Abstract

We introduce a three-dimensional (3D) model of optical media with the quadratic (χ(2)) nonlinearity and an effective 2D isotropic harmonic-oscillator (HO) potential. While it is well known that 3D χ(2) solitons with embedded vorticity (“vortical light bullets”) are unstable in the free space, we demonstrate that they have a broad stability region in the present model, being supported by the HO potential against the splitting instability. The shape of the vortical solitons may be accurately predicted by the variational approximation (VA). They exist above a threshold value of the total energy (norm) and below another critical value, which determines a stability boundary. The existence threshold vanishes is a part of the parameter space, depending on the mismatch parameter, which is explained by means of the comparison with the 2D counterpart of the system. Above the stability boundary, the vortex features shape oscillations, periodically breaking its axisymmetric form and restoring it. Collisions between vortices moving in the longitudinal direction are studied too. The collision is strongly inelastic at relatively small values of the velocities, breaking the two colliding vortices into three, with the same vorticity. The results suggest a possibility of the creation of stable 3D optical solitons with the intrinsic vorticity.

© 2013 OSA

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  1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron.28, 1691–1740 (1996).
    [CrossRef]
  2. C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Progr. Opt.41, 483–568 (2000).
    [CrossRef]
  3. A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep.370, 63–235 (2002).
    [CrossRef]
  4. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semicl. Opt.7, R53–R72 (2005).
    [CrossRef]
  5. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  6. A. A. Kanashov and A. M. Rubenchik, “On diffraction and dispersion effect on three wave interaction,” Physica D4, 122–134 (1981).
    [CrossRef]
  7. D. Mihalache, “Linear and nonlinear light bullets: Recent theoretical and experimental studies,” Rom. J. Phys.57, 352–371 (2012).
  8. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett.74, 5036–5039 (1995).
    [CrossRef] [PubMed]
  9. B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E56, 4725–4735 (1997).
    [CrossRef]
  10. D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media,” Opt. Commun.148, 79–84 (1998).
    [CrossRef]
  11. D. Mihalache, D. Mazilu, J. Dorring, and L. Torner, “Elliptical light bullets,” Opt. Commun.159, 129–138 (1999).
    [CrossRef]
  12. D. Mihalache, D. Mazilu, L.-C. Crasovan, L. Torner, B. A. Malomed, and F. Lederer, “Three-dimensional walking spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 7340–7347 (2000).
    [CrossRef]
  13. X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett.82, 4631–4634 (1999).
    [CrossRef]
  14. X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 1328–1340 (2000).
    [CrossRef]
  15. H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep.523, 61–126 (2013).
    [CrossRef]
  16. F. A. Bovino, M. Braccini, and C. Sibilia, “Orbital angular momentum in noncollinear second-harmonic generation by off-axis vortex beams,” J. Opt. Soc. A, B28, 2806–2811 (201si1).
  17. W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett.79, 2450–2453 (1997).
    [CrossRef]
  18. L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett.33, 608–610 (1997).
    [CrossRef]
  19. D. V. Skryabin and W. J. Firth, “Instabilities of higher-order parametric solitons: Filamentation versus coalescence,” Phys. Rev. E58, R1252–R1255 (1998).
    [CrossRef]
  20. J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B15, 625–627 (1998).
    [CrossRef]
  21. D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett.23, 1444–1446 (1998).
    [CrossRef]
  22. J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in type II second-harmonic generation,” Opt. Commun.149, 77–83 (1998).
    [CrossRef]
  23. G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett.26, 163–165 (2001).
    [CrossRef]
  24. V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt.39, 2277–2291 (1992).
    [CrossRef]
  25. C. J. Pethick and H. Smith, Bose-Einstein condensate in dilute gas (Cambridge University Press, 2008).
    [CrossRef]
  26. F. Dalfovo and S. Stringari, “Bosons in anisotropic traps: Ground state and vortices,” Phys. Rev. A53, 2477–2485 (1996).
    [CrossRef] [PubMed]
  27. R. J. Dodd, “Approximate solutions of the nonlinear Schrödinger equation for ground and excited states of Bose-Einstein condensates,” J. Res. Natl. Inst. Stand. Technol.101, 545–552 (1996).
    [CrossRef]
  28. T. J. Alexander and L. Bergé, “Ground states and vortices of matter-wave condensates and optical guided waves,” Phys. Rev. E65, 026611 (2002).
    [CrossRef]
  29. L. D. Carr and C. W. Clark, “Vortices in attractive Bose-Einstein condensates in two dimensions,” Phys. Rev. Lett.97, 010403 (2006).
    [CrossRef] [PubMed]
  30. D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A73, 043615 (2006).
    [CrossRef]
  31. L. D. Carr and C. W. Clark, “Vortices and ring solitons in Bose-Einstein condensates,” Phys. Rev. A74, 043613 (2006).
    [CrossRef]
  32. G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A77, 043607 (2008).
    [CrossRef]
  33. H. Sakaguchi and B. A. Malomed, “Stabilizing single- and two-color vortex beams in quadratic media by a trapping potential,” J. Opt. Soc. Am. B29, 2741–2748 (2012).
    [CrossRef]
  34. F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett.85, 2181–2183 (2004).
    [CrossRef]
  35. F. Luan, A. K. George, T. D. Hedeley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. S. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett.29, 2369–2371 (2004).
    [CrossRef] [PubMed]
  36. P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates,” Phys. Rev. Lett.81, 3055–3058 (1998).
    [CrossRef]
  37. D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett.84, 5029–5033 (2000).
    [CrossRef] [PubMed]
  38. J. J. Hope and M. K. Olsen, “Quantum superchemistry: Dynamical quantum effects in coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett.86, 3220–3223 (2001).
    [CrossRef] [PubMed]
  39. T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A66, 043607 (2002).
    [CrossRef]
  40. D. L. Feder, C. W. Clark, and B. I. Schneider, “Vortex stability of interacting Bose-Einstein condensates confined in anisotropic harmonic traps,” Phys. Rev. Lett.82, 4956 (1999).
    [CrossRef]
  41. I. N. Towers, B. A. Malomed, and F. W. Wise, “Light bullets in quadratic media with normal dispersion at the second harmonic,” Phys. Rev. Lett.90, 123902 (2003).
    [CrossRef] [PubMed]
  42. Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E71, 016616 (2005).
    [CrossRef]
  43. B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials”. Europhys. Lett.63, 642–648 (2003).
    [CrossRef]
  44. J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett.28, 2094–2096 (2003).
    [CrossRef] [PubMed]
  45. D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E70, 055603 (2004).
    [CrossRef]
  46. D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E66, 016613 (2002).
    [CrossRef]

2013 (1)

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep.523, 61–126 (2013).
[CrossRef]

2012 (2)

D. Mihalache, “Linear and nonlinear light bullets: Recent theoretical and experimental studies,” Rom. J. Phys.57, 352–371 (2012).

H. Sakaguchi and B. A. Malomed, “Stabilizing single- and two-color vortex beams in quadratic media by a trapping potential,” J. Opt. Soc. Am. B29, 2741–2748 (2012).
[CrossRef]

2008 (1)

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A77, 043607 (2008).
[CrossRef]

2006 (3)

L. D. Carr and C. W. Clark, “Vortices in attractive Bose-Einstein condensates in two dimensions,” Phys. Rev. Lett.97, 010403 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A73, 043615 (2006).
[CrossRef]

L. D. Carr and C. W. Clark, “Vortices and ring solitons in Bose-Einstein condensates,” Phys. Rev. A74, 043613 (2006).
[CrossRef]

2005 (2)

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semicl. Opt.7, R53–R72 (2005).
[CrossRef]

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E71, 016616 (2005).
[CrossRef]

2004 (3)

F. Luan, A. K. George, T. D. Hedeley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. S. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett.29, 2369–2371 (2004).
[CrossRef] [PubMed]

F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett.85, 2181–2183 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E70, 055603 (2004).
[CrossRef]

2003 (3)

J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett.28, 2094–2096 (2003).
[CrossRef] [PubMed]

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials”. Europhys. Lett.63, 642–648 (2003).
[CrossRef]

I. N. Towers, B. A. Malomed, and F. W. Wise, “Light bullets in quadratic media with normal dispersion at the second harmonic,” Phys. Rev. Lett.90, 123902 (2003).
[CrossRef] [PubMed]

2002 (4)

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E66, 016613 (2002).
[CrossRef]

T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A66, 043607 (2002).
[CrossRef]

T. J. Alexander and L. Bergé, “Ground states and vortices of matter-wave condensates and optical guided waves,” Phys. Rev. E65, 026611 (2002).
[CrossRef]

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep.370, 63–235 (2002).
[CrossRef]

2001 (2)

J. J. Hope and M. K. Olsen, “Quantum superchemistry: Dynamical quantum effects in coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett.86, 3220–3223 (2001).
[CrossRef] [PubMed]

G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett.26, 163–165 (2001).
[CrossRef]

2000 (4)

D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett.84, 5029–5033 (2000).
[CrossRef] [PubMed]

X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 1328–1340 (2000).
[CrossRef]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Progr. Opt.41, 483–568 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, L. Torner, B. A. Malomed, and F. Lederer, “Three-dimensional walking spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 7340–7347 (2000).
[CrossRef]

1999 (3)

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett.82, 4631–4634 (1999).
[CrossRef]

D. Mihalache, D. Mazilu, J. Dorring, and L. Torner, “Elliptical light bullets,” Opt. Commun.159, 129–138 (1999).
[CrossRef]

D. L. Feder, C. W. Clark, and B. I. Schneider, “Vortex stability of interacting Bose-Einstein condensates confined in anisotropic harmonic traps,” Phys. Rev. Lett.82, 4956 (1999).
[CrossRef]

1998 (6)

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B15, 625–627 (1998).
[CrossRef]

D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett.23, 1444–1446 (1998).
[CrossRef]

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates,” Phys. Rev. Lett.81, 3055–3058 (1998).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media,” Opt. Commun.148, 79–84 (1998).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Instabilities of higher-order parametric solitons: Filamentation versus coalescence,” Phys. Rev. E58, R1252–R1255 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in type II second-harmonic generation,” Opt. Commun.149, 77–83 (1998).
[CrossRef]

1997 (3)

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett.79, 2450–2453 (1997).
[CrossRef]

L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett.33, 608–610 (1997).
[CrossRef]

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E56, 4725–4735 (1997).
[CrossRef]

1996 (3)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron.28, 1691–1740 (1996).
[CrossRef]

F. Dalfovo and S. Stringari, “Bosons in anisotropic traps: Ground state and vortices,” Phys. Rev. A53, 2477–2485 (1996).
[CrossRef] [PubMed]

R. J. Dodd, “Approximate solutions of the nonlinear Schrödinger equation for ground and excited states of Bose-Einstein condensates,” J. Res. Natl. Inst. Stand. Technol.101, 545–552 (1996).
[CrossRef]

1995 (1)

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett.74, 5036–5039 (1995).
[CrossRef] [PubMed]

1992 (1)

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt.39, 2277–2291 (1992).
[CrossRef]

1981 (1)

A. A. Kanashov and A. M. Rubenchik, “On diffraction and dispersion effect on three wave interaction,” Physica D4, 122–134 (1981).
[CrossRef]

Agrawal, G. P.

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

Alexander, T. J.

T. J. Alexander and L. Bergé, “Ground states and vortices of matter-wave condensates and optical guided waves,” Phys. Rev. E65, 026611 (2002).
[CrossRef]

Anderson, D.

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E56, 4725–4735 (1997).
[CrossRef]

Baizakov, B. B.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials”. Europhys. Lett.63, 642–648 (2003).
[CrossRef]

Beckwitt, K.

X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 1328–1340 (2000).
[CrossRef]

Bergé, L.

T. J. Alexander and L. Bergé, “Ground states and vortices of matter-wave condensates and optical guided waves,” Phys. Rev. E65, 026611 (2002).
[CrossRef]

Berntson, A.

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E56, 4725–4735 (1997).
[CrossRef]

Bird, D. M.

Bovino, F. A.

F. A. Bovino, M. Braccini, and C. Sibilia, “Orbital angular momentum in noncollinear second-harmonic generation by off-axis vortex beams,” J. Opt. Soc. A, B28, 2806–2811 (201si1).

Braccini, M.

F. A. Bovino, M. Braccini, and C. Sibilia, “Orbital angular momentum in noncollinear second-harmonic generation by off-axis vortex beams,” J. Opt. Soc. A, B28, 2806–2811 (201si1).

Buryak, A. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep.370, 63–235 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E66, 016613 (2002).
[CrossRef]

Carr, L. D.

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A77, 043607 (2008).
[CrossRef]

L. D. Carr and C. W. Clark, “Vortices in attractive Bose-Einstein condensates in two dimensions,” Phys. Rev. Lett.97, 010403 (2006).
[CrossRef] [PubMed]

L. D. Carr and C. W. Clark, “Vortices and ring solitons in Bose-Einstein condensates,” Phys. Rev. A74, 043613 (2006).
[CrossRef]

Carretero-González, R.

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A77, 043607 (2008).
[CrossRef]

Clark, C. W.

L. D. Carr and C. W. Clark, “Vortices in attractive Bose-Einstein condensates in two dimensions,” Phys. Rev. Lett.97, 010403 (2006).
[CrossRef] [PubMed]

L. D. Carr and C. W. Clark, “Vortices and ring solitons in Bose-Einstein condensates,” Phys. Rev. A74, 043613 (2006).
[CrossRef]

D. L. Feder, C. W. Clark, and B. I. Schneider, “Vortex stability of interacting Bose-Einstein condensates confined in anisotropic harmonic traps,” Phys. Rev. Lett.82, 4956 (1999).
[CrossRef]

Cojocaru, C.

Crasovan, L. C.

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E71, 016616 (2005).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E66, 016613 (2002).
[CrossRef]

Crasovan, L.-C.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E70, 055603 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, L. Torner, B. A. Malomed, and F. Lederer, “Three-dimensional walking spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 7340–7347 (2000).
[CrossRef]

Dalfovo, F.

F. Dalfovo and S. Stringari, “Bosons in anisotropic traps: Ground state and vortices,” Phys. Rev. A53, 2477–2485 (1996).
[CrossRef] [PubMed]

de Vivie-Riedle, R.

T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A66, 043607 (2002).
[CrossRef]

Di Trapani, P.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep.370, 63–235 (2002).
[CrossRef]

Dodd, R. J.

R. J. Dodd, “Approximate solutions of the nonlinear Schrödinger equation for ground and excited states of Bose-Einstein condensates,” J. Res. Natl. Inst. Stand. Technol.101, 545–552 (1996).
[CrossRef]

Dorring, J.

D. Mihalache, D. Mazilu, J. Dorring, and L. Torner, “Elliptical light bullets,” Opt. Commun.159, 129–138 (1999).
[CrossRef]

Drummond, P.

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E56, 4725–4735 (1997).
[CrossRef]

Drummond, P. D.

D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett.84, 5029–5033 (2000).
[CrossRef] [PubMed]

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates,” Phys. Rev. Lett.81, 3055–3058 (1998).
[CrossRef]

Du, F.

F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett.85, 2181–2183 (2004).
[CrossRef]

Etrich, C.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Progr. Opt.41, 483–568 (2000).
[CrossRef]

Feder, D. L.

D. L. Feder, C. W. Clark, and B. I. Schneider, “Vortex stability of interacting Bose-Einstein condensates confined in anisotropic harmonic traps,” Phys. Rev. Lett.82, 4956 (1999).
[CrossRef]

Firth, W. J.

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media,” Opt. Commun.148, 79–84 (1998).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Instabilities of higher-order parametric solitons: Filamentation versus coalescence,” Phys. Rev. E58, R1252–R1255 (1998).
[CrossRef]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett.79, 2450–2453 (1997).
[CrossRef]

Frantzeskakis, D. J.

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A77, 043607 (2008).
[CrossRef]

George, A. K.

Gordienko, S.

T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A66, 043607 (2002).
[CrossRef]

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron.28, 1691–1740 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett.74, 5036–5039 (1995).
[CrossRef] [PubMed]

He, H.

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates,” Phys. Rev. Lett.81, 3055–3058 (1998).
[CrossRef]

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E56, 4725–4735 (1997).
[CrossRef]

Hedeley, T. D.

Heinzen, D. J.

D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett.84, 5029–5033 (2000).
[CrossRef] [PubMed]

Herring, G.

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A77, 043607 (2008).
[CrossRef]

Hope, J. J.

J. J. Hope and M. K. Olsen, “Quantum superchemistry: Dynamical quantum effects in coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett.86, 3220–3223 (2001).
[CrossRef] [PubMed]

Hornung, T.

T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A66, 043607 (2002).
[CrossRef]

Kanashov, A. A.

A. A. Kanashov and A. M. Rubenchik, “On diffraction and dispersion effect on three wave interaction,” Physica D4, 122–134 (1981).
[CrossRef]

Kartashov, Y. V.

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E71, 016616 (2005).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E70, 055603 (2004).
[CrossRef]

Kevrekidis, P. G.

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A77, 043607 (2008).
[CrossRef]

Kheruntsyan, K. V.

D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett.84, 5029–5033 (2000).
[CrossRef] [PubMed]

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates,” Phys. Rev. Lett.81, 3055–3058 (1998).
[CrossRef]

Kivshar, Y. S.

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

Knight, J. C.

Kruglov, V. I.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt.39, 2277–2291 (1992).
[CrossRef]

Leblond, H.

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep.523, 61–126 (2013).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A73, 043615 (2006).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E70, 055603 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E66, 016613 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, L. Torner, B. A. Malomed, and F. Lederer, “Three-dimensional walking spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 7340–7347 (2000).
[CrossRef]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Progr. Opt.41, 483–568 (2000).
[CrossRef]

Lisak, M.

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E56, 4725–4735 (1997).
[CrossRef]

Liu, X.

X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 1328–1340 (2000).
[CrossRef]

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett.82, 4631–4634 (1999).
[CrossRef]

Logvin, Y. A.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt.39, 2277–2291 (1992).
[CrossRef]

Lu, Y. W.

F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett.85, 2181–2183 (2004).
[CrossRef]

Luan, F.

Malomed, B. A.

H. Sakaguchi and B. A. Malomed, “Stabilizing single- and two-color vortex beams in quadratic media by a trapping potential,” J. Opt. Soc. Am. B29, 2741–2748 (2012).
[CrossRef]

D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A73, 043615 (2006).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semicl. Opt.7, R53–R72 (2005).
[CrossRef]

I. N. Towers, B. A. Malomed, and F. W. Wise, “Light bullets in quadratic media with normal dispersion at the second harmonic,” Phys. Rev. Lett.90, 123902 (2003).
[CrossRef] [PubMed]

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials”. Europhys. Lett.63, 642–648 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E66, 016613 (2002).
[CrossRef]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Progr. Opt.41, 483–568 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, L. Torner, B. A. Malomed, and F. Lederer, “Three-dimensional walking spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 7340–7347 (2000).
[CrossRef]

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E56, 4725–4735 (1997).
[CrossRef]

Martorell, J.

Mazilu, D.

D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A73, 043615 (2006).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E70, 055603 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E66, 016613 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, L. Torner, B. A. Malomed, and F. Lederer, “Three-dimensional walking spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 7340–7347 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, J. Dorring, and L. Torner, “Elliptical light bullets,” Opt. Commun.159, 129–138 (1999).
[CrossRef]

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett.74, 5036–5039 (1995).
[CrossRef] [PubMed]

Mihalache, D.

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep.523, 61–126 (2013).
[CrossRef]

D. Mihalache, “Linear and nonlinear light bullets: Recent theoretical and experimental studies,” Rom. J. Phys.57, 352–371 (2012).

D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A73, 043615 (2006).
[CrossRef]

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E71, 016616 (2005).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semicl. Opt.7, R53–R72 (2005).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E70, 055603 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E66, 016613 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, L. Torner, B. A. Malomed, and F. Lederer, “Three-dimensional walking spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 7340–7347 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, J. Dorring, and L. Torner, “Elliptical light bullets,” Opt. Commun.159, 129–138 (1999).
[CrossRef]

Molina-Terriza, G.

Musslimani, Z. H.

Olsen, M. K.

J. J. Hope and M. K. Olsen, “Quantum superchemistry: Dynamical quantum effects in coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett.86, 3220–3223 (2001).
[CrossRef] [PubMed]

Pearce, G. J.

Peschel, T.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Progr. Opt.41, 483–568 (2000).
[CrossRef]

Peschel, U.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Progr. Opt.41, 483–568 (2000).
[CrossRef]

Pethick, C. J.

C. J. Pethick and H. Smith, Bose-Einstein condensate in dilute gas (Cambridge University Press, 2008).
[CrossRef]

Petrov, D. V.

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in type II second-harmonic generation,” Opt. Commun.149, 77–83 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B15, 625–627 (1998).
[CrossRef]

D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett.23, 1444–1446 (1998).
[CrossRef]

L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett.33, 608–610 (1997).
[CrossRef]

Qian, L. J.

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett.82, 4631–4634 (1999).
[CrossRef]

Rubenchik, A. M.

A. A. Kanashov and A. M. Rubenchik, “On diffraction and dispersion effect on three wave interaction,” Physica D4, 122–134 (1981).
[CrossRef]

Russell, P. S. J.

Sakaguchi, H.

Salerno, M.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials”. Europhys. Lett.63, 642–648 (2003).
[CrossRef]

Schneider, B. I.

D. L. Feder, C. W. Clark, and B. I. Schneider, “Vortex stability of interacting Bose-Einstein condensates confined in anisotropic harmonic traps,” Phys. Rev. Lett.82, 4956 (1999).
[CrossRef]

Sibilia, C.

F. A. Bovino, M. Braccini, and C. Sibilia, “Orbital angular momentum in noncollinear second-harmonic generation by off-axis vortex beams,” J. Opt. Soc. A, B28, 2806–2811 (201si1).

Skryabin, D. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep.370, 63–235 (2002).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Instabilities of higher-order parametric solitons: Filamentation versus coalescence,” Phys. Rev. E58, R1252–R1255 (1998).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media,” Opt. Commun.148, 79–84 (1998).
[CrossRef]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett.79, 2450–2453 (1997).
[CrossRef]

Smith, H.

C. J. Pethick and H. Smith, Bose-Einstein condensate in dilute gas (Cambridge University Press, 2008).
[CrossRef]

Soto-Crespo, J. M.

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B15, 625–627 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in type II second-harmonic generation,” Opt. Commun.149, 77–83 (1998).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron.28, 1691–1740 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett.74, 5036–5039 (1995).
[CrossRef] [PubMed]

Stringari, S.

F. Dalfovo and S. Stringari, “Bosons in anisotropic traps: Ground state and vortices,” Phys. Rev. A53, 2477–2485 (1996).
[CrossRef] [PubMed]

Torner, L.

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E71, 016616 (2005).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semicl. Opt.7, R53–R72 (2005).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E70, 055603 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E66, 016613 (2002).
[CrossRef]

G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett.26, 163–165 (2001).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, L. Torner, B. A. Malomed, and F. Lederer, “Three-dimensional walking spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 7340–7347 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, J. Dorring, and L. Torner, “Elliptical light bullets,” Opt. Commun.159, 129–138 (1999).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in type II second-harmonic generation,” Opt. Commun.149, 77–83 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B15, 625–627 (1998).
[CrossRef]

D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett.23, 1444–1446 (1998).
[CrossRef]

L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett.33, 608–610 (1997).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron.28, 1691–1740 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett.74, 5036–5039 (1995).
[CrossRef] [PubMed]

Torres, J. P.

Torruellas, W. E.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett.74, 5036–5039 (1995).
[CrossRef] [PubMed]

Towers, I.

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E66, 016613 (2002).
[CrossRef]

Towers, I. N.

I. N. Towers, B. A. Malomed, and F. W. Wise, “Light bullets in quadratic media with normal dispersion at the second harmonic,” Phys. Rev. Lett.90, 123902 (2003).
[CrossRef] [PubMed]

Trillo, S.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep.370, 63–235 (2002).
[CrossRef]

VanStryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett.74, 5036–5039 (1995).
[CrossRef] [PubMed]

Verhaar, B. J.

T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A66, 043607 (2002).
[CrossRef]

Vilaseca, R.

Volkov, V. M.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt.39, 2277–2291 (1992).
[CrossRef]

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett.74, 5036–5039 (1995).
[CrossRef] [PubMed]

Wise, F.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semicl. Opt.7, R53–R72 (2005).
[CrossRef]

X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E62, 1328–1340 (2000).
[CrossRef]

Wise, F. W.

I. N. Towers, B. A. Malomed, and F. W. Wise, “Light bullets in quadratic media with normal dispersion at the second harmonic,” Phys. Rev. Lett.90, 123902 (2003).
[CrossRef] [PubMed]

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett.82, 4631–4634 (1999).
[CrossRef]

Wright, E. M.

Wu, S. T.

F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett.85, 2181–2183 (2004).
[CrossRef]

Wynar, R.

D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett.84, 5029–5033 (2000).
[CrossRef] [PubMed]

Xu, Z. Y.

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E71, 016616 (2005).
[CrossRef]

Yang, J.

Appl. Phys. Lett. (1)

F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett.85, 2181–2183 (2004).
[CrossRef]

Electron. Lett. (1)

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Figures (7)

Fig. 1
Fig. 1

Plots of |u(x, y, τ)| in the cross section drawn through τ = T/2 ≡ 25 (a) and y = 12.5 (b).

Fig. 2
Fig. 2

(a) The plot of |u(xL/2, y, τ)| at y = L/2, τ = T/2 obtained from the direct numerical solution (the solid curve) and its VA-predicted counterpart, u0|x|exp (−α1x2), with u0 = 0.704, α1 = 0.268 (the dashed curve). (b) The same for |u(x, y, τT/2)| at x = 1.663, y = L/2, the respective VA-predicted profile being u0|x|exp (−α1x2) sech(βτ) with u0 = 0.704, α1 = 0.268, β = 0.438.

Fig. 3
Fig. 3

(a) The slope of the vortical profile at the center, defined as per Eq. (13), produced by the numerical results (rhombuses), and by the VA (dashed curves) at q = 1 and 0.6 for D = 1. (b) The critical value, Nc1, as predicted by the VA (the solid curve) and found from the numerical results (rhombuses), for D = 1. The vortex solitons do not exist at N < Nc. (c) The same as in (a), but for D = 0 (the case of zero GVD of the SH component).

Fig. 4
Fig. 4

(a) The perturbed evolution of the Fourier amplitudes u1 and u−1, defined as per Eq. (14), for D = 1. (b) The same for D = 0.

Fig. 5
Fig. 5

Plots of |u(x, y)| in the cross section drawn through τ = T/2 at z = 110 (a) and z = 150 (b), for the oscillatory unstable vortex soliton with N = 185.

Fig. 6
Fig. 6

The threshold for the onset of the oscillatory instability of the axisymmetric vortex soliton, shown along with the existence threshold, Nc1, for D = 1 (a) and D = 0. Dependence Nc1(q) in panel (a) is tantamount to that shown in Fig. 3(b).

Fig. 7
Fig. 7

Collisions between moving vortex solitons with identical vorticities, under the action of kicks ω = ±0.5 (a); identical vorticities and ω = ±1 (b); opposite vorticities, ±1, and kicks ω = ±0.5 (c) and ω = ±1 (d). Shown are profiles in the cross section drawn in the longitudinal (τ) direction through xL/2 = 1.66 and y = L/2.

Equations (14)

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i u z + 1 2 ( 2 x 2 + 2 y 2 + 2 τ 2 ) u + u * v U ( x , y ) u = 0 , 2 i v z + 1 2 ( 2 x 2 + 2 y 2 + D 2 τ 2 ) v q v + 1 2 u 2 4 U ( x , y ) v = 0 ,
N = ( | u | 2 + 4 | v | 2 ) d x d y d τ ,
M = Im { u * ( y u x x u y ) } d x d y d τ ,
P = Im { u * u τ } d x d y d τ ,
H = { [ 1 2 ( | u x | 2 + | u y | 2 + | u τ | 2 + | v x | 2 + | v y | 2 + D | v τ | 2 ) ] + U ( r ) ( | u | 2 + 4 | v | 2 ) + q | v | 2 1 2 ( u 2 v * + u 2 * v ) } d x d y d τ .
L = { [ i u z u * + 2 i v z v * 1 2 ( | u x | 2 + | u y | 2 + | u τ | 2 + | v x | 2 + | v y | 2 + D | v τ | 2 ) ] U ( r ) ( | u | 2 + 4 | v | 2 ) q | v | 2 + 1 2 ( u 2 v * + u 2 * v ) } d x d y d τ .
u = u 0 r exp ( i k z α 1 r 2 + i θ ) cosh ( β τ ) , v = v 0 r 2 exp ( i k z α 2 r 2 + 2 i θ ) cosh ( β τ ) .
N = 2 π + d τ 0 r d r ( | u | 2 + 4 | v | 2 ) = 4 π β ( u 0 2 8 α 1 3 + v 0 2 2 α 2 3 )
L 2 π = k N 2 π u 0 2 2 α 1 β 3 v 0 2 4 α 2 β u 0 2 β 24 α 1 2 D v 0 2 β 24 α 2 3 Ω 2 u 0 2 8 α 1 3 β 3 Ω 2 v 0 2 4 α 2 4 β q v 0 2 4 α 2 3 β + π u 0 2 v 0 2 β ( 2 α 1 + α 2 ) 3 .
L 2 π = k N 0 u 0 2 2 α 1 β 3 α 2 4 β ( N 0 β u 0 2 4 α 1 2 ) u 0 2 β 24 α 1 2 D β 24 ( N 0 β u 0 2 4 α 1 2 ) Ω 2 u 0 2 8 α 1 3 β ( 3 Ω 2 4 α 2 β + q 4 β ) ( N 0 β u 0 2 4 α 1 2 ) + π u 0 2 α 2 3 / 2 2 β ( 2 α 1 + α 2 ) 3 ( N 0 β u 0 2 4 α 1 2 ) 1 / 2 .
u 0 α 1 β + 3 u 0 α 2 8 β α 1 2 u 0 β 12 α 1 2 + D β u 0 48 α 1 2 Ω 2 u 0 4 α 1 3 β + 3 Ω 2 u 0 8 α 2 β α 1 2 + q u 0 8 β α 1 2 + π u 0 α 2 3 / 2 β ( 2 α 1 + α 2 ) 3 ( N 0 β u 0 2 4 α 1 2 ) 1 / 2 π u 0 3 α 2 3 / 2 8 β α 1 2 ( 2 α 1 + α 2 ) 3 ( N 0 β u 0 2 / ( 4 α 1 2 ) ) 1 / 2 = 0 , u 0 2 2 α 1 2 β ( 3 α 2 2 β + D β 12 + 3 Ω 2 2 α 2 β ) u 0 2 4 α 1 3 + u 0 2 β 12 α 1 3 + 3 Ω 2 u 0 2 8 Ω 1 4 β 3 π u 0 2 α 2 3 / 2 β ( 2 α 1 + α 2 ) 4 ( N 0 β u 0 2 4 α 1 2 ) 1 / 2 + π u 0 4 8 β α 1 3 ( 2 α 1 + α 2 ) 3 ( N 0 β u 0 2 4 α 1 ) 1 / 2 = 0 , ( 3 4 β + 3 Ω 2 4 α 2 2 β ) ( N 0 β u 0 2 4 α 1 2 ) + ( 3 π u 0 2 α 2 3 / 2 2 β ( 2 α 1 + α 2 ) 4 + 3 π u 0 2 α 2 1 / 2 4 β ( 2 α 1 + α 2 ) 3 ) ( N 0 β u 0 2 4 α 1 2 ) 1 / 2 = 0 , u 0 2 2 α 1 β 2 ( 3 α 2 2 β + D β 12 + 3 Ω 2 2 α 2 β ) N 0 2 + ( 3 α 2 4 β D 24 + 3 Ω 2 4 α 2 β 2 ) ( N 0 β u 0 2 4 α 1 2 ) u 0 2 24 α 1 2 + Ω 2 u 0 2 8 α 1 3 β 2 q u 0 2 16 β 2 α 1 2 π u 0 2 α 1 3 / 2 2 β 2 ( 2 α 1 + α 2 ) 3 ( N 0 β u 0 2 4 α 1 2 ) 1 / 2 + π u 0 2 N 0 α 2 3 / 2 4 β ( 2 α 1 + α 2 ) 3 ( N 0 β u 0 2 4 α 1 2 ) 1 / 2 = 0.
u ( x , y , τ ) = ( r / 2 ) exp [ ( r / 2 ) 2 ( τ / 2 ) 2 + i θ ] , v ( x , y , τ ) = 0.
slope r 1 | u ( r , τ ) | at r 0 ,
u ± 1 ( z ) = | u ( x , y , z , τ ) e i θ d x d y d τ | ,

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