Abstract

We show the formation of bright, dark, bistable bright, and vortex spatial-optical solitons in a lifetime-broadened three-state atomic medium. We demonstrate, both analytically and numerically, that these solitons can exist under appropriate conditions. This work may provide other research opportunities in nonlinear optical experiments and may result in a substantial impact on technology.

© 2006 Optical Society of America

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  1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).
  2. R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
    [CrossRef]
  3. Y. S. Kivshar and E. A. Ostrovskaya, "Spatial optical solitons: guiding light for future technologies," Opt. Photonics News 13, 59-63 (2002).
    [CrossRef]
  4. M. Segev and G. I. Stegeman, "Self-trapping of optical beams: spatial solitons," Phys. Today 51, 43-48 (1998).
    [CrossRef]
  5. G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
    [CrossRef] [PubMed]
  6. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-state spatial screening solitons in photorefractive materials with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
    [CrossRef] [PubMed]
  7. M. Shih, P. Lwach, M. Segev, M. H. Garrett, G. Salamo, and G. C. Valley, "Two-dimensional steady-state photorefractive screening solitons," Opt. Lett. 21, 324-326 (1996).
    [CrossRef] [PubMed]
  8. M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, "Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity," Appl. Phys. Lett. 64, 408-410 (1994).
    [CrossRef]
  9. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
    [CrossRef]
  10. S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
    [CrossRef]
  11. H. Schmidt and A. Imamoglu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21, 1936-1938 (1996).
    [CrossRef] [PubMed]
  12. M. D. Lukin and A. Imamoglu, "Nonlinear optics and quantum entanglement of ultraslow single photons," Phys. Rev. Lett. 84, 1419-1422 (2000).
    [CrossRef] [PubMed]
  13. Y. Wu and L. Deng, "Ultraslow bright and dark optical solitons in a cold three-state medium," Opt. Lett. 29, 2064-2066 (2004).
    [CrossRef] [PubMed]
  14. Y. Wu and L. Deng, "Ultraslow optical solitons in a cold four-state medium," Phys. Rev. Lett. 93, 143904 (2004).
    [CrossRef] [PubMed]
  15. Y. Wu, "Two-color ultraslow optical solitons via four-wave mixing in cold-atom media," Phys. Rev. A 71, 053820 (2005).
    [CrossRef]
  16. U. Rathe, M. Fleischhauer, S. Y. Zhu, T. W. Hansch, and M. O. Scully, "Nonlinear theory of index enhancement via quantum coherence and interference," Phys. Rev. A 47, 4994-5002 (1993).
    [CrossRef] [PubMed]
  17. Y. S. Kivshar, "Bright and dark spatial solitons in non-Kerr media," Opt. Quantum Electron. 30, 571-614 (1998).
    [CrossRef]
  18. B. L. Davies, J. Christou, V. Tikhonenko, and Y. S. Kivshar, "Optical vortex solitons: experiment versus theory," J. Opt. Soc. Am. B 14, 3045-3053 (1997).
    [CrossRef]
  19. S. Trillo and W. Torruellas, Spatial Solitons (Springer-Verlag, 2001).
  20. Y. Li and M. Xiao, "Electromagnetically induced transparency in a three-level Lambda-type system in rubidium atoms," Phys. Rev. A 51, R2703-R2706 (1995).
    [CrossRef] [PubMed]
  21. A. E. Kaplan, "Bistable solitons," Phys. Rev. Lett. 55, 1291-1294 (1985).
    [CrossRef] [PubMed]
  22. A. V. Buryak and Y. S. Kivshar, "Multistability of three-wave parametric self-trapping," Phys. Rev. Lett. 78, 3286-3289 (1997).
    [CrossRef]
  23. S. Gatz and J. Herrmann, "Soliton propagation in materials with saturable nonlinearity," J. Opt. Soc. Am. B 8, 2296-2302 (1991).
    [CrossRef]
  24. E. W. Laedke, K. H. Spatschek, and L. Stenflo, "Evolution theorem for a class of perturbed envelope soliton solutions," J. Math. Phys. 24, 2764-2769 (1983).
    [CrossRef]
  25. E. M. Lifshitz and L. P. Pitaevsky, Statistical Physics (Nauka, 1978).
  26. A. E. Gill, Atmosphere-Ocean Dynamics (Academic, 1982).
  27. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999).
    [CrossRef]
  28. G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
    [CrossRef] [PubMed]
  29. Y. S. Kivshar and E. A. Ostrovskaya, "Optical vortices: folding and twisting waves of light," Opt. Photon. News 12, 24-28 (2001).
    [CrossRef]
  30. A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optically written waveguide in an atomic vapor," Phys. Rev. Lett. 82, 1438-1441 (1999).
    [CrossRef]
  31. C. T. Law, X. Zhang, and G. A. Swartzlander, "Waveguiding properties of optical vortex solitons," Opt. Lett. 25, 55-58 (2000).
    [CrossRef]
  32. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular-momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
    [CrossRef] [PubMed]
  33. G. A. Swartzlander, Jr. and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Phys. Rev. Lett. 69, 2503-2506 (1992).
    [CrossRef] [PubMed]
  34. G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, "Dark photorefractive spatial solitons and photorefractive vortex solitons," Phys. Rev. Lett. 74, 1978-1981 (1995).
    [CrossRef] [PubMed]
  35. C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
    [CrossRef] [PubMed]
  36. B. Luther-Davies, R. Powels, and V. Tikhonenko, "Nonlinear rotation of three-dimensional dark spatial solitons in a Gaussian laser beam," Opt. Lett. 19, 1816-1818 (1994).
    [CrossRef] [PubMed]
  37. Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Self-trapping of an optical vortex by use of the bulk photovoltaic effect," Phys. Rev. Lett. 78, 2948-2951 (1997).
    [CrossRef]
  38. Y. S. Kivshar and B. Luther-Davies, "Dark optical solitons: physics and applications," Phys. Rep. 298, 81-197 (1998).
    [CrossRef]
  39. V. Tikhonenko, Y. S. Kivsha, V. V. Steblina, and A. A. Zozulya, "Vortex solitons in a saturable optical medium," J. Opt. Soc. Am. B 15, 79-86 (1998).
    [CrossRef]

2005 (1)

Y. Wu, "Two-color ultraslow optical solitons via four-wave mixing in cold-atom media," Phys. Rev. A 71, 053820 (2005).
[CrossRef]

2004 (3)

Y. Wu and L. Deng, "Ultraslow optical solitons in a cold four-state medium," Phys. Rev. Lett. 93, 143904 (2004).
[CrossRef] [PubMed]

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Y. Wu and L. Deng, "Ultraslow bright and dark optical solitons in a cold three-state medium," Opt. Lett. 29, 2064-2066 (2004).
[CrossRef] [PubMed]

2002 (1)

Y. S. Kivshar and E. A. Ostrovskaya, "Spatial optical solitons: guiding light for future technologies," Opt. Photonics News 13, 59-63 (2002).
[CrossRef]

2001 (1)

Y. S. Kivshar and E. A. Ostrovskaya, "Optical vortices: folding and twisting waves of light," Opt. Photon. News 12, 24-28 (2001).
[CrossRef]

2000 (2)

M. D. Lukin and A. Imamoglu, "Nonlinear optics and quantum entanglement of ultraslow single photons," Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

C. T. Law, X. Zhang, and G. A. Swartzlander, "Waveguiding properties of optical vortex solitons," Opt. Lett. 25, 55-58 (2000).
[CrossRef]

1999 (3)

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optically written waveguide in an atomic vapor," Phys. Rev. Lett. 82, 1438-1441 (1999).
[CrossRef]

S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999).
[CrossRef]

1998 (4)

Y. S. Kivshar, "Bright and dark spatial solitons in non-Kerr media," Opt. Quantum Electron. 30, 571-614 (1998).
[CrossRef]

M. Segev and G. I. Stegeman, "Self-trapping of optical beams: spatial solitons," Phys. Today 51, 43-48 (1998).
[CrossRef]

Y. S. Kivshar and B. Luther-Davies, "Dark optical solitons: physics and applications," Phys. Rep. 298, 81-197 (1998).
[CrossRef]

V. Tikhonenko, Y. S. Kivsha, V. V. Steblina, and A. A. Zozulya, "Vortex solitons in a saturable optical medium," J. Opt. Soc. Am. B 15, 79-86 (1998).
[CrossRef]

1997 (4)

B. L. Davies, J. Christou, V. Tikhonenko, and Y. S. Kivshar, "Optical vortex solitons: experiment versus theory," J. Opt. Soc. Am. B 14, 3045-3053 (1997).
[CrossRef]

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Self-trapping of an optical vortex by use of the bulk photovoltaic effect," Phys. Rev. Lett. 78, 2948-2951 (1997).
[CrossRef]

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

A. V. Buryak and Y. S. Kivshar, "Multistability of three-wave parametric self-trapping," Phys. Rev. Lett. 78, 3286-3289 (1997).
[CrossRef]

1996 (2)

1995 (3)

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, "Dark photorefractive spatial solitons and photorefractive vortex solitons," Phys. Rev. Lett. 74, 1978-1981 (1995).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular-momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Y. Li and M. Xiao, "Electromagnetically induced transparency in a three-level Lambda-type system in rubidium atoms," Phys. Rev. A 51, R2703-R2706 (1995).
[CrossRef] [PubMed]

1994 (3)

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-state spatial screening solitons in photorefractive materials with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, "Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity," Appl. Phys. Lett. 64, 408-410 (1994).
[CrossRef]

B. Luther-Davies, R. Powels, and V. Tikhonenko, "Nonlinear rotation of three-dimensional dark spatial solitons in a Gaussian laser beam," Opt. Lett. 19, 1816-1818 (1994).
[CrossRef] [PubMed]

1993 (2)

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

U. Rathe, M. Fleischhauer, S. Y. Zhu, T. W. Hansch, and M. O. Scully, "Nonlinear theory of index enhancement via quantum coherence and interference," Phys. Rev. A 47, 4994-5002 (1993).
[CrossRef] [PubMed]

1992 (1)

G. A. Swartzlander, Jr. and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Phys. Rev. Lett. 69, 2503-2506 (1992).
[CrossRef] [PubMed]

1991 (2)

S. Gatz and J. Herrmann, "Soliton propagation in materials with saturable nonlinearity," J. Opt. Soc. Am. B 8, 2296-2302 (1991).
[CrossRef]

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

1985 (1)

A. E. Kaplan, "Bistable solitons," Phys. Rev. Lett. 55, 1291-1294 (1985).
[CrossRef] [PubMed]

1983 (1)

E. W. Laedke, K. H. Spatschek, and L. Stenflo, "Evolution theorem for a class of perturbed envelope soliton solutions," J. Math. Phys. 24, 2764-2769 (1983).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Andersen, D. R.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

Buryak, A. V.

A. V. Buryak and Y. S. Kivshar, "Multistability of three-wave parametric self-trapping," Phys. Rev. Lett. 78, 3286-3289 (1997).
[CrossRef]

Chen, Z.

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Self-trapping of an optical vortex by use of the bulk photovoltaic effect," Phys. Rev. Lett. 78, 2948-2951 (1997).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Christou, J.

Crosignani, B.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, "Dark photorefractive spatial solitons and photorefractive vortex solitons," Phys. Rev. Lett. 74, 1978-1981 (1995).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-state spatial screening solitons in photorefractive materials with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

Dalfovo, F.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999).
[CrossRef]

Davies, B. L.

Deng, L.

Y. Wu and L. Deng, "Ultraslow optical solitons in a cold four-state medium," Phys. Rev. Lett. 93, 143904 (2004).
[CrossRef] [PubMed]

Y. Wu and L. Deng, "Ultraslow bright and dark optical solitons in a cold three-state medium," Opt. Lett. 29, 2064-2066 (2004).
[CrossRef] [PubMed]

Di Porto, P.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, "Dark photorefractive spatial solitons and photorefractive vortex solitons," Phys. Rev. Lett. 74, 1978-1981 (1995).
[CrossRef] [PubMed]

DiPorto, P.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-state spatial screening solitons in photorefractive materials with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

Duree, G.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, "Dark photorefractive spatial solitons and photorefractive vortex solitons," Phys. Rev. Lett. 74, 1978-1981 (1995).
[CrossRef] [PubMed]

Duree, G. C.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

Fleischhauer, M.

U. Rathe, M. Fleischhauer, S. Y. Zhu, T. W. Hansch, and M. O. Scully, "Nonlinear theory of index enhancement via quantum coherence and interference," Phys. Rev. A 47, 4994-5002 (1993).
[CrossRef] [PubMed]

Friese, M. E. J.

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optically written waveguide in an atomic vapor," Phys. Rev. Lett. 82, 1438-1441 (1999).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular-momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Garrett, M. H.

Gatz, S.

Gill, A. E.

A. E. Gill, Atmosphere-Ocean Dynamics (Academic, 1982).

Giorgini, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999).
[CrossRef]

Hansch, T. W.

U. Rathe, M. Fleischhauer, S. Y. Zhu, T. W. Hansch, and M. O. Scully, "Nonlinear theory of index enhancement via quantum coherence and interference," Phys. Rev. A 47, 4994-5002 (1993).
[CrossRef] [PubMed]

Harris, S. E.

S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

Hau, L. V.

S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular-momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optically written waveguide in an atomic vapor," Phys. Rev. Lett. 82, 1438-1441 (1999).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular-momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Herrmann, J.

Imamoglu, A.

M. D. Lukin and A. Imamoglu, "Nonlinear optics and quantum entanglement of ultraslow single photons," Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

H. Schmidt and A. Imamoglu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21, 1936-1938 (1996).
[CrossRef] [PubMed]

Iturbe-Castillo, M. D.

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, "Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity," Appl. Phys. Lett. 64, 408-410 (1994).
[CrossRef]

Jeng, C. C.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Kaplan, A. E.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

A. E. Kaplan, "Bistable solitons," Phys. Rev. Lett. 55, 1291-1294 (1985).
[CrossRef] [PubMed]

Kivsha, Y. S.

Kivshar, Y.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Kivshar, Y. S.

Y. S. Kivshar and E. A. Ostrovskaya, "Spatial optical solitons: guiding light for future technologies," Opt. Photonics News 13, 59-63 (2002).
[CrossRef]

Y. S. Kivshar and E. A. Ostrovskaya, "Optical vortices: folding and twisting waves of light," Opt. Photon. News 12, 24-28 (2001).
[CrossRef]

Y. S. Kivshar, "Bright and dark spatial solitons in non-Kerr media," Opt. Quantum Electron. 30, 571-614 (1998).
[CrossRef]

Y. S. Kivshar and B. Luther-Davies, "Dark optical solitons: physics and applications," Phys. Rep. 298, 81-197 (1998).
[CrossRef]

B. L. Davies, J. Christou, V. Tikhonenko, and Y. S. Kivshar, "Optical vortex solitons: experiment versus theory," J. Opt. Soc. Am. B 14, 3045-3053 (1997).
[CrossRef]

A. V. Buryak and Y. S. Kivshar, "Multistability of three-wave parametric self-trapping," Phys. Rev. Lett. 78, 3286-3289 (1997).
[CrossRef]

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Laedke, E. W.

E. W. Laedke, K. H. Spatschek, and L. Stenflo, "Evolution theorem for a class of perturbed envelope soliton solutions," J. Math. Phys. 24, 2764-2769 (1983).
[CrossRef]

Law, C. T.

C. T. Law, X. Zhang, and G. A. Swartzlander, "Waveguiding properties of optical vortex solitons," Opt. Lett. 25, 55-58 (2000).
[CrossRef]

G. A. Swartzlander, Jr. and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Phys. Rev. Lett. 69, 2503-2506 (1992).
[CrossRef] [PubMed]

Li, Y.

Y. Li and M. Xiao, "Electromagnetically induced transparency in a three-level Lambda-type system in rubidium atoms," Phys. Rev. A 51, R2703-R2706 (1995).
[CrossRef] [PubMed]

Lifshitz, E. M.

E. M. Lifshitz and L. P. Pitaevsky, Statistical Physics (Nauka, 1978).

Lukin, M. D.

M. D. Lukin and A. Imamoglu, "Nonlinear optics and quantum entanglement of ultraslow single photons," Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

Luther-Davies, B.

Lwach, P.

Maker, P. D.

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Self-trapping of an optical vortex by use of the bulk photovoltaic effect," Phys. Rev. Lett. 78, 2948-2951 (1997).
[CrossRef]

Marquez-Aguilar, P. A.

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, "Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity," Appl. Phys. Lett. 64, 408-410 (1994).
[CrossRef]

Morin, M.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, "Dark photorefractive spatial solitons and photorefractive vortex solitons," Phys. Rev. Lett. 74, 1978-1981 (1995).
[CrossRef] [PubMed]

Motzek, K.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Muller, R. E.

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Self-trapping of an optical vortex by use of the bulk photovoltaic effect," Phys. Rev. Lett. 78, 2948-2951 (1997).
[CrossRef]

Neurgaonkar, R. R.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

Ostrovskaya, E. A.

Y. S. Kivshar and E. A. Ostrovskaya, "Spatial optical solitons: guiding light for future technologies," Opt. Photonics News 13, 59-63 (2002).
[CrossRef]

Y. S. Kivshar and E. A. Ostrovskaya, "Optical vortices: folding and twisting waves of light," Opt. Photon. News 12, 24-28 (2001).
[CrossRef]

Pitaevskii, L. P.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999).
[CrossRef]

Pitaevsky, L. P.

E. M. Lifshitz and L. P. Pitaevsky, Statistical Physics (Nauka, 1978).

Powels, R.

Rathe, U.

U. Rathe, M. Fleischhauer, S. Y. Zhu, T. W. Hansch, and M. O. Scully, "Nonlinear theory of index enhancement via quantum coherence and interference," Phys. Rev. A 47, 4994-5002 (1993).
[CrossRef] [PubMed]

Regan, J. J.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optically written waveguide in an atomic vapor," Phys. Rev. Lett. 82, 1438-1441 (1999).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular-momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Salamo, G.

M. Shih, P. Lwach, M. Segev, M. H. Garrett, G. Salamo, and G. C. Valley, "Two-dimensional steady-state photorefractive screening solitons," Opt. Lett. 21, 324-326 (1996).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, "Dark photorefractive spatial solitons and photorefractive vortex solitons," Phys. Rev. Lett. 74, 1978-1981 (1995).
[CrossRef] [PubMed]

Salamo, G. J.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

Sanchez-Mondragon, J. J.

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, "Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity," Appl. Phys. Lett. 64, 408-410 (1994).
[CrossRef]

Schmidt, H.

Scully, M. O.

U. Rathe, M. Fleischhauer, S. Y. Zhu, T. W. Hansch, and M. O. Scully, "Nonlinear theory of index enhancement via quantum coherence and interference," Phys. Rev. A 47, 4994-5002 (1993).
[CrossRef] [PubMed]

Segev, M.

M. Segev and G. I. Stegeman, "Self-trapping of optical beams: spatial solitons," Phys. Today 51, 43-48 (1998).
[CrossRef]

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Self-trapping of an optical vortex by use of the bulk photovoltaic effect," Phys. Rev. Lett. 78, 2948-2951 (1997).
[CrossRef]

M. Shih, P. Lwach, M. Segev, M. H. Garrett, G. Salamo, and G. C. Valley, "Two-dimensional steady-state photorefractive screening solitons," Opt. Lett. 21, 324-326 (1996).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, "Dark photorefractive spatial solitons and photorefractive vortex solitons," Phys. Rev. Lett. 74, 1978-1981 (1995).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-state spatial screening solitons in photorefractive materials with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

Sharp, E.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, "Dark photorefractive spatial solitons and photorefractive vortex solitons," Phys. Rev. Lett. 74, 1978-1981 (1995).
[CrossRef] [PubMed]

Sharp, E. J.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

Shih, M.

Shih, M. F.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Shultz, J. L.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

Spatschek, K. H.

E. W. Laedke, K. H. Spatschek, and L. Stenflo, "Evolution theorem for a class of perturbed envelope soliton solutions," J. Math. Phys. 24, 2764-2769 (1983).
[CrossRef]

Steblina, V. V.

Stegeman, G. I.

M. Segev and G. I. Stegeman, "Self-trapping of optical beams: spatial solitons," Phys. Today 51, 43-48 (1998).
[CrossRef]

Stenflo, L.

E. W. Laedke, K. H. Spatschek, and L. Stenflo, "Evolution theorem for a class of perturbed envelope soliton solutions," J. Math. Phys. 24, 2764-2769 (1983).
[CrossRef]

Stepanov, S.

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, "Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity," Appl. Phys. Lett. 64, 408-410 (1994).
[CrossRef]

Stringari, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999).
[CrossRef]

Swartzlander, G. A.

C. T. Law, X. Zhang, and G. A. Swartzlander, "Waveguiding properties of optical vortex solitons," Opt. Lett. 25, 55-58 (2000).
[CrossRef]

G. A. Swartzlander, Jr. and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Phys. Rev. Lett. 69, 2503-2506 (1992).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

Tikhonenko, V.

Torruellas, W.

S. Trillo and W. Torruellas, Spatial Solitons (Springer-Verlag, 2001).

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Trillo, S.

S. Trillo and W. Torruellas, Spatial Solitons (Springer-Verlag, 2001).

Truscott, A. G.

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optically written waveguide in an atomic vapor," Phys. Rev. Lett. 82, 1438-1441 (1999).
[CrossRef]

Valley, G. C.

M. Shih, P. Lwach, M. Segev, M. H. Garrett, G. Salamo, and G. C. Valley, "Two-dimensional steady-state photorefractive screening solitons," Opt. Lett. 21, 324-326 (1996).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-state spatial screening solitons in photorefractive materials with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

Vysloukh, V.

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, "Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity," Appl. Phys. Lett. 64, 408-410 (1994).
[CrossRef]

Wilson, D. W.

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Self-trapping of an optical vortex by use of the bulk photovoltaic effect," Phys. Rev. Lett. 78, 2948-2951 (1997).
[CrossRef]

Wu, Y.

Y. Wu, "Two-color ultraslow optical solitons via four-wave mixing in cold-atom media," Phys. Rev. A 71, 053820 (2005).
[CrossRef]

Y. Wu and L. Deng, "Ultraslow optical solitons in a cold four-state medium," Phys. Rev. Lett. 93, 143904 (2004).
[CrossRef] [PubMed]

Y. Wu and L. Deng, "Ultraslow bright and dark optical solitons in a cold three-state medium," Opt. Lett. 29, 2064-2066 (2004).
[CrossRef] [PubMed]

Xiao, M.

Y. Li and M. Xiao, "Electromagnetically induced transparency in a three-level Lambda-type system in rubidium atoms," Phys. Rev. A 51, R2703-R2706 (1995).
[CrossRef] [PubMed]

Yariv, A.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, "Dark photorefractive spatial solitons and photorefractive vortex solitons," Phys. Rev. Lett. 74, 1978-1981 (1995).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-state spatial screening solitons in photorefractive materials with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

Yin, H.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

Zhang, X.

Zhu, S. Y.

U. Rathe, M. Fleischhauer, S. Y. Zhu, T. W. Hansch, and M. O. Scully, "Nonlinear theory of index enhancement via quantum coherence and interference," Phys. Rev. A 47, 4994-5002 (1993).
[CrossRef] [PubMed]

Zozulya, A. A.

Appl. Phys. Lett. (1)

M. D. Iturbe-Castillo, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, "Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity," Appl. Phys. Lett. 64, 408-410 (1994).
[CrossRef]

J. Math. Phys. (1)

E. W. Laedke, K. H. Spatschek, and L. Stenflo, "Evolution theorem for a class of perturbed envelope soliton solutions," J. Math. Phys. 24, 2764-2769 (1983).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (5)

Opt. Photon. News (1)

Y. S. Kivshar and E. A. Ostrovskaya, "Optical vortices: folding and twisting waves of light," Opt. Photon. News 12, 24-28 (2001).
[CrossRef]

Opt. Photonics News (1)

Y. S. Kivshar and E. A. Ostrovskaya, "Spatial optical solitons: guiding light for future technologies," Opt. Photonics News 13, 59-63 (2002).
[CrossRef]

Opt. Quantum Electron. (1)

Y. S. Kivshar, "Bright and dark spatial solitons in non-Kerr media," Opt. Quantum Electron. 30, 571-614 (1998).
[CrossRef]

Phys. Rep. (1)

Y. S. Kivshar and B. Luther-Davies, "Dark optical solitons: physics and applications," Phys. Rep. 298, 81-197 (1998).
[CrossRef]

Phys. Rev. A (3)

Y. Li and M. Xiao, "Electromagnetically induced transparency in a three-level Lambda-type system in rubidium atoms," Phys. Rev. A 51, R2703-R2706 (1995).
[CrossRef] [PubMed]

Y. Wu, "Two-color ultraslow optical solitons via four-wave mixing in cold-atom media," Phys. Rev. A 71, 053820 (2005).
[CrossRef]

U. Rathe, M. Fleischhauer, S. Y. Zhu, T. W. Hansch, and M. O. Scully, "Nonlinear theory of index enhancement via quantum coherence and interference," Phys. Rev. A 47, 4994-5002 (1993).
[CrossRef] [PubMed]

Phys. Rev. Lett. (15)

S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

Y. Wu and L. Deng, "Ultraslow optical solitons in a cold four-state medium," Phys. Rev. Lett. 93, 143904 (2004).
[CrossRef] [PubMed]

A. E. Kaplan, "Bistable solitons," Phys. Rev. Lett. 55, 1291-1294 (1985).
[CrossRef] [PubMed]

A. V. Buryak and Y. S. Kivshar, "Multistability of three-wave parametric self-trapping," Phys. Rev. Lett. 78, 3286-3289 (1997).
[CrossRef]

R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-state spatial screening solitons in photorefractive materials with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optically written waveguide in an atomic vapor," Phys. Rev. Lett. 82, 1438-1441 (1999).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular-momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr. and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Phys. Rev. Lett. 69, 2503-2506 (1992).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, "Dark photorefractive spatial solitons and photorefractive vortex solitons," Phys. Rev. Lett. 74, 1978-1981 (1995).
[CrossRef] [PubMed]

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

M. D. Lukin and A. Imamoglu, "Nonlinear optics and quantum entanglement of ultraslow single photons," Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Self-trapping of an optical vortex by use of the bulk photovoltaic effect," Phys. Rev. Lett. 78, 2948-2951 (1997).
[CrossRef]

Phys. Today (2)

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

M. Segev and G. I. Stegeman, "Self-trapping of optical beams: spatial solitons," Phys. Today 51, 43-48 (1998).
[CrossRef]

Rev. Mod. Phys. (1)

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999).
[CrossRef]

Other (4)

E. M. Lifshitz and L. P. Pitaevsky, Statistical Physics (Nauka, 1978).

A. E. Gill, Atmosphere-Ocean Dynamics (Academic, 1982).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

S. Trillo and W. Torruellas, Spatial Solitons (Springer-Verlag, 2001).

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Figures (6)

Fig. 1
Fig. 1

Lifetime-broadened three-state atomic system interacting with a strong control field (Rabi frequency 2 Ω c ) and a probe field (Rabi frequency 2 Ω p ).

Fig. 2
Fig. 2

Bright and dark spatial soliton shapes of the probe field; the propagation distance is L z = 1 ( k p n 2 I 0 ) 1.0 cm and the transverse distance is L x = 1 ( n 0 n 2 I 0 k p 2 ) 1 2 3.50 × 10 3 cm .

Fig. 3
Fig. 3

Bistable soliton shapes of the probe field; the propagation distance is L z = 1 ( k p n 2 I 0 ) 0.1 cm and the transverse distance is L x = 1 ( n 0 n 2 I 0 k p 2 ) 1 2 3.50 × 10 4 cm .

Fig. 4
Fig. 4

Soliton energy E relative to the wavenumber shift β n l . According to condition (15), solitons are stable.

Fig. 5
Fig. 5

Soliton energy E relative to the wavenumber shift β n l . According to condition (15), solitons are stable.

Fig. 6
Fig. 6

Vortex spatial soliton shape to the propagation constant β (the values of β are indicated near the curves). Here m = 2 , the propagation distance L z = 1 ( k p n 2 I 0 ) 0.1 cm , and the transverse distance L x = 1 ( n 0 n 2 I 0 k p 2 ) 1 2 3.50 × 10 4 cm .

Equations (35)

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

ρ a a t = γ a ρ a a i ( Ω p ρ b a + Ω c ρ c a c.c. ) ,
ρ c c t = γ c ρ c c + i ( Ω c ρ c a c.c. ) ,
ρ a b t = ( i Δ + γ a b ) ρ a b + i Ω p ( ρ a a ρ b b ) i Ω c ρ c b ,
ρ c b t = ( i Δ + γ c b ) ρ c b + i Ω p ρ c a i Ω c * ρ a b ,
ρ a c t = γ a c ρ a c + i Ω c ( ρ a a ρ c c ) i Ω p ρ b c ,
1 = ρ a a + ρ b b + ρ c c .
χ a b = D a b 2 N 2 ϵ 0 A + B Ω p 2 + C Ω p 4 a + b Ω p 2 + c Ω p 4 + d Ω p 6 ,
A = ( 2 Δ i γ c ) ( γ a + γ c ) 2 ( γ a γ c + 4 Ω c 2 ) [ ( 2 Δ + i γ a ) ( 2 Δ + i γ c ) 4 Ω c 2 ] 32 ,
B = γ c ( γ a + γ c ) ( 2 Δ + i γ a ) ( γ a γ c + 4 Ω c 2 ) 4 ,
C = γ a γ c ( 2 Δ + i γ a ) 2 ,
a = ( γ a + γ c ) 2 ( γ a γ c + 4 Ω c 2 ) [ ( 4 Δ 2 + γ a 2 ) ( 4 Δ 2 + γ c 2 ) 8 ( 4 Δ 2 γ a γ c ) Ω c 2 + 16 Ω c 4 ] 64 ,
b = [ 4 Δ 2 ( γ a + 2 γ c ) + γ c ( γ a 2 + γ a γ c + γ c 2 ) + 4 γ a Ω c 2 ] ( γ a + γ c ) ( γ a γ c + 4 Ω c 2 ) 8 ,
c = γ a γ c [ 4 Δ 2 + ( γ a + 2 γ c ) 2 ] 4 + ( γ a 2 + 2 γ a γ c + 2 γ c 2 ) Ω c 2 ,
d = 2 γ a γ c .
χ a b = χ 1 + χ 2 ( Ω p ) = D a b 2 N 2 ϵ 0 A a + D a b 2 N B 2 ϵ 0 a Ω p 2 + C Ω p 4 + D Ω p 6 1 + b Ω p 2 + c Ω p 4 + d Ω p 6 ,
χ 1 = D a b 2 N 2 ϵ 0 A a ,
χ 2 ( Ω p ) = D a b 2 N B 2 ϵ 0 a Ω p 2 + C Ω p 4 + D Ω p 6 1 + b Ω p 2 + c Ω p 4 + d Ω p 6
n = n 0 + n n l ( Ω p ) = 1 + χ a b ( 1 + χ 1 2 ) + χ 2 ( Ω p ) 2 .
n n l ( Ω p ) χ 2 ( Ω p ) 2 = n 2 Ω p 2 + C Ω p 4 + D Ω p 6 1 + b Ω p 2 + c Ω p 4 + d Ω p 6
2 i k p n 0 Ω p Z + Ω p + 2 n 0 k p 2 n n l ( Ω p ) Ω p = 0 .
i Ψ z + 1 2 Ψ ± Ψ 2 + C Ψ 4 + D Ψ 6 1 + b Ψ 2 + c Ψ 4 + d Ψ 6 Ψ = 0 ,
i Ψ z + 1 2 2 Ψ x 2 ± Ψ 2 + C Ψ 4 + D Ψ 6 1 + b Ψ 2 + c Ψ 4 + d Ψ 6 Ψ = 0 ,
i Ψ z + 1 2 ( 2 Ψ x 2 + 2 Ψ y 2 ) ± Ψ 2 + C Ψ 4 + D Ψ 6 1 + b Ψ 2 + c Ψ 4 + d Ψ 6 Ψ = 0 ,
n n l ( Ω p ) n 2 Ω p 2 .
i Ψ z + 1 2 2 Ψ x 2 ± Ψ 2 Ψ = 0 ,
Ψ ( z , x ) = sech ( x ) exp ( i z 2 ) .
Ψ ( z , x ) = tanh ( x ) exp ( i z 2 ) .
Ψ = φ ( x ) exp [ i β n l z ] ,
β n l φ 0 = F ( φ 0 ) ,
φ ( x ) = φ 0 exp ( 0 x d t { 8 [ β n l F ( φ ) φ ] } 1 2 ) ,
F ( φ ) = 0 φ d t t + C t 2 + D t 3 1 + b t + c t 2 + d t 3 , φ 0 = φ ( 0 ) , φ ( 0.88 ) = φ 0 2
E ( β n l ) β n l > 0 ,
d 2 u d r 2 + 1 r d u d r 2 β u m 2 r 2 u + 2 u 2 + C u 4 + D u 6 1 + b u 2 + c u 4 + d u 6 u = 0 ,
u ( r 0 ) u 1 r m ,
u ( r ) u 1 r 1 2 exp [ r 2 β ] ,

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