Abstract

In the search for suitable new media for the propagation of (3+1) D optical light bullets, we show that nonlinear dissipation provides interesting possibilities. Using the complex cubic-quintic Ginzburg-Landau equation model with localized initial conditions, we are able to observe stable light bullet propagation or higher-order transverse pattern formation. The type of evolution depends on the model parameters.

© 2005 Optical Society of America

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  1. Y. Silberberg , “ Collapse of optical pulses ,” Opt. Lett.   15 , 1282 ( 1990 ).
    [CrossRef] [PubMed]
  2. See the review paper : B. A. Malomed , D. Mihalache , F. Wise , and L. Torner , “ Spatiotemporal solitons ,” J. Opt. B   7 , R53 ( 2005 ).,
    [CrossRef]
  3. D. Mihalache , D. Mazilu , L.-C. Crasovan , B. Malomed , F. Lederer , and L. Torner , “ Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities ,” J. Opt. B   6 , S333 ( 2004 ).
    [CrossRef]
  4. N. Akhmediev and J. M. Soto-Crespo , “ Generation of a train of 3D optical solitons in a self-focusing medium ,” Phys. Rev. A   47 , 1358 ( 1993 ).
    [CrossRef] [PubMed]
  5. Y.-F. Chen , K. Beckwitt , F. Wise , and B. A. Malomed , “ Criteria for the experimental observation of multidimensional optical solitons in saturable media ,” Phys. Rev. E   70 , 046610 ( 2004 ).
    [CrossRef]
  6. G. Duree , J. Shultz , G. Salamo , M. Segev , A. Yariv , B. Crosignani , P. Di Porto , E. Sharp , and R. Neurgaonkar , “ Observation of self-trapping of an optical beam due to the photorefractive effect ,” Phys. Rev. Lett.   71 , 533 ( 1993 ).
    [CrossRef] [PubMed]
  7. H. Meng , G. Salamo , M.-F. Shih , and M. Segev , “ Coherent collisions of photorefractive solitons ,” Opt. Lett.   22 , 448 ( 1997 ).
    [CrossRef] [PubMed]
  8. D. Skryabin and W. Firth , “ Generation and stability of optical bullets in quadratic nonlinear media ,” Optics Commun.   148 , 79 ( 1998 ).
    [CrossRef]
  9. X. Liu , L. J. Qian , and F. W. Wise , “ Generation of optical spatiotemporal solitons ,” Phys. Rev. Lett.   82 , 4631 ( 1999 ).
    [CrossRef]
  10. See the review paper: L. Torner and A. Barthélémy , “ Quadratic solitons: recent developments ,” IEEE J. Quantum Electron.   39 , 22 ( 2003 ).
    [CrossRef]
  11. L. Jankovic , P. Aboussan , M. Affolter , and G. Stegeman , “ Quadratic soliton collisions ,” Opt. Express   12 , 5562 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5562 .
    [CrossRef] [PubMed]
  12. A. Buryak , P. Di Trapani , D. Skryabin , and S. Trillo , “ Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications ,” Phys. Rep.   370 , 63 ( 2002 ).
    [CrossRef]
  13. See the Feature Section on Cavity Solitons in IEEE Journal of Quant. Electron. 39, N°2 ( 2003 ).
  14. N. N. Rosanov , “Solitons in laser systems with absorption,” in “ Dissipative solitons, ” N. Akhmediev and A. Ankiewicz eds., ( Springer-Verlag, Berlin 2005 ).
  15. S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
    [CrossRef] [PubMed]
  16. N. Akhmediev and A. Ankiewicz , eds. “Dissipative solitons,” ( Springer-Verlag, Berlin 2005 ).
  17. E. Ultanir , G. Stegeman , Ch. Lange , and F. Lederer , “ Coherent, dissipative spatial soliton interactions in semiconductor optical amplifiers ,” Opt. Lett.   29 , 283 ( 2004 ).
    [CrossRef] [PubMed]
  18. L.-C. Crasovan , B. A. Malomed , and D. Mihalache , “ Erupting, flat-top, and composite spiral solitons in the two-dimensional Ginzburg-Landau equation ,” Phys. Lett. A   289 , 59 ( 2001 ).
    [CrossRef]
  19. N. N. Rosanov , “Spatial Hysteresis and Optical Patterns,” ( Springer, Berlin Heidelberg , 2002 ) section 6.6 and references therein).
  20. W. Firth and A. Scroggie , “ Optical bullet holes: robust controllable states of a nonlinear cavity ,” Phys. Rev. Lett.   76 , 1623 ( 1996 ).
    [CrossRef] [PubMed]
  21. M. Porras , A. Parola , D. Faccio , A. Dubietis , and P. Di Trapani , “ Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses ,” Phys. Rev. Lett.   93 , 153902 ( 2004 ).
    [CrossRef] [PubMed]
  22. N. N. Akhmediev and A. Ankiewicz , “ Solitons: nonlinear pulses and beams, ” ( Chapman & Hall, London , 1997 ).

2005 (1)

See the review paper : B. A. Malomed , D. Mihalache , F. Wise , and L. Torner , “ Spatiotemporal solitons ,” J. Opt. B   7 , R53 ( 2005 ).,
[CrossRef]

2004 (5)

D. Mihalache , D. Mazilu , L.-C. Crasovan , B. Malomed , F. Lederer , and L. Torner , “ Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities ,” J. Opt. B   6 , S333 ( 2004 ).
[CrossRef]

Y.-F. Chen , K. Beckwitt , F. Wise , and B. A. Malomed , “ Criteria for the experimental observation of multidimensional optical solitons in saturable media ,” Phys. Rev. E   70 , 046610 ( 2004 ).
[CrossRef]

L. Jankovic , P. Aboussan , M. Affolter , and G. Stegeman , “ Quadratic soliton collisions ,” Opt. Express   12 , 5562 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5562 .
[CrossRef] [PubMed]

E. Ultanir , G. Stegeman , Ch. Lange , and F. Lederer , “ Coherent, dissipative spatial soliton interactions in semiconductor optical amplifiers ,” Opt. Lett.   29 , 283 ( 2004 ).
[CrossRef] [PubMed]

M. Porras , A. Parola , D. Faccio , A. Dubietis , and P. Di Trapani , “ Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses ,” Phys. Rev. Lett.   93 , 153902 ( 2004 ).
[CrossRef] [PubMed]

2003 (1)

See the review paper: L. Torner and A. Barthélémy , “ Quadratic solitons: recent developments ,” IEEE J. Quantum Electron.   39 , 22 ( 2003 ).
[CrossRef]

2002 (2)

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

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

2001 (1)

L.-C. Crasovan , B. A. Malomed , and D. Mihalache , “ Erupting, flat-top, and composite spiral solitons in the two-dimensional Ginzburg-Landau equation ,” Phys. Lett. A   289 , 59 ( 2001 ).
[CrossRef]

1999 (1)

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

1998 (1)

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

1997 (1)

1996 (1)

W. Firth and A. Scroggie , “ Optical bullet holes: robust controllable states of a nonlinear cavity ,” Phys. Rev. Lett.   76 , 1623 ( 1996 ).
[CrossRef] [PubMed]

1993 (2)

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

N. Akhmediev and J. M. Soto-Crespo , “ Generation of a train of 3D optical solitons in a self-focusing medium ,” Phys. Rev. A   47 , 1358 ( 1993 ).
[CrossRef] [PubMed]

1990 (1)

Aboussan, P.

Affolter, M.

Akhmediev, N.

N. Akhmediev and J. M. Soto-Crespo , “ Generation of a train of 3D optical solitons in a self-focusing medium ,” Phys. Rev. A   47 , 1358 ( 1993 ).
[CrossRef] [PubMed]

Akhmediev, N. N.

N. N. Akhmediev and A. Ankiewicz , “ Solitons: nonlinear pulses and beams, ” ( Chapman & Hall, London , 1997 ).

Ankiewicz, A.

N. N. Akhmediev and A. Ankiewicz , “ Solitons: nonlinear pulses and beams, ” ( Chapman & Hall, London , 1997 ).

Balle, S.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Barland, S.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Barthélémy, A.

See the review paper: L. Torner and A. Barthélémy , “ Quadratic solitons: recent developments ,” IEEE J. Quantum Electron.   39 , 22 ( 2003 ).
[CrossRef]

Beckwitt, K.

Y.-F. Chen , K. Beckwitt , F. Wise , and B. A. Malomed , “ Criteria for the experimental observation of multidimensional optical solitons in saturable media ,” Phys. Rev. E   70 , 046610 ( 2004 ).
[CrossRef]

Brambilla, M.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Buryak, A.

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

Chen, Y.-F.

Y.-F. Chen , K. Beckwitt , F. Wise , and B. A. Malomed , “ Criteria for the experimental observation of multidimensional optical solitons in saturable media ,” Phys. Rev. E   70 , 046610 ( 2004 ).
[CrossRef]

Crasovan, L.-C.

D. Mihalache , D. Mazilu , L.-C. Crasovan , B. Malomed , F. Lederer , and L. Torner , “ Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities ,” J. Opt. B   6 , S333 ( 2004 ).
[CrossRef]

L.-C. Crasovan , B. A. Malomed , and D. Mihalache , “ Erupting, flat-top, and composite spiral solitons in the two-dimensional Ginzburg-Landau equation ,” Phys. Lett. A   289 , 59 ( 2001 ).
[CrossRef]

Crosignani, B.

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

Di Porto, P.

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

Di Trapani, P.

M. Porras , A. Parola , D. Faccio , A. Dubietis , and P. Di Trapani , “ Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses ,” Phys. Rev. Lett.   93 , 153902 ( 2004 ).
[CrossRef] [PubMed]

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

Dubietis, A.

M. Porras , A. Parola , D. Faccio , A. Dubietis , and P. Di Trapani , “ Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses ,” Phys. Rev. Lett.   93 , 153902 ( 2004 ).
[CrossRef] [PubMed]

Duree, G.

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

Faccio, D.

M. Porras , A. Parola , D. Faccio , A. Dubietis , and P. Di Trapani , “ Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses ,” Phys. Rev. Lett.   93 , 153902 ( 2004 ).
[CrossRef] [PubMed]

Firth, W.

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

W. Firth and A. Scroggie , “ Optical bullet holes: robust controllable states of a nonlinear cavity ,” Phys. Rev. Lett.   76 , 1623 ( 1996 ).
[CrossRef] [PubMed]

Giudici, M.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Jäger, R.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Jankovic, L.

Knödl, T.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Lange, Ch.

Lederer, F.

E. Ultanir , G. Stegeman , Ch. Lange , and F. Lederer , “ Coherent, dissipative spatial soliton interactions in semiconductor optical amplifiers ,” Opt. Lett.   29 , 283 ( 2004 ).
[CrossRef] [PubMed]

D. Mihalache , D. Mazilu , L.-C. Crasovan , B. Malomed , F. Lederer , and L. Torner , “ Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities ,” J. Opt. B   6 , S333 ( 2004 ).
[CrossRef]

Liu, X.

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

Lugiato, L.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Maggipinto, T.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Malomed, B.

D. Mihalache , D. Mazilu , L.-C. Crasovan , B. Malomed , F. Lederer , and L. Torner , “ Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities ,” J. Opt. B   6 , S333 ( 2004 ).
[CrossRef]

Malomed, B. A.

See the review paper : B. A. Malomed , D. Mihalache , F. Wise , and L. Torner , “ Spatiotemporal solitons ,” J. Opt. B   7 , R53 ( 2005 ).,
[CrossRef]

Y.-F. Chen , K. Beckwitt , F. Wise , and B. A. Malomed , “ Criteria for the experimental observation of multidimensional optical solitons in saturable media ,” Phys. Rev. E   70 , 046610 ( 2004 ).
[CrossRef]

L.-C. Crasovan , B. A. Malomed , and D. Mihalache , “ Erupting, flat-top, and composite spiral solitons in the two-dimensional Ginzburg-Landau equation ,” Phys. Lett. A   289 , 59 ( 2001 ).
[CrossRef]

Mazilu, D.

D. Mihalache , D. Mazilu , L.-C. Crasovan , B. Malomed , F. Lederer , and L. Torner , “ Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities ,” J. Opt. B   6 , S333 ( 2004 ).
[CrossRef]

Meng, H.

Mihalache, D.

See the review paper : B. A. Malomed , D. Mihalache , F. Wise , and L. Torner , “ Spatiotemporal solitons ,” J. Opt. B   7 , R53 ( 2005 ).,
[CrossRef]

D. Mihalache , D. Mazilu , L.-C. Crasovan , B. Malomed , F. Lederer , and L. Torner , “ Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities ,” J. Opt. B   6 , S333 ( 2004 ).
[CrossRef]

L.-C. Crasovan , B. A. Malomed , and D. Mihalache , “ Erupting, flat-top, and composite spiral solitons in the two-dimensional Ginzburg-Landau equation ,” Phys. Lett. A   289 , 59 ( 2001 ).
[CrossRef]

Miller, M.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Neurgaonkar, R.

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

Parola, A.

M. Porras , A. Parola , D. Faccio , A. Dubietis , and P. Di Trapani , “ Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses ,” Phys. Rev. Lett.   93 , 153902 ( 2004 ).
[CrossRef] [PubMed]

Porras, M.

M. Porras , A. Parola , D. Faccio , A. Dubietis , and P. Di Trapani , “ Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses ,” Phys. Rev. Lett.   93 , 153902 ( 2004 ).
[CrossRef] [PubMed]

Qian, L. J.

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

Rosanov, N. N.

N. N. Rosanov , “Spatial Hysteresis and Optical Patterns,” ( Springer, Berlin Heidelberg , 2002 ) section 6.6 and references therein).

N. N. Rosanov , “Solitons in laser systems with absorption,” in “ Dissipative solitons, ” N. Akhmediev and A. Ankiewicz eds., ( Springer-Verlag, Berlin 2005 ).

Salamo, G.

H. Meng , G. Salamo , M.-F. Shih , and M. Segev , “ Coherent collisions of photorefractive solitons ,” Opt. Lett.   22 , 448 ( 1997 ).
[CrossRef] [PubMed]

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

Scroggie, A.

W. Firth and A. Scroggie , “ Optical bullet holes: robust controllable states of a nonlinear cavity ,” Phys. Rev. Lett.   76 , 1623 ( 1996 ).
[CrossRef] [PubMed]

Segev, M.

H. Meng , G. Salamo , M.-F. Shih , and M. Segev , “ Coherent collisions of photorefractive solitons ,” Opt. Lett.   22 , 448 ( 1997 ).
[CrossRef] [PubMed]

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

Sharp, E.

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

Shih, M.-F.

Shultz, J.

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

Silberberg, Y.

Skryabin, D.

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

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

Soto-Crespo, J. M.

N. Akhmediev and J. M. Soto-Crespo , “ Generation of a train of 3D optical solitons in a self-focusing medium ,” Phys. Rev. A   47 , 1358 ( 1993 ).
[CrossRef] [PubMed]

Spinelli, L.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Stegeman, G.

Tissono, G.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Torner, L.

See the review paper : B. A. Malomed , D. Mihalache , F. Wise , and L. Torner , “ Spatiotemporal solitons ,” J. Opt. B   7 , R53 ( 2005 ).,
[CrossRef]

D. Mihalache , D. Mazilu , L.-C. Crasovan , B. Malomed , F. Lederer , and L. Torner , “ Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities ,” J. Opt. B   6 , S333 ( 2004 ).
[CrossRef]

See the review paper: L. Torner and A. Barthélémy , “ Quadratic solitons: recent developments ,” IEEE J. Quantum Electron.   39 , 22 ( 2003 ).
[CrossRef]

Tredicce, J.

S. Barland , J. Tredicce , M. Brambilla , L. Lugiato , S. Balle , M. Giudici , T. Maggipinto , L. Spinelli , G. Tissono , T. Knödl , M. Miller , and R. Jäger , “ Cavity solitons as pixels in semiconductor microcavities ,” Nature   419 , 699 ( 2002 ).
[CrossRef] [PubMed]

Trillo, S.

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

Ultanir, E.

Wise, F.

See the review paper : B. A. Malomed , D. Mihalache , F. Wise , and L. Torner , “ Spatiotemporal solitons ,” J. Opt. B   7 , R53 ( 2005 ).,
[CrossRef]

Y.-F. Chen , K. Beckwitt , F. Wise , and B. A. Malomed , “ Criteria for the experimental observation of multidimensional optical solitons in saturable media ,” Phys. Rev. E   70 , 046610 ( 2004 ).
[CrossRef]

Wise, F. W.

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

Yariv, A.

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

IEEE J. Quantum Electron. (1)

See the review paper: L. Torner and A. Barthélémy , “ Quadratic solitons: recent developments ,” IEEE J. Quantum Electron.   39 , 22 ( 2003 ).
[CrossRef]

J. Opt. B (2)

See the review paper : B. A. Malomed , D. Mihalache , F. Wise , and L. Torner , “ Spatiotemporal solitons ,” J. Opt. B   7 , R53 ( 2005 ).,
[CrossRef]

D. Mihalache , D. Mazilu , L.-C. Crasovan , B. Malomed , F. Lederer , and L. Torner , “ Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities ,” J. Opt. B   6 , S333 ( 2004 ).
[CrossRef]

Nature (1)

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Supplementary Material (3)

» Media 1: MOV (301 KB)     
» Media 2: MOV (470 KB)     
» Media 3: MOV (356 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

(a) Pulse energy Q versus propagation distance z for cylindrically-symmetric initial condition (2). (b) The evolution for the initial condition (3) with elliptic symmetry. The parameter ε in the equation (1) is different for each curve. Other parameters are fixed for all simulations. Their values are shown inside each plot.

Fig. 2.
Fig. 2.

(a) The shape of the optical bullet (stationary solution) in the (x,y) plane (b) The profile of the same bullet along the t-axis.

Fig. 3.
Fig. 3.

The shape of the stable optical bullet in t (upper row) and in the (x,y) plane (lower row). The three plots (a), (b) and (c) for various ε, correspond to the three curves with the same color coding in Fig. 1. (b).

Fig. 4.
Fig. 4.

On the left (a), we show pulse energy Q versus propagation distance z for two different elliptically-symmetric initial conditions. The equation parameters are the same in each case. The initial condition for the blue curve provides convergence to an optical bullet, while that for the red curve does not. The movie on the right demonstrates the field evolution for the case of the red line in the left diagram. It should be noted that when the field expansion reaches the boundaries, the simulation corresponds to a collision between the central light pulse and ghost neighbors. [Media 1]

Fig. 5.
Fig. 5.

Movies of pulse evolution and pattern formation when stable optical bullets do not exist. The cubic gain is ε = 1.6 (left), ε = 1.8 (right). As in Fig. 4(b), the simulations describe the evolution of an isolated pulse, valid as long as the field does not reach significantly the boundaries. [Media 2] [Media 3]

Fig. 6.
Fig. 6.

Pulse evolution in the t-domain, in a parameter domain where stable optical bullets do not exist. The two cases are for ε = 1.6 (left) and ε = 1.8 (right).

Equations (5)

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i ψ z + D 2 ψ t t + 1 2 ψ x x + 1 2 ψ y y + ψ 2 ψ + ν ψ 4 ψ = iδψ + i ε ψ 2 ψ + ψ t t + ψ 4 ψ .
ψ t x y 0 = 1.8 exp ( t 2 x 2 y 2 ) .
Q ( z ) = + ψ t x y z 2 dx dy dt .
ψ t x y 0 = 4.0 exp ( ( t 1.3 ) 2 x 2 ( y 0.9 ) 2 ) .
ψ t x y 0 = 4.0 exp ( t 2 x 2 ( y 0.8 ) 2 ) .

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