Abstract

We present an experimental observation of an oscillating Kerr cavity soliton, i.e., a time-periodic oscillating one-dimensional temporally localized structure excited in a driven nonlinear fiber cavity with a Kerr-type nonlinearity. More generally, these oscillations result from a Hopf bifurcation of a (spatially or temporally) localized state in the generic class of driven dissipative systems close to the 1 : 1 resonance tongue. Furthermore, we theoretically analyze dynamical instabilities of the one-dimensional cavity soliton, revealing oscillations and different chaotic states in previously unexplored regions of parameter space. As cavity solitons are closely related to Kerr frequency combs, we expect these dynamical regimes to be highly relevant for the field of microresonator-based frequency combs.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. A. Lugiato, “Introduction to the feature section on cavity solitons: an overview,” IEEE J. Quantum Elec.39, 193–196 (2003).
    [CrossRef]
  2. G. S. McDonald and W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B7, 1328–1335 (1990).
    [CrossRef]
  3. S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
    [CrossRef] [PubMed]
  4. F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon.4, 471–476 (2010).
    [CrossRef]
  5. V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys.13, 113026/1–13 (2011).
    [CrossRef]
  6. S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model,” Opt. Lett.38, 37–39 (2013).
    [CrossRef] [PubMed]
  7. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys.65, 851–1112 (1993).
    [CrossRef]
  8. J. Wu, R. Keolian, and I. Rudnick, “Observation of a nonpropagating hydrodynamic soliton,” Phys. Rev. Lett.52, 1421–1424 (1984).
    [CrossRef]
  9. H. C. Kim, R. L. Stenzel, and A. Y. Wong, “Development of ‘cavitons’ and trapping of RF field,” Phys. Rev. Lett.33, 886–889 (1974).
    [CrossRef]
  10. R. Richter and I. V. Barashenkov, “Two-dimensional solitons on the surface of magnetic fluids,” Phys. Rev. Lett.94, 184503/1–4 (2005).
    [CrossRef]
  11. P. B. Umbanhowar, F. Melo, and H. L. Swinney, “Localized excitations in a vertically vibrated granular layer,” Nature382, 793–796 (1996).
    [CrossRef]
  12. A. Ustinov, “Solitons in Josephson junctions,” Physica D123, 315–329 (1998).
    [CrossRef]
  13. B. Ermentrout, X. Chen, and Z. Chen, “Transition fronts and localized structures in bistable reaction-diffusion equations,” Physica D108, 147–167 (1997).
    [CrossRef]
  14. V. K. Vanag, A. M. Zhabotinsky, and I. R. Epstein, “Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction,” Phys. Rev. Lett.86, 552–555 (2001).
    [CrossRef] [PubMed]
  15. O. Lejeune, M. Tlidi, and P. Couteron, “Localized vegetation patches: A self-organized response to resource scarcity,” Phys. Rev. E66, 010901 (2002).
    [CrossRef]
  16. B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett.85, 748–751 (2000).
    [CrossRef] [PubMed]
  17. S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett.101, 253902 (2008).
    [CrossRef] [PubMed]
  18. O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, and J. Fineberg, “Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension,” Phys. Rev. Lett.83, 3190–3193 (1999).
    [CrossRef]
  19. C. Elphick, G. Iooss, and E. Tirapegui, “Normal form reduction for time-periodically driven differential equations,” Phys. Lett. A120, 459–463 (1987).
    [CrossRef]
  20. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett.58, 2209–2211 (1987).
    [CrossRef] [PubMed]
  21. K. Nozaki and N. Bekki, “Chaotic solitons in a plasma driven by an RF field,” J. Phys. Soc. Jpn.54, 2363–2366 (1985); ibid. Physica D 21, 381 (1986)
    [CrossRef]
  22. D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett.108, 263906/1–5 (2012).
    [CrossRef]
  23. A. B. Matsko, A. A. Savchenkov, and L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett.37, 4856–4858 (2012).
    [CrossRef] [PubMed]
  24. D. Gomila, A. Scroggie, and W. Firth, “Bifurcation structure of dissipative solitons,” Physica D227, 70–77 (2007).
    [CrossRef]
  25. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
    [CrossRef]
  26. M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express19, 14233–14239 (2011).
    [CrossRef] [PubMed]
  27. A. Tierno, F. Gustave, and S. Barland, “Class A mode-locked semiconductor ring laser,” Opt. Lett.37, 2004–2006 (2012).
    [CrossRef] [PubMed]
  28. W. J. Firth, G. K. Harkness, A. Lord, J. M. McSloy, D. Gomila, and P. Colet, “Dynamical properties of two-dimensional Kerr cavity solitons,” J. Opt. Soc. Am. B19, 747–752 (2002).
    [CrossRef]
  29. M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun.91, 401–407 (1992).
    [CrossRef]
  30. A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals4, 1323–1354 (1994).
    [CrossRef]
  31. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2006).
  32. K. Wiesenfeld, “Noisy precursors of nonlinear instabilities,” J. Stat. Phys.38, 1071–1097 (1985).
    [CrossRef]
  33. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys.74, 99–143 (2002).
    [CrossRef]
  34. O. Descalzi, C. Cartes, J. Cisternas, and H. R. Brand, “Exploding dissipative solitons: The analog of the Ruelle-Takens route for spatially localized solutions,” Phys. Rev. E83, 056214/1–6 (2011).
    [CrossRef]
  35. J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett.85, 2937–2940 (2000).
    [CrossRef] [PubMed]
  36. T. Kapitula and B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,” Physica D124, 58–103 (1998).
    [CrossRef]
  37. P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photon.6, 84–92 (2012).
    [CrossRef]
  38. L. Gelens and E. Knobloch, “Traveling waves and defects in the complex Swift-Hohenberg equation,” Phys. Rev. E84, 056203/1–22 (2011).
    [CrossRef]
  39. A. G. Vladimirov, S. V. Fedorov, N. A. Kaliteevskii, G. V. Khodova, and N. N. Rosanov, “Numerical investigation of laser localized structures,” J. Opt. B: Quantum Semiclass. Opt.1, 101–106 (1999).
    [CrossRef]
  40. N. V. Alexeeva, I. V. Barashenkov, and D. E. Pelinovsky, “Dynamics of the parametrically driven NLS solitons beyond the onset of the oscillatory instability,” Nonlinearity12, 103–140 (1999).
    [CrossRef]
  41. J. Burke, A. Yochelis, and E. Knobloch, “Classification of spatially localized oscillations in periodically forced dissipative systems,” SIAM J. Appl. Dyn. Syst.7, 651–711 (2008).
    [CrossRef]
  42. Y.-P. Ma, J. Burke, and E. Knobloch, “Defect-mediated snaking: A new growth mechanism for localized structures,” Physica D239, 1867–1883 (2010).
    [CrossRef]
  43. D. Gomila, A. Jacobo, M. A. Matías, and P. Colet, “Phase-space structure of two-dimensional excitable localized structures,” Phys. Rev. E75, 026217/1–10 (2007).
    [CrossRef]
  44. C. Grebogi, E. Ott, and J. A. Yorke, “Crises, sudden changes in chaotic attractors, and transient chaos,” Physica D7, 181–200 (1983).
    [CrossRef]
  45. E. M. Izhikevich, “Neural excitability, spiking and bursting,” Int. J. Bifurcation Chaos10, 1171–1266 (2000).
    [CrossRef]
  46. L. Gelens, L. Mashal, S. Beri, W. Coomans, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Excitability in semiconductor microring lasers: Experimental and theoretical pulse characterization,” Phys. Rev. A82, 063841/1–9 (2010).
    [CrossRef]
  47. W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 036209/1–8 (2011).
    [CrossRef]
  48. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
    [CrossRef] [PubMed]

2013 (1)

2012 (4)

A. Tierno, F. Gustave, and S. Barland, “Class A mode-locked semiconductor ring laser,” Opt. Lett.37, 2004–2006 (2012).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett.37, 4856–4858 (2012).
[CrossRef] [PubMed]

D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett.108, 263906/1–5 (2012).
[CrossRef]

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photon.6, 84–92 (2012).
[CrossRef]

2011 (6)

L. Gelens and E. Knobloch, “Traveling waves and defects in the complex Swift-Hohenberg equation,” Phys. Rev. E84, 056203/1–22 (2011).
[CrossRef]

O. Descalzi, C. Cartes, J. Cisternas, and H. R. Brand, “Exploding dissipative solitons: The analog of the Ruelle-Takens route for spatially localized solutions,” Phys. Rev. E83, 056214/1–6 (2011).
[CrossRef]

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 036209/1–8 (2011).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
[CrossRef] [PubMed]

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express19, 14233–14239 (2011).
[CrossRef] [PubMed]

V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys.13, 113026/1–13 (2011).
[CrossRef]

2010 (3)

L. Gelens, L. Mashal, S. Beri, W. Coomans, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Excitability in semiconductor microring lasers: Experimental and theoretical pulse characterization,” Phys. Rev. A82, 063841/1–9 (2010).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon.4, 471–476 (2010).
[CrossRef]

Y.-P. Ma, J. Burke, and E. Knobloch, “Defect-mediated snaking: A new growth mechanism for localized structures,” Physica D239, 1867–1883 (2010).
[CrossRef]

2008 (2)

J. Burke, A. Yochelis, and E. Knobloch, “Classification of spatially localized oscillations in periodically forced dissipative systems,” SIAM J. Appl. Dyn. Syst.7, 651–711 (2008).
[CrossRef]

S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett.101, 253902 (2008).
[CrossRef] [PubMed]

2007 (3)

D. Gomila, A. Jacobo, M. A. Matías, and P. Colet, “Phase-space structure of two-dimensional excitable localized structures,” Phys. Rev. E75, 026217/1–10 (2007).
[CrossRef]

D. Gomila, A. Scroggie, and W. Firth, “Bifurcation structure of dissipative solitons,” Physica D227, 70–77 (2007).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

2005 (1)

R. Richter and I. V. Barashenkov, “Two-dimensional solitons on the surface of magnetic fluids,” Phys. Rev. Lett.94, 184503/1–4 (2005).
[CrossRef]

2003 (1)

L. A. Lugiato, “Introduction to the feature section on cavity solitons: an overview,” IEEE J. Quantum Elec.39, 193–196 (2003).
[CrossRef]

2002 (4)

W. J. Firth, G. K. Harkness, A. Lord, J. M. McSloy, D. Gomila, and P. Colet, “Dynamical properties of two-dimensional Kerr cavity solitons,” J. Opt. Soc. Am. B19, 747–752 (2002).
[CrossRef]

O. Lejeune, M. Tlidi, and P. Couteron, “Localized vegetation patches: A self-organized response to resource scarcity,” Phys. Rev. E66, 010901 (2002).
[CrossRef]

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys.74, 99–143 (2002).
[CrossRef]

2001 (1)

V. K. Vanag, A. M. Zhabotinsky, and I. R. Epstein, “Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction,” Phys. Rev. Lett.86, 552–555 (2001).
[CrossRef] [PubMed]

2000 (3)

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett.85, 2937–2940 (2000).
[CrossRef] [PubMed]

E. M. Izhikevich, “Neural excitability, spiking and bursting,” Int. J. Bifurcation Chaos10, 1171–1266 (2000).
[CrossRef]

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett.85, 748–751 (2000).
[CrossRef] [PubMed]

1999 (3)

O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, and J. Fineberg, “Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension,” Phys. Rev. Lett.83, 3190–3193 (1999).
[CrossRef]

A. G. Vladimirov, S. V. Fedorov, N. A. Kaliteevskii, G. V. Khodova, and N. N. Rosanov, “Numerical investigation of laser localized structures,” J. Opt. B: Quantum Semiclass. Opt.1, 101–106 (1999).
[CrossRef]

N. V. Alexeeva, I. V. Barashenkov, and D. E. Pelinovsky, “Dynamics of the parametrically driven NLS solitons beyond the onset of the oscillatory instability,” Nonlinearity12, 103–140 (1999).
[CrossRef]

1998 (2)

T. Kapitula and B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,” Physica D124, 58–103 (1998).
[CrossRef]

A. Ustinov, “Solitons in Josephson junctions,” Physica D123, 315–329 (1998).
[CrossRef]

1997 (1)

B. Ermentrout, X. Chen, and Z. Chen, “Transition fronts and localized structures in bistable reaction-diffusion equations,” Physica D108, 147–167 (1997).
[CrossRef]

1996 (1)

P. B. Umbanhowar, F. Melo, and H. L. Swinney, “Localized excitations in a vertically vibrated granular layer,” Nature382, 793–796 (1996).
[CrossRef]

1994 (1)

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals4, 1323–1354 (1994).
[CrossRef]

1993 (1)

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys.65, 851–1112 (1993).
[CrossRef]

1992 (1)

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun.91, 401–407 (1992).
[CrossRef]

1990 (1)

1987 (2)

C. Elphick, G. Iooss, and E. Tirapegui, “Normal form reduction for time-periodically driven differential equations,” Phys. Lett. A120, 459–463 (1987).
[CrossRef]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett.58, 2209–2211 (1987).
[CrossRef] [PubMed]

1985 (2)

K. Nozaki and N. Bekki, “Chaotic solitons in a plasma driven by an RF field,” J. Phys. Soc. Jpn.54, 2363–2366 (1985); ibid. Physica D 21, 381 (1986)
[CrossRef]

K. Wiesenfeld, “Noisy precursors of nonlinear instabilities,” J. Stat. Phys.38, 1071–1097 (1985).
[CrossRef]

1984 (1)

J. Wu, R. Keolian, and I. Rudnick, “Observation of a nonpropagating hydrodynamic soliton,” Phys. Rev. Lett.52, 1421–1424 (1984).
[CrossRef]

1983 (1)

C. Grebogi, E. Ott, and J. A. Yorke, “Crises, sudden changes in chaotic attractors, and transient chaos,” Physica D7, 181–200 (1983).
[CrossRef]

1974 (1)

H. C. Kim, R. L. Stenzel, and A. Y. Wong, “Development of ‘cavitons’ and trapping of RF field,” Phys. Rev. Lett.33, 886–889 (1974).
[CrossRef]

Ackemann, T.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett.85, 748–751 (2000).
[CrossRef] [PubMed]

Agnon, A.

O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, and J. Fineberg, “Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension,” Phys. Rev. Lett.83, 3190–3193 (1999).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2006).

Akhmediev, N.

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photon.6, 84–92 (2012).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett.85, 2937–2940 (2000).
[CrossRef] [PubMed]

Alexeeva, N. V.

N. V. Alexeeva, I. V. Barashenkov, and D. E. Pelinovsky, “Dynamics of the parametrically driven NLS solitons beyond the onset of the oscillatory instability,” Nonlinearity12, 103–140 (1999).
[CrossRef]

Ankiewicz, A.

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett.85, 2937–2940 (2000).
[CrossRef] [PubMed]

Aranson, I. S.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys.74, 99–143 (2002).
[CrossRef]

Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Balle, S.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Barashenkov, I. V.

R. Richter and I. V. Barashenkov, “Two-dimensional solitons on the surface of magnetic fluids,” Phys. Rev. Lett.94, 184503/1–4 (2005).
[CrossRef]

N. V. Alexeeva, I. V. Barashenkov, and D. E. Pelinovsky, “Dynamics of the parametrically driven NLS solitons beyond the onset of the oscillatory instability,” Nonlinearity12, 103–140 (1999).
[CrossRef]

Barbay, S.

S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett.101, 253902 (2008).
[CrossRef] [PubMed]

Barland, S.

A. Tierno, F. Gustave, and S. Barland, “Class A mode-locked semiconductor ring laser,” Opt. Lett.37, 2004–2006 (2012).
[CrossRef] [PubMed]

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Bekki, N.

K. Nozaki and N. Bekki, “Chaotic solitons in a plasma driven by an RF field,” J. Phys. Soc. Jpn.54, 2363–2366 (1985); ibid. Physica D 21, 381 (1986)
[CrossRef]

Beri, S.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 036209/1–8 (2011).
[CrossRef]

L. Gelens, L. Mashal, S. Beri, W. Coomans, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Excitability in semiconductor microring lasers: Experimental and theoretical pulse characterization,” Phys. Rev. A82, 063841/1–9 (2010).
[CrossRef]

Brambilla, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Brand, H. R.

O. Descalzi, C. Cartes, J. Cisternas, and H. R. Brand, “Exploding dissipative solitons: The analog of the Ruelle-Takens route for spatially localized solutions,” Phys. Rev. E83, 056214/1–6 (2011).
[CrossRef]

Burke, J.

Y.-P. Ma, J. Burke, and E. Knobloch, “Defect-mediated snaking: A new growth mechanism for localized structures,” Physica D239, 1867–1883 (2010).
[CrossRef]

J. Burke, A. Yochelis, and E. Knobloch, “Classification of spatially localized oscillations in periodically forced dissipative systems,” SIAM J. Appl. Dyn. Syst.7, 651–711 (2008).
[CrossRef]

Cartes, C.

O. Descalzi, C. Cartes, J. Cisternas, and H. R. Brand, “Exploding dissipative solitons: The analog of the Ruelle-Takens route for spatially localized solutions,” Phys. Rev. E83, 056214/1–6 (2011).
[CrossRef]

Chen, X.

B. Ermentrout, X. Chen, and Z. Chen, “Transition fronts and localized structures in bistable reaction-diffusion equations,” Physica D108, 147–167 (1997).
[CrossRef]

Chen, Z.

B. Ermentrout, X. Chen, and Z. Chen, “Transition fronts and localized structures in bistable reaction-diffusion equations,” Physica D108, 147–167 (1997).
[CrossRef]

Cisternas, J.

O. Descalzi, C. Cartes, J. Cisternas, and H. R. Brand, “Exploding dissipative solitons: The analog of the Ruelle-Takens route for spatially localized solutions,” Phys. Rev. E83, 056214/1–6 (2011).
[CrossRef]

Coen, S.

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model,” Opt. Lett.38, 37–39 (2013).
[CrossRef] [PubMed]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon.4, 471–476 (2010).
[CrossRef]

Colet, P.

D. Gomila, A. Jacobo, M. A. Matías, and P. Colet, “Phase-space structure of two-dimensional excitable localized structures,” Phys. Rev. E75, 026217/1–10 (2007).
[CrossRef]

W. J. Firth, G. K. Harkness, A. Lord, J. M. McSloy, D. Gomila, and P. Colet, “Dynamical properties of two-dimensional Kerr cavity solitons,” J. Opt. Soc. Am. B19, 747–752 (2002).
[CrossRef]

Coomans, W.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 036209/1–8 (2011).
[CrossRef]

L. Gelens, L. Mashal, S. Beri, W. Coomans, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Excitability in semiconductor microring lasers: Experimental and theoretical pulse characterization,” Phys. Rev. A82, 063841/1–9 (2010).
[CrossRef]

Couteron, P.

O. Lejeune, M. Tlidi, and P. Couteron, “Localized vegetation patches: A self-organized response to resource scarcity,” Phys. Rev. E66, 010901 (2002).
[CrossRef]

Cross, M. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys.65, 851–1112 (1993).
[CrossRef]

Danckaert, J.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 036209/1–8 (2011).
[CrossRef]

L. Gelens, L. Mashal, S. Beri, W. Coomans, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Excitability in semiconductor microring lasers: Experimental and theoretical pulse characterization,” Phys. Rev. A82, 063841/1–9 (2010).
[CrossRef]

Del’Haye, P.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Descalzi, O.

O. Descalzi, C. Cartes, J. Cisternas, and H. R. Brand, “Exploding dissipative solitons: The analog of the Ruelle-Takens route for spatially localized solutions,” Phys. Rev. E83, 056214/1–6 (2011).
[CrossRef]

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
[CrossRef] [PubMed]

Elphick, C.

C. Elphick, G. Iooss, and E. Tirapegui, “Normal form reduction for time-periodically driven differential equations,” Phys. Lett. A120, 459–463 (1987).
[CrossRef]

Elsass, T.

S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett.101, 253902 (2008).
[CrossRef] [PubMed]

Emplit, Ph.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon.4, 471–476 (2010).
[CrossRef]

Epstein, I. R.

V. K. Vanag, A. M. Zhabotinsky, and I. R. Epstein, “Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction,” Phys. Rev. Lett.86, 552–555 (2001).
[CrossRef] [PubMed]

Erkintalo, M.

Ermentrout, B.

B. Ermentrout, X. Chen, and Z. Chen, “Transition fronts and localized structures in bistable reaction-diffusion equations,” Physica D108, 147–167 (1997).
[CrossRef]

Fedorov, S. V.

A. G. Vladimirov, S. V. Fedorov, N. A. Kaliteevskii, G. V. Khodova, and N. N. Rosanov, “Numerical investigation of laser localized structures,” J. Opt. B: Quantum Semiclass. Opt.1, 101–106 (1999).
[CrossRef]

Feldmann, M.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett.85, 748–751 (2000).
[CrossRef] [PubMed]

Fineberg, J.

O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, and J. Fineberg, “Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension,” Phys. Rev. Lett.83, 3190–3193 (1999).
[CrossRef]

Firth, W.

D. Gomila, A. Scroggie, and W. Firth, “Bifurcation structure of dissipative solitons,” Physica D227, 70–77 (2007).
[CrossRef]

Firth, W. J.

Foster, M. A.

Gaeta, A. L.

Gelens, L.

L. Gelens and E. Knobloch, “Traveling waves and defects in the complex Swift-Hohenberg equation,” Phys. Rev. E84, 056203/1–22 (2011).
[CrossRef]

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 036209/1–8 (2011).
[CrossRef]

L. Gelens, L. Mashal, S. Beri, W. Coomans, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Excitability in semiconductor microring lasers: Experimental and theoretical pulse characterization,” Phys. Rev. A82, 063841/1–9 (2010).
[CrossRef]

Giudici, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Gomila, D.

D. Gomila, A. Scroggie, and W. Firth, “Bifurcation structure of dissipative solitons,” Physica D227, 70–77 (2007).
[CrossRef]

D. Gomila, A. Jacobo, M. A. Matías, and P. Colet, “Phase-space structure of two-dimensional excitable localized structures,” Phys. Rev. E75, 026217/1–10 (2007).
[CrossRef]

W. J. Firth, G. K. Harkness, A. Lord, J. M. McSloy, D. Gomila, and P. Colet, “Dynamical properties of two-dimensional Kerr cavity solitons,” J. Opt. Soc. Am. B19, 747–752 (2002).
[CrossRef]

Gorza, S.-P.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon.4, 471–476 (2010).
[CrossRef]

Grebogi, C.

C. Grebogi, E. Ott, and J. A. Yorke, “Crises, sudden changes in chaotic attractors, and transient chaos,” Physica D7, 181–200 (1983).
[CrossRef]

Grelu, P.

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photon.6, 84–92 (2012).
[CrossRef]

Gustave, F.

Hachair, X.

S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett.101, 253902 (2008).
[CrossRef] [PubMed]

Haelterman, M.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon.4, 471–476 (2010).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun.91, 401–407 (1992).
[CrossRef]

Hamiel, Y.

O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, and J. Fineberg, “Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension,” Phys. Rev. Lett.83, 3190–3193 (1999).
[CrossRef]

Harkness, G. K.

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys.65, 851–1112 (1993).
[CrossRef]

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
[CrossRef] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Iooss, G.

C. Elphick, G. Iooss, and E. Tirapegui, “Normal form reduction for time-periodically driven differential equations,” Phys. Lett. A120, 459–463 (1987).
[CrossRef]

Izhikevich, E. M.

E. M. Izhikevich, “Neural excitability, spiking and bursting,” Int. J. Bifurcation Chaos10, 1171–1266 (2000).
[CrossRef]

Jacobo, A.

D. Gomila, A. Jacobo, M. A. Matías, and P. Colet, “Phase-space structure of two-dimensional excitable localized structures,” Phys. Rev. E75, 026217/1–10 (2007).
[CrossRef]

Jäger, R.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Kaliteevskii, N. A.

A. G. Vladimirov, S. V. Fedorov, N. A. Kaliteevskii, G. V. Khodova, and N. N. Rosanov, “Numerical investigation of laser localized structures,” J. Opt. B: Quantum Semiclass. Opt.1, 101–106 (1999).
[CrossRef]

Kapitula, T.

T. Kapitula and B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,” Physica D124, 58–103 (1998).
[CrossRef]

Keolian, R.

J. Wu, R. Keolian, and I. Rudnick, “Observation of a nonpropagating hydrodynamic soliton,” Phys. Rev. Lett.52, 1421–1424 (1984).
[CrossRef]

Khodova, G. V.

A. G. Vladimirov, S. V. Fedorov, N. A. Kaliteevskii, G. V. Khodova, and N. N. Rosanov, “Numerical investigation of laser localized structures,” J. Opt. B: Quantum Semiclass. Opt.1, 101–106 (1999).
[CrossRef]

Kim, H. C.

H. C. Kim, R. L. Stenzel, and A. Y. Wong, “Development of ‘cavitons’ and trapping of RF field,” Phys. Rev. Lett.33, 886–889 (1974).
[CrossRef]

Kippenberg, T. J.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
[CrossRef] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Knobloch, E.

L. Gelens and E. Knobloch, “Traveling waves and defects in the complex Swift-Hohenberg equation,” Phys. Rev. E84, 056203/1–22 (2011).
[CrossRef]

Y.-P. Ma, J. Burke, and E. Knobloch, “Defect-mediated snaking: A new growth mechanism for localized structures,” Physica D239, 1867–1883 (2010).
[CrossRef]

J. Burke, A. Yochelis, and E. Knobloch, “Classification of spatially localized oscillations in periodically forced dissipative systems,” SIAM J. Appl. Dyn. Syst.7, 651–711 (2008).
[CrossRef]

Knodl, T.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Kockaert, P.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon.4, 471–476 (2010).
[CrossRef]

Kramer, L.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys.74, 99–143 (2002).
[CrossRef]

Kuszelewicz, R.

S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett.101, 253902 (2008).
[CrossRef] [PubMed]

Kuzucu, O.

Lange, W.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett.85, 748–751 (2000).
[CrossRef] [PubMed]

Lefever, R.

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals4, 1323–1354 (1994).
[CrossRef]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett.58, 2209–2211 (1987).
[CrossRef] [PubMed]

Lejeune, O.

O. Lejeune, M. Tlidi, and P. Couteron, “Localized vegetation patches: A self-organized response to resource scarcity,” Phys. Rev. E66, 010901 (2002).
[CrossRef]

Leo, F.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon.4, 471–476 (2010).
[CrossRef]

Levy, J. S.

Lioubashevski, O.

O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, and J. Fineberg, “Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension,” Phys. Rev. Lett.83, 3190–3193 (1999).
[CrossRef]

Lipson, M.

Lord, A.

Louvergneaux, E.

V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys.13, 113026/1–13 (2011).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, “Introduction to the feature section on cavity solitons: an overview,” IEEE J. Quantum Elec.39, 193–196 (2003).
[CrossRef]

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals4, 1323–1354 (1994).
[CrossRef]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett.58, 2209–2211 (1987).
[CrossRef] [PubMed]

Ma, Y.-P.

Y.-P. Ma, J. Burke, and E. Knobloch, “Defect-mediated snaking: A new growth mechanism for localized structures,” Physica D239, 1867–1883 (2010).
[CrossRef]

Maggipinto, T.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Maleki, L.

Mashal, L.

L. Gelens, L. Mashal, S. Beri, W. Coomans, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Excitability in semiconductor microring lasers: Experimental and theoretical pulse characterization,” Phys. Rev. A82, 063841/1–9 (2010).
[CrossRef]

Matías, M. A.

D. Gomila, A. Jacobo, M. A. Matías, and P. Colet, “Phase-space structure of two-dimensional excitable localized structures,” Phys. Rev. E75, 026217/1–10 (2007).
[CrossRef]

Matsko, A. B.

McDonald, G. S.

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals4, 1323–1354 (1994).
[CrossRef]

G. S. McDonald and W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B7, 1328–1335 (1990).
[CrossRef]

McSloy, J. M.

Melo, F.

P. B. Umbanhowar, F. Melo, and H. L. Swinney, “Localized excitations in a vertically vibrated granular layer,” Nature382, 793–796 (1996).
[CrossRef]

Miller, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Nozaki, K.

K. Nozaki and N. Bekki, “Chaotic solitons in a plasma driven by an RF field,” J. Phys. Soc. Jpn.54, 2363–2366 (1985); ibid. Physica D 21, 381 (1986)
[CrossRef]

Odent, V.

V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys.13, 113026/1–13 (2011).
[CrossRef]

Ott, E.

C. Grebogi, E. Ott, and J. A. Yorke, “Crises, sudden changes in chaotic attractors, and transient chaos,” Physica D7, 181–200 (1983).
[CrossRef]

Pelinovsky, D. E.

N. V. Alexeeva, I. V. Barashenkov, and D. E. Pelinovsky, “Dynamics of the parametrically driven NLS solitons beyond the onset of the oscillatory instability,” Nonlinearity12, 103–140 (1999).
[CrossRef]

Randle, H. G.

Reches, Z.

O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, and J. Fineberg, “Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension,” Phys. Rev. Lett.83, 3190–3193 (1999).
[CrossRef]

Richter, R.

R. Richter and I. V. Barashenkov, “Two-dimensional solitons on the surface of magnetic fluids,” Phys. Rev. Lett.94, 184503/1–4 (2005).
[CrossRef]

Rosanov, N. N.

A. G. Vladimirov, S. V. Fedorov, N. A. Kaliteevskii, G. V. Khodova, and N. N. Rosanov, “Numerical investigation of laser localized structures,” J. Opt. B: Quantum Semiclass. Opt.1, 101–106 (1999).
[CrossRef]

Rudnick, I.

J. Wu, R. Keolian, and I. Rudnick, “Observation of a nonpropagating hydrodynamic soliton,” Phys. Rev. Lett.52, 1421–1424 (1984).
[CrossRef]

Sagnes, I.

S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett.101, 253902 (2008).
[CrossRef] [PubMed]

Saha, K.

Sandstede, B.

T. Kapitula and B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,” Physica D124, 58–103 (1998).
[CrossRef]

Savchenkov, A. A.

Schäpers, B.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett.85, 748–751 (2000).
[CrossRef] [PubMed]

Schliesser, A.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Scroggie, A.

D. Gomila, A. Scroggie, and W. Firth, “Bifurcation structure of dissipative solitons,” Physica D227, 70–77 (2007).
[CrossRef]

Scroggie, A. J.

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals4, 1323–1354 (1994).
[CrossRef]

Soto-Crespo, J. M.

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett.85, 2937–2940 (2000).
[CrossRef] [PubMed]

Spinelli, L.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Stenzel, R. L.

H. C. Kim, R. L. Stenzel, and A. Y. Wong, “Development of ‘cavitons’ and trapping of RF field,” Phys. Rev. Lett.33, 886–889 (1974).
[CrossRef]

Swinney, H. L.

P. B. Umbanhowar, F. Melo, and H. L. Swinney, “Localized excitations in a vertically vibrated granular layer,” Nature382, 793–796 (1996).
[CrossRef]

Sylvestre, T.

Taki, M.

V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys.13, 113026/1–13 (2011).
[CrossRef]

Tierno, A.

Tirapegui, E.

C. Elphick, G. Iooss, and E. Tirapegui, “Normal form reduction for time-periodically driven differential equations,” Phys. Lett. A120, 459–463 (1987).
[CrossRef]

Tissoni, G.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Tlidi, M.

O. Lejeune, M. Tlidi, and P. Couteron, “Localized vegetation patches: A self-organized response to resource scarcity,” Phys. Rev. E66, 010901 (2002).
[CrossRef]

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals4, 1323–1354 (1994).
[CrossRef]

Tredicce, J. R.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Trillo, S.

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun.91, 401–407 (1992).
[CrossRef]

Turaev, D.

D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett.108, 263906/1–5 (2012).
[CrossRef]

Umbanhowar, P. B.

P. B. Umbanhowar, F. Melo, and H. L. Swinney, “Localized excitations in a vertically vibrated granular layer,” Nature382, 793–796 (1996).
[CrossRef]

Ustinov, A.

A. Ustinov, “Solitons in Josephson junctions,” Physica D123, 315–329 (1998).
[CrossRef]

Van der Sande, G.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 036209/1–8 (2011).
[CrossRef]

L. Gelens, L. Mashal, S. Beri, W. Coomans, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Excitability in semiconductor microring lasers: Experimental and theoretical pulse characterization,” Phys. Rev. A82, 063841/1–9 (2010).
[CrossRef]

Vanag, V. K.

V. K. Vanag, A. M. Zhabotinsky, and I. R. Epstein, “Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction,” Phys. Rev. Lett.86, 552–555 (2001).
[CrossRef] [PubMed]

Verschaffelt, G.

L. Gelens, L. Mashal, S. Beri, W. Coomans, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Excitability in semiconductor microring lasers: Experimental and theoretical pulse characterization,” Phys. Rev. A82, 063841/1–9 (2010).
[CrossRef]

Vladimirov, A. G.

D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett.108, 263906/1–5 (2012).
[CrossRef]

A. G. Vladimirov, S. V. Fedorov, N. A. Kaliteevskii, G. V. Khodova, and N. N. Rosanov, “Numerical investigation of laser localized structures,” J. Opt. B: Quantum Semiclass. Opt.1, 101–106 (1999).
[CrossRef]

Wabnitz, S.

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun.91, 401–407 (1992).
[CrossRef]

Wiesenfeld, K.

K. Wiesenfeld, “Noisy precursors of nonlinear instabilities,” J. Stat. Phys.38, 1071–1097 (1985).
[CrossRef]

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Wong, A. Y.

H. C. Kim, R. L. Stenzel, and A. Y. Wong, “Development of ‘cavitons’ and trapping of RF field,” Phys. Rev. Lett.33, 886–889 (1974).
[CrossRef]

Wu, J.

J. Wu, R. Keolian, and I. Rudnick, “Observation of a nonpropagating hydrodynamic soliton,” Phys. Rev. Lett.52, 1421–1424 (1984).
[CrossRef]

Yochelis, A.

J. Burke, A. Yochelis, and E. Knobloch, “Classification of spatially localized oscillations in periodically forced dissipative systems,” SIAM J. Appl. Dyn. Syst.7, 651–711 (2008).
[CrossRef]

Yorke, J. A.

C. Grebogi, E. Ott, and J. A. Yorke, “Crises, sudden changes in chaotic attractors, and transient chaos,” Physica D7, 181–200 (1983).
[CrossRef]

Zelik, S.

D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett.108, 263906/1–5 (2012).
[CrossRef]

Zhabotinsky, A. M.

V. K. Vanag, A. M. Zhabotinsky, and I. R. Epstein, “Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction,” Phys. Rev. Lett.86, 552–555 (2001).
[CrossRef] [PubMed]

Chaos, Solitons & Fractals (1)

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals4, 1323–1354 (1994).
[CrossRef]

IEEE J. Quantum Elec. (1)

L. A. Lugiato, “Introduction to the feature section on cavity solitons: an overview,” IEEE J. Quantum Elec.39, 193–196 (2003).
[CrossRef]

Int. J. Bifurcation Chaos (1)

E. M. Izhikevich, “Neural excitability, spiking and bursting,” Int. J. Bifurcation Chaos10, 1171–1266 (2000).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (1)

A. G. Vladimirov, S. V. Fedorov, N. A. Kaliteevskii, G. V. Khodova, and N. N. Rosanov, “Numerical investigation of laser localized structures,” J. Opt. B: Quantum Semiclass. Opt.1, 101–106 (1999).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. Soc. Jpn. (1)

K. Nozaki and N. Bekki, “Chaotic solitons in a plasma driven by an RF field,” J. Phys. Soc. Jpn.54, 2363–2366 (1985); ibid. Physica D 21, 381 (1986)
[CrossRef]

J. Stat. Phys. (1)

K. Wiesenfeld, “Noisy precursors of nonlinear instabilities,” J. Stat. Phys.38, 1071–1097 (1985).
[CrossRef]

Nat. Photon. (2)

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photon.6, 84–92 (2012).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon.4, 471–476 (2010).
[CrossRef]

Nature (3)

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

P. B. Umbanhowar, F. Melo, and H. L. Swinney, “Localized excitations in a vertically vibrated granular layer,” Nature382, 793–796 (1996).
[CrossRef]

New J. Phys. (1)

V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys.13, 113026/1–13 (2011).
[CrossRef]

Nonlinearity (1)

N. V. Alexeeva, I. V. Barashenkov, and D. E. Pelinovsky, “Dynamics of the parametrically driven NLS solitons beyond the onset of the oscillatory instability,” Nonlinearity12, 103–140 (1999).
[CrossRef]

Opt. Commun. (1)

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun.91, 401–407 (1992).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Lett. A (1)

C. Elphick, G. Iooss, and E. Tirapegui, “Normal form reduction for time-periodically driven differential equations,” Phys. Lett. A120, 459–463 (1987).
[CrossRef]

Phys. Rev. A (1)

L. Gelens, L. Mashal, S. Beri, W. Coomans, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Excitability in semiconductor microring lasers: Experimental and theoretical pulse characterization,” Phys. Rev. A82, 063841/1–9 (2010).
[CrossRef]

Phys. Rev. E (5)

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 036209/1–8 (2011).
[CrossRef]

D. Gomila, A. Jacobo, M. A. Matías, and P. Colet, “Phase-space structure of two-dimensional excitable localized structures,” Phys. Rev. E75, 026217/1–10 (2007).
[CrossRef]

L. Gelens and E. Knobloch, “Traveling waves and defects in the complex Swift-Hohenberg equation,” Phys. Rev. E84, 056203/1–22 (2011).
[CrossRef]

O. Descalzi, C. Cartes, J. Cisternas, and H. R. Brand, “Exploding dissipative solitons: The analog of the Ruelle-Takens route for spatially localized solutions,” Phys. Rev. E83, 056214/1–6 (2011).
[CrossRef]

O. Lejeune, M. Tlidi, and P. Couteron, “Localized vegetation patches: A self-organized response to resource scarcity,” Phys. Rev. E66, 010901 (2002).
[CrossRef]

Phys. Rev. Lett. (10)

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett.85, 748–751 (2000).
[CrossRef] [PubMed]

S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett.101, 253902 (2008).
[CrossRef] [PubMed]

O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, and J. Fineberg, “Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension,” Phys. Rev. Lett.83, 3190–3193 (1999).
[CrossRef]

J. Wu, R. Keolian, and I. Rudnick, “Observation of a nonpropagating hydrodynamic soliton,” Phys. Rev. Lett.52, 1421–1424 (1984).
[CrossRef]

H. C. Kim, R. L. Stenzel, and A. Y. Wong, “Development of ‘cavitons’ and trapping of RF field,” Phys. Rev. Lett.33, 886–889 (1974).
[CrossRef]

R. Richter and I. V. Barashenkov, “Two-dimensional solitons on the surface of magnetic fluids,” Phys. Rev. Lett.94, 184503/1–4 (2005).
[CrossRef]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett.58, 2209–2211 (1987).
[CrossRef] [PubMed]

V. K. Vanag, A. M. Zhabotinsky, and I. R. Epstein, “Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction,” Phys. Rev. Lett.86, 552–555 (2001).
[CrossRef] [PubMed]

D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett.108, 263906/1–5 (2012).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett.85, 2937–2940 (2000).
[CrossRef] [PubMed]

Physica D (6)

T. Kapitula and B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,” Physica D124, 58–103 (1998).
[CrossRef]

C. Grebogi, E. Ott, and J. A. Yorke, “Crises, sudden changes in chaotic attractors, and transient chaos,” Physica D7, 181–200 (1983).
[CrossRef]

Y.-P. Ma, J. Burke, and E. Knobloch, “Defect-mediated snaking: A new growth mechanism for localized structures,” Physica D239, 1867–1883 (2010).
[CrossRef]

A. Ustinov, “Solitons in Josephson junctions,” Physica D123, 315–329 (1998).
[CrossRef]

B. Ermentrout, X. Chen, and Z. Chen, “Transition fronts and localized structures in bistable reaction-diffusion equations,” Physica D108, 147–167 (1997).
[CrossRef]

D. Gomila, A. Scroggie, and W. Firth, “Bifurcation structure of dissipative solitons,” Physica D227, 70–77 (2007).
[CrossRef]

Rev. Mod. Phys. (2)

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys.65, 851–1112 (1993).
[CrossRef]

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys.74, 99–143 (2002).
[CrossRef]

Science (1)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science332, 555–559 (2011).
[CrossRef] [PubMed]

SIAM J. Appl. Dyn. Syst. (1)

J. Burke, A. Yochelis, and E. Knobloch, “Classification of spatially localized oscillations in periodically forced dissipative systems,” SIAM J. Appl. Dyn. Syst.7, 651–711 (2008).
[CrossRef]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2006).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Figure 1
Figure 1

Experimental setup: Our cavity is made up of 380 m of standard single mode fiber. The driving and addressing beams are combined through a wavelength division multiplexer (WDM). Part of the output beam is used to actively stabilize the cavity lenth, while the rest is directed towards an oscilloscope. PC: Polarization controller. PD: Photodiode. BPF: Optical band pass filter

Figure 2
Figure 2

Oscilloscope traces (linear scaling) at the output of the cavity a few milliseconds after launching the writing pulse. The delay between subsequent pulses in the output sequences is equal to the 1.85 μs cavity round-trip time. (a) Pin = 200 mW and Pout = 184 mW, corresponding to |S|2 = 6.2 and Δ = 3.8. (b) Pin = 236 mW and Pout = 219 mW, corresponding to |S|2 = 7.3 and Δ = 4.1. (c) Pin = 274 mW and Pout = 254 mW, corresponding to |S|2 = 8.5 and Δ = 4.1.

Figure 3
Figure 3

(a) Theoretical bifurcation diagrams corresponding to the two values of the detuning Δ used for the experimental measurements of Fig. 2. The light (dark) shades correspond to Δ = 3.8 (Δ = 4.1) respectively. The top curves represent the CS peak power (blue: stable CS; dotted-red: unstable saddle CS; dashed-purple: oscillatory CS), with the filled gray squares highlighting the experimental points, while the bottom S-shaped curves show the bistable behavior of the homogeneous state (where the solid (dotted) lines correspond to stable (unstable) states). (b)–(d) provide more detailed numerical results for the parameters corresponding to Fig. 2(c): |S|2 = 8.5 and Δ = 4.1. (b) shows the theoretical temporal intensity profile of the CS circulating inside the fiber cavity when it reaches its maximum peak power. The slow-time evolution of the CS peak power is depicted in (c). (d) shows, as a contour plot, the corresponding slow-time evolution of the temporal intensity profile of the CS.

Figure 4
Figure 4

Panels on the left show a projection of the time-evolution of the LLE onto the 2D phase-space (φ, R) with φ and R the phase and amplitude of the center of the CS (note that the phase φ wraps at 2π). The middle panels show the corresponding time traces for the center intensity R2. Two different initial conditions are chosen (shown in dashed and solid line): Sa ± 0.005 × r⃗u. The right panels depict a contour plot showing the time evolution of the temporal intensity profile of the CS. From top to bottom the driving S is increased from 6 to 7.1, showing period-1 oscillations, period-2 oscillations, temporal chaos and transient chaos, respectively. Δ = 10.

Figure 5
Figure 5

Contour plot showing the time evolution of a CS profile for S = 10, Δ = 10. The white dashed line shows the moment the system loses its left-right symmetry and the system shows spatio-temporal chaos.

Figure 6
Figure 6

Different dynamical regimes of operation in the LLE for a detuning Δ > 3.

Equations (6)

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

E ( t , τ ) t = [ 1 + i ( | E ( t , τ ) | 2 Δ ) i η 2 τ 2 ] E ( t , τ ) + S ,
t α t t R ,
τ τ 2 α | β 2 | L ,
E E γ L α ,
S = E in γ L θ α 3 ,
E out = α θ γ L ( E κ S ) ,

Metrics