Abstract

We report on the existence, stability and dynamical properties of two-dimensional self-localized vortices with azimuthal numbers up to 4 in a simple model for lasers with frequency-selective feedback. We build the full bifurcation diagram for vortex solutions and characterize the different dynamical regimes. The mathematical model used, which consists of a laser rate equation coupled to a linear equation for the feedback field, can describe the spatiotemporal dynamics of broad area vertical cavity surface emitting lasers with external frequency selective feedback in the limit of zero delay.

© 2010 Optical Society of America

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References

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  1. T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of Spatial Dissipative Solitons in Photonic Devices,” Advances in Atomic, Molecular, and Optical Physics,  57, 323, (2009).
  2. S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).
  3. Y. Tanguy,, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100, 013907(2008).
  4. P. Genevet, S. Barland, M. Guidici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Let.. 101, 123905 (2008).
  5. A. G. Vladimirov, N. N. Rosanov, S. V. Fedorov, and G. V. Khodova, “Bifurcation analysis of laser autosolitons,” Quantum Electronics 27, 949–952 (1997).
  6. T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327 (2010).
  7. P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Bistable and addressable localized vortices in semiconductor lasers,” submitted (2009) http://hal.archives-ouvertes.fr/docs/00/43/59/20/PDF/localizedvortices.pdf
  8. N. N. Rosanov, “Solitons in laser systems with saturable absorption,” in Dissipative Solitons, edited by N. Akhmediev and A. Ankiewicz, Lect. Notes Phys. 661, 101–130 (2004).
  9. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett. 95, 053903 (2005).
    [PubMed]
  10. L.-C Crasovan, B. A. Malomed, and D. Michalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2000).
  11. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).
  12. W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450 (1997).
  13. J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E 70, 056613 (2004).
  14. A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Progress in Optics, ed. E. Wolf,  47, 291 (2005).
  15. P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E 78, 016212 (2008).
  16. P. V. Paulau, D. Gomila, P. Colet, M. A. Matias, N. A. Loiko, and W. J. Firth “Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A 80, 023808 (2009).
  17. J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg-Landau equations,” Phys. Lett. A,  246,412 (1998).
  18. W. J. Firth and P. V. Paulau, “Soliton lasers stabilized by coupling to a resonant linear system,” submitted (2009).
  19. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995).
    [PubMed]
  20. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser. and Photon Rev. 2, 299 (2008).
  21. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys. 3, 305 (2007).
  22. A. P. A. Fisher, O. K. Andersen, M. Yousefi, S. Stolte, and D. Lenstra, “Experimental and theoretical study of filtered optical feedback in a semiconductor laser,” IEEE Journal of Quantum Electronics 36, 375 (2000).
  23. M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, “Spontaneous motion of cavity solitons induced by a delayed feedback,” Phys. Rev. Lett, 103,103904 (2009).

2010 (1)

T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327 (2010).

2009 (3)

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of Spatial Dissipative Solitons in Photonic Devices,” Advances in Atomic, Molecular, and Optical Physics,  57, 323, (2009).

P. V. Paulau, D. Gomila, P. Colet, M. A. Matias, N. A. Loiko, and W. J. Firth “Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A 80, 023808 (2009).

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, “Spontaneous motion of cavity solitons induced by a delayed feedback,” Phys. Rev. Lett, 103,103904 (2009).

2008 (5)

P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E 78, 016212 (2008).

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser. and Photon Rev. 2, 299 (2008).

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).

Y. Tanguy,, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100, 013907(2008).

P. Genevet, S. Barland, M. Guidici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Let.. 101, 123905 (2008).

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys. 3, 305 (2007).

2005 (2)

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Progress in Optics, ed. E. Wolf,  47, 291 (2005).

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett. 95, 053903 (2005).
[PubMed]

2004 (2)

N. N. Rosanov, “Solitons in laser systems with saturable absorption,” in Dissipative Solitons, edited by N. Akhmediev and A. Ankiewicz, Lect. Notes Phys. 661, 101–130 (2004).

J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E 70, 056613 (2004).

2002 (1)

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

2000 (2)

L.-C Crasovan, B. A. Malomed, and D. Michalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2000).

A. P. A. Fisher, O. K. Andersen, M. Yousefi, S. Stolte, and D. Lenstra, “Experimental and theoretical study of filtered optical feedback in a semiconductor laser,” IEEE Journal of Quantum Electronics 36, 375 (2000).

1998 (1)

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg-Landau equations,” Phys. Lett. A,  246,412 (1998).

1997 (2)

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

A. G. Vladimirov, N. N. Rosanov, S. V. Fedorov, and G. V. Khodova, “Bifurcation analysis of laser autosolitons,” Quantum Electronics 27, 949–952 (1997).

1995 (1)

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

Ackemann, T.

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of Spatial Dissipative Solitons in Photonic Devices,” Advances in Atomic, Molecular, and Optical Physics,  57, 323, (2009).

Y. Tanguy,, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100, 013907(2008).

P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E 78, 016212 (2008).

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser. and Photon Rev. 2, 299 (2008).

Andersen, O. K.

A. P. A. Fisher, O. K. Andersen, M. Yousefi, S. Stolte, and D. Lenstra, “Experimental and theoretical study of filtered optical feedback in a semiconductor laser,” IEEE Journal of Quantum Electronics 36, 375 (2000).

Atai, J.

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg-Landau equations,” Phys. Lett. A,  246,412 (1998).

Balle, S.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

Barbay, S.

T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327 (2010).

Barland, S.

P. Genevet, S. Barland, M. Guidici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Let.. 101, 123905 (2008).

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Bistable and addressable localized vortices in semiconductor lasers,” submitted (2009) http://hal.archives-ouvertes.fr/docs/00/43/59/20/PDF/localizedvortices.pdf

Beaudoin, G.

T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327 (2010).

Brambilla, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

Colet, P.

P. V. Paulau, D. Gomila, P. Colet, M. A. Matias, N. A. Loiko, and W. J. Firth “Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A 80, 023808 (2009).

Crasovan, L.-C

L.-C Crasovan, B. A. Malomed, and D. Michalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2000).

Desyatnikov, A. S.

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Progress in Optics, ed. E. Wolf,  47, 291 (2005).

Elsass, T.

T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327 (2010).

Fedorov, S. V.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett. 95, 053903 (2005).
[PubMed]

A. G. Vladimirov, N. N. Rosanov, S. V. Fedorov, and G. V. Khodova, “Bifurcation analysis of laser autosolitons,” Quantum Electronics 27, 949–952 (1997).

Firth, W. J.

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of Spatial Dissipative Solitons in Photonic Devices,” Advances in Atomic, Molecular, and Optical Physics,  57, 323, (2009).

P. V. Paulau, D. Gomila, P. Colet, M. A. Matias, N. A. Loiko, and W. J. Firth “Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A 80, 023808 (2009).

P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E 78, 016212 (2008).

Y. Tanguy,, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100, 013907(2008).

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

W. J. Firth and P. V. Paulau, “Soliton lasers stabilized by coupling to a resonant linear system,” submitted (2009).

Fisher, A. P. A.

A. P. A. Fisher, O. K. Andersen, M. Yousefi, S. Stolte, and D. Lenstra, “Experimental and theoretical study of filtered optical feedback in a semiconductor laser,” IEEE Journal of Quantum Electronics 36, 375 (2000).

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser. and Photon Rev. 2, 299 (2008).

Friese, M. E. J.

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

Gauthron, K.

T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327 (2010).

Genevet, P.

P. Genevet, S. Barland, M. Guidici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Let.. 101, 123905 (2008).

P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Bistable and addressable localized vortices in semiconductor lasers,” submitted (2009) http://hal.archives-ouvertes.fr/docs/00/43/59/20/PDF/localizedvortices.pdf

Giudici, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Bistable and addressable localized vortices in semiconductor lasers,” submitted (2009) http://hal.archives-ouvertes.fr/docs/00/43/59/20/PDF/localizedvortices.pdf

Gomila, D.

P. V. Paulau, D. Gomila, P. Colet, M. A. Matias, N. A. Loiko, and W. J. Firth “Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A 80, 023808 (2009).

P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E 78, 016212 (2008).

Guidici, M.

P. Genevet, S. Barland, M. Guidici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Let.. 101, 123905 (2008).

He, H.

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

Heckenberg, N. R.

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

Jager, R.

Y. Tanguy,, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100, 013907(2008).

Jagerk, R.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

Khodova, G. V.

A. G. Vladimirov, N. N. Rosanov, S. V. Fedorov, and G. V. Khodova, “Bifurcation analysis of laser autosolitons,” Quantum Electronics 27, 949–952 (1997).

Kivshar, Y. S.

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Progress in Optics, ed. E. Wolf,  47, 291 (2005).

J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E 70, 056613 (2004).

Knodlk, T.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

Kuszelewicz, R.

T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327 (2010).

Leblond, H.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).

Lenstra, D.

A. P. A. Fisher, O. K. Andersen, M. Yousefi, S. Stolte, and D. Lenstra, “Experimental and theoretical study of filtered optical feedback in a semiconductor laser,” IEEE Journal of Quantum Electronics 36, 375 (2000).

Loiko, N. A.

P. V. Paulau, D. Gomila, P. Colet, M. A. Matias, N. A. Loiko, and W. J. Firth “Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A 80, 023808 (2009).

P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E 78, 016212 (2008).

Lugiato, L. A.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

Maggipinto, T.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

Malomed, B. A.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).

L.-C Crasovan, B. A. Malomed, and D. Michalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2000).

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg-Landau equations,” Phys. Lett. A,  246,412 (1998).

Matias, M. A.

P. V. Paulau, D. Gomila, P. Colet, M. A. Matias, N. A. Loiko, and W. J. Firth “Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A 80, 023808 (2009).

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).

Michalache, D.

L.-C Crasovan, B. A. Malomed, and D. Michalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2000).

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).

Millerk, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys. 3, 305 (2007).

Oppo, G.-L.

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of Spatial Dissipative Solitons in Photonic Devices,” Advances in Atomic, Molecular, and Optical Physics,  57, 323, (2009).

Padgett, M.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser. and Photon Rev. 2, 299 (2008).

Paulau, P. V.

P. V. Paulau, D. Gomila, P. Colet, M. A. Matias, N. A. Loiko, and W. J. Firth “Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A 80, 023808 (2009).

P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E 78, 016212 (2008).

W. J. Firth and P. V. Paulau, “Soliton lasers stabilized by coupling to a resonant linear system,” submitted (2009).

Pieroux, D.

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, “Spontaneous motion of cavity solitons induced by a delayed feedback,” Phys. Rev. Lett, 103,103904 (2009).

Rosanov, N. N.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett. 95, 053903 (2005).
[PubMed]

N. N. Rosanov, “Solitons in laser systems with saturable absorption,” in Dissipative Solitons, edited by N. Akhmediev and A. Ankiewicz, Lect. Notes Phys. 661, 101–130 (2004).

A. G. Vladimirov, N. N. Rosanov, S. V. Fedorov, and G. V. Khodova, “Bifurcation analysis of laser autosolitons,” Quantum Electronics 27, 949–952 (1997).

Rubinsztein-Dunlop, H.

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

Sagnes, I.

T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327 (2010).

Salgueiro, J. R.

J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E 70, 056613 (2004).

Shatsev, A. N.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett. 95, 053903 (2005).
[PubMed]

Skryabin, D. V.

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

Spinelli, L.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

Stolte, S.

A. P. A. Fisher, O. K. Andersen, M. Yousefi, S. Stolte, and D. Lenstra, “Experimental and theoretical study of filtered optical feedback in a semiconductor laser,” IEEE Journal of Quantum Electronics 36, 375 (2000).

Tanguy,, Y.

Y. Tanguy,, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100, 013907(2008).

Tis-soni, G.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

Tlidi, M.

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, “Spontaneous motion of cavity solitons induced by a delayed feedback,” Phys. Rev. Lett, 103,103904 (2009).

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys. 3, 305 (2007).

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Progress in Optics, ed. E. Wolf,  47, 291 (2005).

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys. 3, 305 (2007).

Tredicce, J. R.

P. Genevet, S. Barland, M. Guidici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Let.. 101, 123905 (2008).

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Bistable and addressable localized vortices in semiconductor lasers,” submitted (2009) http://hal.archives-ouvertes.fr/docs/00/43/59/20/PDF/localizedvortices.pdf

Turaev, D.

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, “Spontaneous motion of cavity solitons induced by a delayed feedback,” Phys. Rev. Lett, 103,103904 (2009).

Vladimirov, A. G.

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, “Spontaneous motion of cavity solitons induced by a delayed feedback,” Phys. Rev. Lett, 103,103904 (2009).

A. G. Vladimirov, N. N. Rosanov, S. V. Fedorov, and G. V. Khodova, “Bifurcation analysis of laser autosolitons,” Quantum Electronics 27, 949–952 (1997).

Yousefi, M.

A. P. A. Fisher, O. K. Andersen, M. Yousefi, S. Stolte, and D. Lenstra, “Experimental and theoretical study of filtered optical feedback in a semiconductor laser,” IEEE Journal of Quantum Electronics 36, 375 (2000).

Advances in Atomic, Molecular, and Optical Physics (1)

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of Spatial Dissipative Solitons in Photonic Devices,” Advances in Atomic, Molecular, and Optical Physics,  57, 323, (2009).

Appl. Phys. B (1)

T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327 (2010).

IEEE Journal of Quantum Electronics (1)

A. P. A. Fisher, O. K. Andersen, M. Yousefi, S. Stolte, and D. Lenstra, “Experimental and theoretical study of filtered optical feedback in a semiconductor laser,” IEEE Journal of Quantum Electronics 36, 375 (2000).

Laser. and Photon Rev. (1)

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser. and Photon Rev. 2, 299 (2008).

Nature (London) (1)

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) 419, 699 (2002).

Nature Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys. 3, 305 (2007).

Notes Phys. (1)

N. N. Rosanov, “Solitons in laser systems with saturable absorption,” in Dissipative Solitons, edited by N. Akhmediev and A. Ankiewicz, Lect. Notes Phys. 661, 101–130 (2004).

Phys. Lett. A (1)

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg-Landau equations,” Phys. Lett. A,  246,412 (1998).

Phys. Rev. A (2)

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).

P. V. Paulau, D. Gomila, P. Colet, M. A. Matias, N. A. Loiko, and W. J. Firth “Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A 80, 023808 (2009).

Phys. Rev. E (3)

J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E 70, 056613 (2004).

L.-C Crasovan, B. A. Malomed, and D. Michalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2000).

P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E 78, 016212 (2008).

Phys. Rev. Let.. (1)

P. Genevet, S. Barland, M. Guidici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Let.. 101, 123905 (2008).

Phys. Rev. Lett, (1)

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, “Spontaneous motion of cavity solitons induced by a delayed feedback,” Phys. Rev. Lett, 103,103904 (2009).

Phys. Rev. Lett. (4)

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

Y. Tanguy,, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100, 013907(2008).

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett. 95, 053903 (2005).
[PubMed]

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

Progress in Optics (1)

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Progress in Optics, ed. E. Wolf,  47, 291 (2005).

Quantum Electronics (1)

A. G. Vladimirov, N. N. Rosanov, S. V. Fedorov, and G. V. Khodova, “Bifurcation analysis of laser autosolitons,” Quantum Electronics 27, 949–952 (1997).

Other (2)

P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Bistable and addressable localized vortices in semiconductor lasers,” submitted (2009) http://hal.archives-ouvertes.fr/docs/00/43/59/20/PDF/localizedvortices.pdf

W. J. Firth and P. V. Paulau, “Soliton lasers stabilized by coupling to a resonant linear system,” submitted (2009).

Supplementary Material (5)

» Media 1: AVI (428 KB)     
» Media 2: AVI (705 KB)     
» Media 3: AVI (2455 KB)     
» Media 4: AVI (1622 KB)     
» Media 5: AVI (701 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Coexistence of the fundamental soliton and vortices with m = 1,2: (a) the stationary transverse amplitude distribution; and (b) the instantaneous distribution of real part of the field. Black (white) corresponds to the minimum (maximum) value. Here 1 2 T = 2.7 ns 1 , ω S = 250 2 π ns 1 , α=5.0, σ=60ns -1,κ=100ns -1, and μ=0.66.

Fig. 2.
Fig. 2.

Transverse sections through the center of stable self-localized states. (a) Amplitude in semi-logarithmic scale, and (b) phase. Parameters as in Fig. 1. Dots correspond to m = 0, circles to m = 1, asterisks to m = 2, triangles to m = 3, and crosses to m = 4.

Fig. 3.
Fig. 3.

Bifurcation diagrams of self-localized states with m = 0,1,2. Total power as a function of the pump current μ. Other parameters are as in Fig. 1 and symbols as in Fig. 2. Instabilities below bifurcation points M and M 1 are illustrated in movies (Media 1) and (Media 2) correspondingly. Other labeled dots are explained in Section 5. Dash-dotted vertical lines A,B indicate the bifurcation points of the homogeneous solutions.

Fig. 4.
Fig. 4.

Additional features of the m = 0 (dots), 1 (circles) branches presented in Fig. 3: (a) maximum intensity; (b) frequency; and (c) radius of the m = 1 vortex ring (in arbitrary units); as a function of the pump current μ. Dashed thin lines represent the homogeneous steady states. Dash-dotted vertical lines A,B indicate the bifurcation points of the homogeneous solutions.

Fig. 5.
Fig. 5.

Dependence on pump current of instability growth rates for the m = 2 vortex soliton. The real part of the relevant perturbation eigenvalues is plotted for that part of the upper m = 2 branch in Fig. 3 for which the E = 0 state is stable, i.e. between B and C 2. The thin line indicates the mode with bifurcation point G 1 in Fig. 3. The real part of this mode is shown in lower-right inset, and the corresponding dynamical evolution in (Media 3). The medium-thickness line indicates the mode with bifurcation point G 2 in Fig.3. The real part of this mode is shown in middle inset, and the corresponding dynamical evolution in (Media 4). The thickest line indicates the mode with bifurcation point G 3 in Fig. 3. The real part of this mode is shown in the upper-left inset, and the corresponding dynamical evolution in (Media 5).

Equations (3)

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E t = κ ( 1 + ) ( E μE 1 + | E 2 ) i Δ E + F + i ω s E ,
F dt = - 1 2 T F + σ 1 2 T E ,
A E 0 + B E 0 1 + E 0 2 + Δ E 0 = 0 ,

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