Abstract

We have investigated the dynamics of optical beams in a cubic (focusing)–quintic (defocusing) nonlinear medium. In particular, we have found that strong beams can show long-lived elliptical oscillations, whereas in other cases, i.e., for weak beams or cylindrical oscillations, the dynamics decay quickly. This finding explains the observed higher efficiency in the fusion of two strong beams. We have also investigated, numerically and analytically, the robustness of the beams to an initial phase chirp.

© 1999 Optical Society of America

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  1. W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
    [Crossref]
  2. J. H. Marburger and E. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556–558 (1968).
    [Crossref]
  3. K. Konno and H. Suzuki, “Self-focusing of laser beam in nonlinear media,” Phys. Scr. 20, 382–386 (1979).
    [Crossref]
  4. J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
    [Crossref] [PubMed]
  5. V. G. Makhankov, G. Kummer, and A. B. Shvachka, “Interaction of two-space-dimensional classical Q solitons,” Phys. Lett. A 70, 171–176 (1979).
    [Crossref]
  6. V. G. Makhankov, “Computer and solitons,” Phys. Scr. 20, 558–562 (1979).
    [Crossref]
  7. J. Oficjalski and I. Bialynicki-Birula, “Collisions of gaussons,” Acta Phys. Pol. B 9, 759–775 (1978).
  8. V. G. Makhankov, G. Kummer, and A. B. Shvachka, “Many dimensional U(1) solitons, their interactions, resonances and bound states,” Phys. Scr. 20, 454–461 (1979).
    [Crossref]
  9. M. L. Quiroga-Teixeiro, D. Anderson, A. Berntson, M. Lisak, E. V. Vanin, and A. V. Kim, “Interaction and collision between two optical beams in a saturable nonlinear medium,” presented at the Symposium on Progress in Electromagnetics Research, Innsbruck, Austria, July 8–12, 1996.
  10. V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan’kov, “Soliton turbulence,” JETP Lett. 48, 82–87 (1988).
  11. J. M. Soto-Crespo, D. R. Heartley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
    [Crossref] [PubMed]
  12. V. I. Kruglov and R. A. Vlasov, “Spiral self-trapping propagation of optical beams in media with cubic nonlinearity,” Phys. Lett. A 111, 401–404 (1985).
    [Crossref]
  13. B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 889–900 (1994).
    [Crossref]
  14. C. Josserand and S. Rica, “Coalescence and droplets in the subcritical nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 1215–1218 (1997).
    [Crossref]
  15. A. H. Piekara, J. S. Moore, and M. S. Feld, “Analysis of self-trapping using the wave equation with high-order nonlinear electric permittivity,” Phys. Rev. A 9, 1403–1407 (1974).
    [Crossref]
  16. E. M. Wright, B. I. Lawrence, W. Torruellas, and G. I. Stegeman, “Stable self-trapping and ring formation in polydiacetylene para-toluene sulfonate,” Opt. Lett. 20, 2481–2483 (1996).
    [Crossref]
  17. K. Dimitrievski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
    [Crossref]
  18. M. L. Quiroga-Teixeiro and H. Michinel, “Stable azimuthal stationary state in quintic nonlinear optical media,” J. Opt. Soc. Am. B 14, 2004–2009 (1997).
    [Crossref]
  19. H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. 8, 128–129 (1966).
    [Crossref]
  20. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–104 (1992).
    [Crossref]
  21. H. H. Kuehl and C. Y. Zhang, “Effects of higher-order dispersion on envelope solitons,” Phys. Fluids B 2, 889–900 (1990).
    [Crossref]
  22. B. A. Malomed, D. Anderson, M. Lisak, M. L. Quiroga-Teixeiro, and L. Stenflo, “Internal dynamics of a soliton in the Zakharov model equations,” Phys. Rev. E 55, 962–968 (1997).
    [Crossref]
  23. M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A 46, 2726 (1992).
    [Crossref] [PubMed]
  24. D. Anderson, M. Bonnedal, and M. Lisak, “Nonlinear propagation of elliptically shaped Gaussian laser beams,” J. Plasma Phys. 23, 115–127 (1980).
    [Crossref]

1998 (1)

K. Dimitrievski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

1997 (3)

M. L. Quiroga-Teixeiro and H. Michinel, “Stable azimuthal stationary state in quintic nonlinear optical media,” J. Opt. Soc. Am. B 14, 2004–2009 (1997).
[Crossref]

C. Josserand and S. Rica, “Coalescence and droplets in the subcritical nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 1215–1218 (1997).
[Crossref]

B. A. Malomed, D. Anderson, M. Lisak, M. L. Quiroga-Teixeiro, and L. Stenflo, “Internal dynamics of a soliton in the Zakharov model equations,” Phys. Rev. E 55, 962–968 (1997).
[Crossref]

1996 (1)

1994 (1)

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 889–900 (1994).
[Crossref]

1992 (3)

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[Crossref] [PubMed]

M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A 46, 2726 (1992).
[Crossref] [PubMed]

J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–104 (1992).
[Crossref]

1991 (1)

J. M. Soto-Crespo, D. R. Heartley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

1990 (1)

H. H. Kuehl and C. Y. Zhang, “Effects of higher-order dispersion on envelope solitons,” Phys. Fluids B 2, 889–900 (1990).
[Crossref]

1988 (1)

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan’kov, “Soliton turbulence,” JETP Lett. 48, 82–87 (1988).

1985 (1)

V. I. Kruglov and R. A. Vlasov, “Spiral self-trapping propagation of optical beams in media with cubic nonlinearity,” Phys. Lett. A 111, 401–404 (1985).
[Crossref]

1980 (1)

D. Anderson, M. Bonnedal, and M. Lisak, “Nonlinear propagation of elliptically shaped Gaussian laser beams,” J. Plasma Phys. 23, 115–127 (1980).
[Crossref]

1979 (4)

V. G. Makhankov, G. Kummer, and A. B. Shvachka, “Interaction of two-space-dimensional classical Q solitons,” Phys. Lett. A 70, 171–176 (1979).
[Crossref]

V. G. Makhankov, “Computer and solitons,” Phys. Scr. 20, 558–562 (1979).
[Crossref]

K. Konno and H. Suzuki, “Self-focusing of laser beam in nonlinear media,” Phys. Scr. 20, 382–386 (1979).
[Crossref]

V. G. Makhankov, G. Kummer, and A. B. Shvachka, “Many dimensional U(1) solitons, their interactions, resonances and bound states,” Phys. Scr. 20, 454–461 (1979).
[Crossref]

1978 (1)

J. Oficjalski and I. Bialynicki-Birula, “Collisions of gaussons,” Acta Phys. Pol. B 9, 759–775 (1978).

1974 (1)

A. H. Piekara, J. S. Moore, and M. S. Feld, “Analysis of self-trapping using the wave equation with high-order nonlinear electric permittivity,” Phys. Rev. A 9, 1403–1407 (1974).
[Crossref]

1968 (2)

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[Crossref]

J. H. Marburger and E. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556–558 (1968).
[Crossref]

1966 (1)

H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. 8, 128–129 (1966).
[Crossref]

Akhmediev, N. N.

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[Crossref] [PubMed]

J. M. Soto-Crespo, D. R. Heartley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

Anderson, D.

K. Dimitrievski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

B. A. Malomed, D. Anderson, M. Lisak, M. L. Quiroga-Teixeiro, and L. Stenflo, “Internal dynamics of a soliton in the Zakharov model equations,” Phys. Rev. E 55, 962–968 (1997).
[Crossref]

D. Anderson, M. Bonnedal, and M. Lisak, “Nonlinear propagation of elliptically shaped Gaussian laser beams,” J. Plasma Phys. 23, 115–127 (1980).
[Crossref]

M. L. Quiroga-Teixeiro, D. Anderson, A. Berntson, M. Lisak, E. V. Vanin, and A. V. Kim, “Interaction and collision between two optical beams in a saturable nonlinear medium,” presented at the Symposium on Progress in Electromagnetics Research, Innsbruck, Austria, July 8–12, 1996.

Baker, G.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 889–900 (1994).
[Crossref]

Berntson, A.

K. Dimitrievski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

M. L. Quiroga-Teixeiro, D. Anderson, A. Berntson, M. Lisak, E. V. Vanin, and A. V. Kim, “Interaction and collision between two optical beams in a saturable nonlinear medium,” presented at the Symposium on Progress in Electromagnetics Research, Innsbruck, Austria, July 8–12, 1996.

Bialynicki-Birula, I.

J. Oficjalski and I. Bialynicki-Birula, “Collisions of gaussons,” Acta Phys. Pol. B 9, 759–775 (1978).

Bonnedal, M.

D. Anderson, M. Bonnedal, and M. Lisak, “Nonlinear propagation of elliptically shaped Gaussian laser beams,” J. Plasma Phys. 23, 115–127 (1980).
[Crossref]

Cha, M.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 889–900 (1994).
[Crossref]

Dawes, E.

J. H. Marburger and E. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556–558 (1968).
[Crossref]

Dimitrievski, K.

K. Dimitrievski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

Etemad, S.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 889–900 (1994).
[Crossref]

Feld, M. S.

A. H. Piekara, J. S. Moore, and M. S. Feld, “Analysis of self-trapping using the wave equation with high-order nonlinear electric permittivity,” Phys. Rev. A 9, 1403–1407 (1974).
[Crossref]

Gordon, J. P.

Haus, H. A.

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[Crossref]

H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. 8, 128–129 (1966).
[Crossref]

Heartley, D. R.

J. M. Soto-Crespo, D. R. Heartley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

Josserand, C.

C. Josserand and S. Rica, “Coalescence and droplets in the subcritical nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 1215–1218 (1997).
[Crossref]

Kang, J. U.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 889–900 (1994).
[Crossref]

Karlsson, M.

M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A 46, 2726 (1992).
[Crossref] [PubMed]

Kim, A. V.

M. L. Quiroga-Teixeiro, D. Anderson, A. Berntson, M. Lisak, E. V. Vanin, and A. V. Kim, “Interaction and collision between two optical beams in a saturable nonlinear medium,” presented at the Symposium on Progress in Electromagnetics Research, Innsbruck, Austria, July 8–12, 1996.

Konno, K.

K. Konno and H. Suzuki, “Self-focusing of laser beam in nonlinear media,” Phys. Scr. 20, 382–386 (1979).
[Crossref]

Kruglov, V. I.

V. I. Kruglov and R. A. Vlasov, “Spiral self-trapping propagation of optical beams in media with cubic nonlinearity,” Phys. Lett. A 111, 401–404 (1985).
[Crossref]

Kuehl, H. H.

H. H. Kuehl and C. Y. Zhang, “Effects of higher-order dispersion on envelope solitons,” Phys. Fluids B 2, 889–900 (1990).
[Crossref]

Kummer, G.

V. G. Makhankov, G. Kummer, and A. B. Shvachka, “Many dimensional U(1) solitons, their interactions, resonances and bound states,” Phys. Scr. 20, 454–461 (1979).
[Crossref]

V. G. Makhankov, G. Kummer, and A. B. Shvachka, “Interaction of two-space-dimensional classical Q solitons,” Phys. Lett. A 70, 171–176 (1979).
[Crossref]

Lawrence, B. I.

Lawrence, B. L.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 889–900 (1994).
[Crossref]

Lisak, M.

K. Dimitrievski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

B. A. Malomed, D. Anderson, M. Lisak, M. L. Quiroga-Teixeiro, and L. Stenflo, “Internal dynamics of a soliton in the Zakharov model equations,” Phys. Rev. E 55, 962–968 (1997).
[Crossref]

D. Anderson, M. Bonnedal, and M. Lisak, “Nonlinear propagation of elliptically shaped Gaussian laser beams,” J. Plasma Phys. 23, 115–127 (1980).
[Crossref]

M. L. Quiroga-Teixeiro, D. Anderson, A. Berntson, M. Lisak, E. V. Vanin, and A. V. Kim, “Interaction and collision between two optical beams in a saturable nonlinear medium,” presented at the Symposium on Progress in Electromagnetics Research, Innsbruck, Austria, July 8–12, 1996.

Makhankov, V. G.

V. G. Makhankov, “Computer and solitons,” Phys. Scr. 20, 558–562 (1979).
[Crossref]

V. G. Makhankov, G. Kummer, and A. B. Shvachka, “Many dimensional U(1) solitons, their interactions, resonances and bound states,” Phys. Scr. 20, 454–461 (1979).
[Crossref]

V. G. Makhankov, G. Kummer, and A. B. Shvachka, “Interaction of two-space-dimensional classical Q solitons,” Phys. Lett. A 70, 171–176 (1979).
[Crossref]

Malomed, B. A.

B. A. Malomed, D. Anderson, M. Lisak, M. L. Quiroga-Teixeiro, and L. Stenflo, “Internal dynamics of a soliton in the Zakharov model equations,” Phys. Rev. E 55, 962–968 (1997).
[Crossref]

Marburger, J. H.

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[Crossref]

J. H. Marburger and E. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556–558 (1968).
[Crossref]

Meth, J.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 889–900 (1994).
[Crossref]

Michinel, H.

Moore, J. S.

A. H. Piekara, J. S. Moore, and M. S. Feld, “Analysis of self-trapping using the wave equation with high-order nonlinear electric permittivity,” Phys. Rev. A 9, 1403–1407 (1974).
[Crossref]

Oficjalski, J.

J. Oficjalski and I. Bialynicki-Birula, “Collisions of gaussons,” Acta Phys. Pol. B 9, 759–775 (1978).

Ohgren, A.

K. Dimitrievski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

Piekara, A. H.

A. H. Piekara, J. S. Moore, and M. S. Feld, “Analysis of self-trapping using the wave equation with high-order nonlinear electric permittivity,” Phys. Rev. A 9, 1403–1407 (1974).
[Crossref]

Pushkarev, A. N.

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan’kov, “Soliton turbulence,” JETP Lett. 48, 82–87 (1988).

Quiroga-Teixeiro, M. L.

K. Dimitrievski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

B. A. Malomed, D. Anderson, M. Lisak, M. L. Quiroga-Teixeiro, and L. Stenflo, “Internal dynamics of a soliton in the Zakharov model equations,” Phys. Rev. E 55, 962–968 (1997).
[Crossref]

M. L. Quiroga-Teixeiro and H. Michinel, “Stable azimuthal stationary state in quintic nonlinear optical media,” J. Opt. Soc. Am. B 14, 2004–2009 (1997).
[Crossref]

M. L. Quiroga-Teixeiro, D. Anderson, A. Berntson, M. Lisak, E. V. Vanin, and A. V. Kim, “Interaction and collision between two optical beams in a saturable nonlinear medium,” presented at the Symposium on Progress in Electromagnetics Research, Innsbruck, Austria, July 8–12, 1996.

Reimhult, E.

K. Dimitrievski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

Rica, S.

C. Josserand and S. Rica, “Coalescence and droplets in the subcritical nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 1215–1218 (1997).
[Crossref]

Shvachka, A. B.

V. G. Makhankov, G. Kummer, and A. B. Shvachka, “Interaction of two-space-dimensional classical Q solitons,” Phys. Lett. A 70, 171–176 (1979).
[Crossref]

V. G. Makhankov, G. Kummer, and A. B. Shvachka, “Many dimensional U(1) solitons, their interactions, resonances and bound states,” Phys. Scr. 20, 454–461 (1979).
[Crossref]

Shvets, V. F.

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan’kov, “Soliton turbulence,” JETP Lett. 48, 82–87 (1988).

Soto-Crespo, J. M.

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[Crossref] [PubMed]

J. M. Soto-Crespo, D. R. Heartley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

Stegeman, G.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 889–900 (1994).
[Crossref]

Stegeman, G. I.

Stenflo, L.

B. A. Malomed, D. Anderson, M. Lisak, M. L. Quiroga-Teixeiro, and L. Stenflo, “Internal dynamics of a soliton in the Zakharov model equations,” Phys. Rev. E 55, 962–968 (1997).
[Crossref]

Suzuki, H.

K. Konno and H. Suzuki, “Self-focusing of laser beam in nonlinear media,” Phys. Scr. 20, 382–386 (1979).
[Crossref]

Svensson, E.

K. Dimitrievski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

Torruellas, W.

Toruellas, W.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 889–900 (1994).
[Crossref]

Vanin, E. V.

M. L. Quiroga-Teixeiro, D. Anderson, A. Berntson, M. Lisak, E. V. Vanin, and A. V. Kim, “Interaction and collision between two optical beams in a saturable nonlinear medium,” presented at the Symposium on Progress in Electromagnetics Research, Innsbruck, Austria, July 8–12, 1996.

Vlasov, R. A.

V. I. Kruglov and R. A. Vlasov, “Spiral self-trapping propagation of optical beams in media with cubic nonlinearity,” Phys. Lett. A 111, 401–404 (1985).
[Crossref]

Wagner, W. G.

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[Crossref]

Wright, E. M.

E. M. Wright, B. I. Lawrence, W. Torruellas, and G. I. Stegeman, “Stable self-trapping and ring formation in polydiacetylene para-toluene sulfonate,” Opt. Lett. 20, 2481–2483 (1996).
[Crossref]

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Other (1)

M. L. Quiroga-Teixeiro, D. Anderson, A. Berntson, M. Lisak, E. V. Vanin, and A. V. Kim, “Interaction and collision between two optical beams in a saturable nonlinear medium,” presented at the Symposium on Progress in Electromagnetics Research, Innsbruck, Austria, July 8–12, 1996.

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Figures (8)

Fig. 1
Fig. 1

Amplitude profiles for strong and weak beams (2k0δ=0.3 and 2k0δ=1.6, respectively). Dashed curve, a phase chirp with C=0.1.

Fig. 2
Fig. 2

Comparison of the observed frequencies 2k0ω0, 2k0ω, and 2k0δ as functions of beam flux N; numerical measurements. A large gap between 2k0ω and 2k0δ can be seen for NNc2=56.

Fig. 3
Fig. 3

(a) Evolution of the rms widths σx and σy and (b) beam flux N for a ground state 2k0δ=1.0 (N=23.7). The value of the initial chirp was C=0.05; L=8, γ0=5, and α=3. In (a), lower curves represent evolution for a cylindrically symmetric chirp; cf. Eq. (4). The lines that represent the evolution when the elliptic oscillation mode is excited [cf. Eq. (5)] have been shifted upward for clarity. Dashed curves, the elongations of the beam width on the x axis σx; continuous curves, the elongations along the y axis σy. It can be seen that for both types of oscillation mode the dynamics decreases until the beam adopts a dynamics-free state and that the variations of N along the beam have similar ratios during the initial stage.

Fig. 4
Fig. 4

Same as in Fig. 3 but for 2k0δ=1.65 (N=225) and L=12. In this case the elliptic oscillation mode survives for longer distances and the radiation of flux is much slower than in the cylindrical oscillation mode.

Fig. 5
Fig. 5

Same as in Fig. 3 but for 2k0δ=1.7 (N=368) and L=12. The radiation rate is even slower in this case. The small residual oscillations observed for the cylindrical mode are due to the small perturbation that the grid imposes on the beam whose width in this case is much larger than in the preceding cases.

Fig. 6
Fig. 6

Fusion of two identical strong parallel beams (2k0δ=1.65) and l=8. Distribution of the amplitude |u(x, y, z)| at (a) z=0, (b) z=10, (c) z=90, (d) z=100. The ellipticity of the final beam in this case is apparently conserved for a long time; see Fig. 4.

Fig. 7
Fig. 7

(a) Asymptotic evolution of a strong beam (2k0δ=1.6) with an initial phase curvature C=0.3. The final values of the amplitudes, the wave-vector shift, and the flux correspond to those obtained for the pure stationary solutions of the ground mode by solution of the eigenvalue problem [Eq. (3)]. (b) Evolution of the beam flux with distance. (c) Comparison of the numerically measured (circles) and analytically calculated [Eq. (11)] critical chirp for beam destruction as a function of the N.

Fig. 8
Fig. 8

Dependence of frequencies 2k0ω0, 2k0ω, and 2k0δ on beam flux N according to Eqs. (12)–(14) with Gaussian model ansatz Eq. (7), which must be compared with the values obtained numerically and plotted in Fig. 2.

Tables (1)

Tables Icon

Table 1 Beam Fusion Efficiencies as Functions of Individual Initial Beam Fluxes, N0

Equations (17)

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2ik0uz+uxx+uyy+(κ1|u|2+κ2|u|4)u=0,
u(r, z; v, δ)=ρ(r-v·r)exp iϕ(z, r; v, δ),
2ρ+(κ1ρ2+κ2ρ4-2k0δ)ρ=0,
σx=S x2|u|2dxdyS|u|2dxdy1/2
σy=S y2|u|2dxdyS|u|2dxdy1/2,
u(z=0, r)=ρ(r, 2k0δ)exp iC(x2+y2).
u(z=0, r)=ρ(r, 2k0δ)exp iC(x2-y2).
u(z=0, r)=ρ0[x-(l/2), y; 2k0δ]+ρ0[x+(l/2), y; 2k0δ],
u(x, y, z)=A exp-12x2σx2+y2σy2+i(ρxx2+ρyy2+ϕ),
L=ik0(u*uz-uz*u)-|ux|2-|uy|2+κ12|u|2+κ23|u|6.
(σx, σy)=12k021σx2+1σy2-κ1χ21σxσy-29κ2χ2σx2σy2,
ρx=k02dσxdz,ρy=k02dσydz.
σ=8κ2N29π2[4-κ1(N/π)]1/2.
Ccrit=ρx=ρy=932π1/2|κ2|(κ1N-4π)3/2N2.
2k0δ=2k0dϕdz=964κ216π2N2-κ12,
2k0ω0=-9π242κ2N2κ1πN-43/2.
2k0ω=9π222κ2N24-κ1πN.

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