2 ( 2 λ 22 | A 2 | 2 μ k 2 ) ]
D c ( ω , k , k ) = [ ( ω υ 1 k ) 2 + μ k 2 ( 2 λ 11 | A 1 | 2 μ k 2 ) ] × [ ( ω υ 2 k ) 2 + μ k 2 ( 2 λ 22 | A 2 | 2 μ k 2 ) ] ( 2 λ 12 | A 1 A 2 | μ k 2 ) ( 2 λ 21 | A 2 A 1 | μ k 2 )
ω λ 11 | A 1 | 2 ω , υ j k λ 11 | A 1 | 2 υ j k , μ k 2 λ 11 | A 1 | 2 k 2 ,
D c ( ω , k , k ) = [ ( ω υ s k υ d k ) 2 + σ k 2 ( 2 σ k 2 ) ] × [ ( ω υ s k υ d k ) 2 + σ k 2 ( 2 r 2 σ k 2 ) ] ( 2 r σ k 2 ) 2 ,
ω = υ s k ± σ k 2 ( 2 δ ± σ k 2 ) 1 / 2 ,
2 δ ± = ( 1 + r 2 ) ± [ ( 1 r 2 ) 2 + ( 2 r ) 2 ] 1 / 2
( ω υ s k ) 2 = ( υ d k ) 1 / 2 [ σ k 2 ( 2 σ k 2 ) ] ± { ( 2 σ k 2 ) 2 4 ( υ d k ) 2 [ σ k 2 ( 2 σ k 2 ) ] } 1 / 2
ω ( υ s ± υ d ) k ± i [ σ k 2 ( 2 σ k 2 ) ] 1 / 2 .
k 2 ( k 2 2 σ 2 | | ) < ( υ d k ) 2 < k 2 ( k 2 2 σ 2 | | ) .
| ( υ 1 υ 2 ) / υ 1 | | k / k 1 | , | ( υ 1 υ 2 ) / υ 2 | | k / k 2 |
| k | ( λ 11 / μ ) 1 / 2 | A 1 |
[ k 2 1 2 σ ( 1 + r 2 ) | | r ] < | υ d k | < [ k 2 1 2 σ ( 1 + r 2 ) + | | r ]
A 1 r ( 1 ) ( 0 ) = 0 , A 1 i ( 1 ) ( 0 ) = 0 , A 2 r ( 1 ) ( l ) = 0 , A 2 i ( 1 ) ( l ) = 0.
[ ( υ d z ) 2 μ k 2 ( 2 Δ + μ k 2 ) ] × [ ( υ d z ) 2 μ k 2 ( 2 Δ μ k 2 ) ] A 1 r ( 1 ) = 0.
( υ k ± ) 2 = μ k 2 ( μ k 2 2 Δ ± ) .
2 cos ( k + l ) cos ( k l ) + ( k + k + k k + ) sin ( k + l ) sin ( k l ) ( α + α + α α + ) = 0 ,
υ 2 α = 2 μ k 2 ( λ 11 | A 1 | 2 Δ ± ) .
α ± 1 δ ± , k ± l ( γ 1 l / υ ) [ σ k 2 ( σ k 2 2 δ ± ) ] 1 / 2 ,
( γ ± l / υ ) ~ ( ν l / υ ) exp ( ν l / υ ) ,
γ 1 = ω e 2 8 ω 1 | υ 1 υ e | 2 , k = ω e 2 c | υ 1 υ e | ,
| υ 1 υ e | 1.9 × 10 8 λ [ μ m ] ( I [ W cm 2 ] ) 1 / 2 ( T e [ keV ] ) 1 / 2 .
γ 1 l υ = ( k 1 l ) ( n e / n c ) 8 [ 1 ( n e / n c ) ] 1 / 2 | υ 1 υ e | 2 ,
μ k l υ = k 2 l 2 k 1 [ 1 ( n e / n c ) ] 1 / 2 ,
n c [ cm 3 ] 1.1 × 10 21 / ( λ [ μ m ] ) 2 .
A 1 ( 1 ) ( t , z ) ϕ ( x A 1 ( + ) ( t , z ) exp ( i k x ) + A 1 ( ) ( t , z ) exp ( i k x ) , A 2 ( 1 ) ( t , z ) ϕ ( x A 2 ( + ) ( t , z ) exp ( i k x ) + A 2 ( ) ( t , z ) exp ( i k x )
( t + υ 1 z + i δ 1 ) A 1 ( + ) i c 12 A 2 ( ) * = 0 , ( t + υ 2 z i δ 2 ) A 2 ( ) * + i c 21 A 1 ( + ) = 0 ,
δ 1 = μ k 2 λ 11 | A 1 | 2 , δ 2 = μ k 2 λ 22 | A 2 | 2
c 12 = λ 12 | A 1 A 2 | , c 21 = λ 21 | A 2 A 1 | ,
A 1 ( + ) ( 0 , z ) = F + ( z ) , A 2 ( ) * ( 0 , z ) = F ( z ) ,
A 1 ( + ) ( t , z ) = F L S + ( ω , k | | , k ) D ( ω , k | | , k ) exp [ i ( k z ω t ) ] d ω d k ,
S + ( ω , k , k ) = i [ ( ω υ 2 k + δ 2 ) F + ( k ) c 12 F ( k ) ] / 2 π ,
D ( ω , k , k ) = ( ω υ 1 k δ 1 ) ( ω υ 2 k + δ 2 ) + γ 2 ,
γ 2 = c 12 c 21 .
2 ω = ( υ 1 k + δ 1 ) + ( υ 2 k δ 2 ) ± { [ ( υ 1 k + δ 1 ) + ( υ 2 k δ 2 ) ] 2 4 γ 2 } 1 / 2 .
( δ 1 + δ 2 ) 2 γ < ( υ 1 υ 2 ) k < ( δ 1 + δ 2 ) + 2 γ .
2 υ 1 υ 2 k = υ 2 ( ω δ 1 ) + υ 1 ( ω + δ 2 ) ± { [ υ 2 ( ω δ 1 ) υ 1 ( ω δ 2 ) ] 2 + 4 υ 1 υ 2 γ 2 } 1 / 2 .
ω * = υ 1 δ 2 + υ 2 δ 1 υ 1 υ 2 ± 2 γ ( υ 1 υ 2 ) 1 / 2 υ 1 υ 2 .
k * = δ 1 + δ 2 υ 1 υ 2 ± 2 γ ( υ 1 + υ 2 ) ( υ 1 υ 2 ) 1 / 2 ( υ 1 υ 2 ) .
υ s = 1 2 ( υ 1 + υ 2 ) , υ d = 1 2 ( υ 1 υ 2 ) ,
ω ¯ * ( υ s + υ ) = 1 2 [ ( δ 1 δ 2 ) + ( υ / υ d ) ( δ 1 + δ 2 ) ] ± i γ [ 1 ( υ / υ d ) 2 ] 1 / 2 , υ d k ¯ * ( υ s + υ ) = 1 2 ( δ 1 + δ 2 ) ± i γ ( υ / υ d ) [ 1 ( υ / υ d ) 2 ] 1 / 2 ,
ω ¯ * ( υ s ) = 1 2 ( δ 1 δ 2 ) ± i γ, υ d k ¯ * ( υ s ) = 1 2 ( δ 1 + δ 2 ) .
( δ 1 + δ 2 ) 2 γ < ( υ 1 υ 2 ) k < ( δ 1 + δ 2 ) + 2 γ .
( t + υ z + i δ 1 ) A 1 ( + ) i c 12 A 2 ( ) * = 0 , ( t υ z i δ 2 ) A 2 ( ) * i c 21 A 1 ( + ) = 0
A 1 ( + ) ( t , 0 ) = 0 , A 2 ( ) * ( t , l ) = 0.
A 1 ( + ) ( t , z ) = B + ( z ) exp [ i ( δ 1 + δ 2 ) z / 2 υ i ω t ] , A 2 ( ) * ( t , z ) = B ( z ) exp [ i ( δ 1 + δ 2 ) z / 2 υ i ω t ] ,
{ ( υ d z ) 2 + γ 2 + [ ω + 1 2 ( δ 2 δ 1 ) ] 2 } B + = 0.
B + ( 0 ) = 0 , { υ d z i [ ω + 1 2 ( δ 2 δ 1 ) ] } B + ( l ) = 0.
υ k cos ( k l ) i [ ω + 1 2 ( δ 2 δ 1 ) ] sin ( k l ) = 0 ,
( υ k ) 2 = γ 2 + [ ω + 1 2 ( δ 2 δ 1 ) ] 2 .
ω r = 1 2 ( δ 1 δ 2 ) .
γ l / υ = ( 2 m + 1 ) π / 2 .
υ k cos ( k l ) + ω i sin ( k l ) = 0 ,
( υ k ) 2 = γ 2 ω i 2

Metrics