Abstract

Using numerical simulations of an extended Lugiato–Lefever equation we analyze the stability and nonlinear dynamics of Kerr frequency combs generated in microresonators and fiber resonators, taking into account third-order dispersion effects. We show that cavity solitons underlying Kerr frequency combs, normally sensitive to oscillatory and chaotic instabilities, are stabilized in a wide range of parameter space by third-order dispersion. Moreover, we demonstrate how the snaking structure organizing compound states of multiple cavity solitons is qualitatively changed by third-order dispersion, promoting an increased stability of Kerr combs underlined by a single cavity soliton.

© 2014 Optical Society of America

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  1. T. Udem, R. Holzwarth, and T. W. Hänsch, Nature 416, 233 (2002).
    [CrossRef]
  2. S. T. Cundiff and J. Ye, Rev. Mod. Phys. 75, 325 (2003).
    [CrossRef]
  3. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
    [CrossRef]
  4. P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
    [CrossRef]
  5. Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, Opt. Lett. 36, 3398 (2011).
    [CrossRef]
  6. S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, Opt. Lett. 38, 37 (2013).
    [CrossRef]
  7. Y. K. Chembo and C. Menyuk, Phys. Rev. A 87, 053852 (2013).
    [CrossRef]
  8. F. Leo, S. Coen, P. Kockaertl, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
    [CrossRef]
  9. F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, Opt. Express 21, 9180 (2013).
    [CrossRef]
  10. A. B. Matsko, W. Liang, A. A. Savchenkov, and L. Maleki, Opt. Lett. 38, 525 (2013).
    [CrossRef]
  11. I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. part II: case of anomalous dispersion,” arXiv:1308.2542 (2013).
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    [CrossRef]
  13. M. Erkintalo and S. Coen, Opt. Lett. 39, 283 (2014).
    [CrossRef]
  14. L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, Phys. Rev. A 77, 033841 (2008).
    [CrossRef]
  15. M. Tlidi and L. Gelens, Opt. Lett. 35, 306 (2010).
    [CrossRef]
  16. C. Milián and D. Skryabin, Opt. Express 22, 3732 (2014).
    [CrossRef]
  17. L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
    [CrossRef]
  18. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
    [CrossRef]
  19. Y. Xu and S. Coen, “Experimental observation of the spontaneous breaking of the time-reversal symmetry in a synchronously-pumped passive Kerr resonator,” arXiv:1401.2972 (2014).
  20. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
    [CrossRef]
  21. D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, Nat. Photonics 7, 597 (2013).
    [CrossRef]
  22. F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
    [CrossRef]
  23. J. K. Jang, S. G. Murdoch, S. Coen, and M. Erkintalo, CLEO/Europe, Munich, Germany, 2013, PD–B.7.
  24. A. Mussot, E. Louvergneaux, N. Akhmediev, F. Reynaud, L. Delage, and M. Taki, Phys. Rev. Lett. 101, 113904 (2008).
    [CrossRef]
  25. J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
    [CrossRef]
  26. M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, Opt. Lett. 32, 662 (2007).
    [CrossRef]
  27. P. D. Woods and A. R. Champneys, Phys. D 129, 147 (1999).
    [CrossRef]
  28. P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. 84, 3069 (2000).
    [CrossRef]
  29. J. Burke, S. M. Houghton, and E. Knobloch, Phys. Rev. E 80, 036202 (2009).
    [CrossRef]
  30. J. Burke and E. Knobloch, Chaos 17, 037102 (2007).
    [CrossRef]
  31. E. Makrides and B. Sandstede, Phys. D 268, 59 (2014).
    [CrossRef]
  32. M. R. E. Lamont, Y. Okawachi, and A. L. Gaeta, Opt. Lett. 38, 3478 (2013).
    [CrossRef]

2014 (4)

P. Parra-Rivas, D. Gomila, M. A. Matías, S. Coen, and L. Gelens, Phys. Rev. A 89, 043813 (2014).
[CrossRef]

M. Erkintalo and S. Coen, Opt. Lett. 39, 283 (2014).
[CrossRef]

C. Milián and D. Skryabin, Opt. Express 22, 3732 (2014).
[CrossRef]

E. Makrides and B. Sandstede, Phys. D 268, 59 (2014).
[CrossRef]

2013 (9)

M. R. E. Lamont, Y. Okawachi, and A. L. Gaeta, Opt. Lett. 38, 3478 (2013).
[CrossRef]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[CrossRef]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, Nat. Photonics 7, 597 (2013).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, Opt. Lett. 38, 37 (2013).
[CrossRef]

Y. K. Chembo and C. Menyuk, Phys. Rev. A 87, 053852 (2013).
[CrossRef]

F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, Opt. Express 21, 9180 (2013).
[CrossRef]

A. B. Matsko, W. Liang, A. A. Savchenkov, and L. Maleki, Opt. Lett. 38, 525 (2013).
[CrossRef]

2011 (4)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, Opt. Lett. 36, 3398 (2011).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
[CrossRef]

2010 (2)

M. Tlidi and L. Gelens, Opt. Lett. 35, 306 (2010).
[CrossRef]

F. Leo, S. Coen, P. Kockaertl, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

2009 (1)

J. Burke, S. M. Houghton, and E. Knobloch, Phys. Rev. E 80, 036202 (2009).
[CrossRef]

2008 (2)

A. Mussot, E. Louvergneaux, N. Akhmediev, F. Reynaud, L. Delage, and M. Taki, Phys. Rev. Lett. 101, 113904 (2008).
[CrossRef]

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, Phys. Rev. A 77, 033841 (2008).
[CrossRef]

2007 (2)

2003 (1)

S. T. Cundiff and J. Ye, Rev. Mod. Phys. 75, 325 (2003).
[CrossRef]

2002 (1)

T. Udem, R. Holzwarth, and T. W. Hänsch, Nature 416, 233 (2002).
[CrossRef]

2000 (1)

P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. 84, 3069 (2000).
[CrossRef]

1999 (1)

P. D. Woods and A. R. Champneys, Phys. D 129, 147 (1999).
[CrossRef]

1987 (1)

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef]

Akhmediev, N.

A. Mussot, E. Louvergneaux, N. Akhmediev, F. Reynaud, L. Delage, and M. Taki, Phys. Rev. Lett. 101, 113904 (2008).
[CrossRef]

Balakireva, I.

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. part II: case of anomalous dispersion,” arXiv:1308.2542 (2013).

Brasch, V.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

Burke, J.

J. Burke, S. M. Houghton, and E. Knobloch, Phys. Rev. E 80, 036202 (2009).
[CrossRef]

J. Burke and E. Knobloch, Chaos 17, 037102 (2007).
[CrossRef]

Champneys, A. R.

P. D. Woods and A. R. Champneys, Phys. D 129, 147 (1999).
[CrossRef]

Chembo, Y. K.

Y. K. Chembo and C. Menyuk, Phys. Rev. A 87, 053852 (2013).
[CrossRef]

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. part II: case of anomalous dispersion,” arXiv:1308.2542 (2013).

Coen, S.

P. Parra-Rivas, D. Gomila, M. A. Matías, S. Coen, and L. Gelens, Phys. Rev. A 89, 043813 (2014).
[CrossRef]

M. Erkintalo and S. Coen, Opt. Lett. 39, 283 (2014).
[CrossRef]

F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, Opt. Express 21, 9180 (2013).
[CrossRef]

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, Opt. Lett. 38, 37 (2013).
[CrossRef]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaertl, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

Y. Xu and S. Coen, “Experimental observation of the spontaneous breaking of the time-reversal symmetry in a synchronously-pumped passive Kerr resonator,” arXiv:1401.2972 (2014).

J. K. Jang, S. G. Murdoch, S. Coen, and M. Erkintalo, CLEO/Europe, Munich, Germany, 2013, PD–B.7.

Coillet, A.

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. part II: case of anomalous dispersion,” arXiv:1308.2542 (2013).

Colet, P.

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, Phys. Rev. A 77, 033841 (2008).
[CrossRef]

Coullet, P.

P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. 84, 3069 (2000).
[CrossRef]

Cundiff, S. T.

S. T. Cundiff and J. Ye, Rev. Mod. Phys. 75, 325 (2003).
[CrossRef]

Danckaert, J.

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, Phys. Rev. A 77, 033841 (2008).
[CrossRef]

Del’Haye, P.

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Delage, L.

A. Mussot, E. Louvergneaux, N. Akhmediev, F. Reynaud, L. Delage, and M. Taki, Phys. Rev. Lett. 101, 113904 (2008).
[CrossRef]

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

Emplit, P.

F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, Opt. Express 21, 9180 (2013).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaertl, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

Erkintalo, M.

M. Erkintalo and S. Coen, Opt. Lett. 39, 283 (2014).
[CrossRef]

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, Opt. Lett. 38, 37 (2013).
[CrossRef]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[CrossRef]

J. K. Jang, S. G. Murdoch, S. Coen, and M. Erkintalo, CLEO/Europe, Munich, Germany, 2013, PD–B.7.

Gaeta, A. L.

Gavartin, E.

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Gelens, L.

P. Parra-Rivas, D. Gomila, M. A. Matías, S. Coen, and L. Gelens, Phys. Rev. A 89, 043813 (2014).
[CrossRef]

F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, Opt. Express 21, 9180 (2013).
[CrossRef]

M. Tlidi and L. Gelens, Opt. Lett. 35, 306 (2010).
[CrossRef]

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, Phys. Rev. A 77, 033841 (2008).
[CrossRef]

Godey, C.

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. part II: case of anomalous dispersion,” arXiv:1308.2542 (2013).

Gomila, D.

P. Parra-Rivas, D. Gomila, M. A. Matías, S. Coen, and L. Gelens, Phys. Rev. A 89, 043813 (2014).
[CrossRef]

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, Phys. Rev. A 77, 033841 (2008).
[CrossRef]

Gorodetsky, M. L.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Gorza, S.-P.

F. Leo, S. Coen, P. Kockaertl, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

Haelterman, M.

F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, Opt. Express 21, 9180 (2013).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaertl, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

Hänsch, T. W.

T. Udem, R. Holzwarth, and T. W. Hänsch, Nature 416, 233 (2002).
[CrossRef]

Herr, T.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Holzwarth, R.

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

T. Udem, R. Holzwarth, and T. W. Hänsch, Nature 416, 233 (2002).
[CrossRef]

Houghton, S. M.

J. Burke, S. M. Houghton, and E. Knobloch, Phys. Rev. E 80, 036202 (2009).
[CrossRef]

Ilchenko, V. S.

Jang, J. K.

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[CrossRef]

J. K. Jang, S. G. Murdoch, S. Coen, and M. Erkintalo, CLEO/Europe, Munich, Germany, 2013, PD–B.7.

Jost, J. D.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

Kippenberg, T. J.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Knobloch, E.

J. Burke, S. M. Houghton, and E. Knobloch, Phys. Rev. E 80, 036202 (2009).
[CrossRef]

J. Burke and E. Knobloch, Chaos 17, 037102 (2007).
[CrossRef]

Kockaert, P.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Kockaertl, P.

F. Leo, S. Coen, P. Kockaertl, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

Kondratiev, N. M.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

Kozyreff, G.

Lamont, M. R. E.

Lefever, R.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef]

Leo, F.

F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, Opt. Express 21, 9180 (2013).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaertl, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

Levy, J. S.

Liang, W.

Lipson, M.

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, Nat. Photonics 7, 597 (2013).
[CrossRef]

Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, Opt. Lett. 36, 3398 (2011).
[CrossRef]

Louvergneaux, E.

A. Mussot, E. Louvergneaux, N. Akhmediev, F. Reynaud, L. Delage, and M. Taki, Phys. Rev. Lett. 101, 113904 (2008).
[CrossRef]

M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, Opt. Lett. 32, 662 (2007).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef]

Makrides, E.

E. Makrides and B. Sandstede, Phys. D 268, 59 (2014).
[CrossRef]

Maleki, L.

Matías, M. A.

P. Parra-Rivas, D. Gomila, M. A. Matías, S. Coen, and L. Gelens, Phys. Rev. A 89, 043813 (2014).
[CrossRef]

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, Phys. Rev. A 77, 033841 (2008).
[CrossRef]

Matsko, A. B.

Menyuk, C.

Y. K. Chembo and C. Menyuk, Phys. Rev. A 87, 053852 (2013).
[CrossRef]

Milián, C.

Morandotti, R.

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, Nat. Photonics 7, 597 (2013).
[CrossRef]

Moss, D. J.

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, Nat. Photonics 7, 597 (2013).
[CrossRef]

Murdoch, S. G.

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[CrossRef]

J. K. Jang, S. G. Murdoch, S. Coen, and M. Erkintalo, CLEO/Europe, Munich, Germany, 2013, PD–B.7.

Mussot, A.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

A. Mussot, E. Louvergneaux, N. Akhmediev, F. Reynaud, L. Delage, and M. Taki, Phys. Rev. Lett. 101, 113904 (2008).
[CrossRef]

M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, Opt. Lett. 32, 662 (2007).
[CrossRef]

Okawachi, Y.

Parra-Rivas, P.

P. Parra-Rivas, D. Gomila, M. A. Matías, S. Coen, and L. Gelens, Phys. Rev. A 89, 043813 (2014).
[CrossRef]

Randle, H. G.

Reynaud, F.

A. Mussot, E. Louvergneaux, N. Akhmediev, F. Reynaud, L. Delage, and M. Taki, Phys. Rev. Lett. 101, 113904 (2008).
[CrossRef]

Riera, C.

P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. 84, 3069 (2000).
[CrossRef]

Saha, K.

Sandstede, B.

E. Makrides and B. Sandstede, Phys. D 268, 59 (2014).
[CrossRef]

Savchenkov, A. A.

Seidel, D.

Skryabin, D.

Sylvestre, T.

Taki, M.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

A. Mussot, E. Louvergneaux, N. Akhmediev, F. Reynaud, L. Delage, and M. Taki, Phys. Rev. Lett. 101, 113904 (2008).
[CrossRef]

M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, Opt. Lett. 32, 662 (2007).
[CrossRef]

Tlidi, M.

Tresser, C.

P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. 84, 3069 (2000).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Evolution of (a) the temporal intensity profile of an oscillating CS over successive round trips (top) and its associated comb spectrum in dB (bottom) in the absence of TOD (d3=0). (b) With d3=0.15, the system is stable. The profiles at time t=5 are shown on top of each graph. θ=6.1, u0=4.

Fig. 2.
Fig. 2.

Same as Fig. 1, but for u0=5.5 for which the solution exhibits spatial-temporal chaos in the absence of TOD.

Fig. 3.
Fig. 3.

Regions of existence of CSs and their instabilities in the (θ,u0) parameter space for (a) d3=0, (b) d3=0.15, (c) d3=0.9. Region I: stable CSs; region II (light-gray): time-oscillation solutions arising through Hopf bifurcation H (red line, between regions I and II); region III: spatiotemporal chaos. The green dotted (blue solid) upper and lower lines denote the SN bifurcations of the HSS (CSs), respectively. (d) Width of the pinning region where CSs exist versus d3 for θ=6.1.

Fig. 4.
Fig. 4.

Snaking bifurcation diagram showing the energy of single and multipeak CS solutions for θ=1.5 versus pump amplitude u0. Plots are made for increasing values of the TOD, d3=0 (solid line), and d3=0.01, 0.05, 0.075, with increasingly shorter and lighter dashed curves. Blue (red) curves are stable (unstable). Insets (i)–(iii), related to the corresponding points on the snaking diagram, are examples of temporal intensity profiles of CSs obtained for d3=0.05.

Fig. 5.
Fig. 5.

Snaking diagrams of the single peak and multipeak CS branches for θ=6.1 for increasing values of the TOD, d3=0.1, 0.15, 0.6, and 0.9 [from (a) to (d)]. Blue (red) curves correspond to stable (unstable) solutions. H: Hopf bifurcations.

Tables (1)

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Table 1. Physical Parameters and Normalized TOD Coefficient d3 for Three Different Optical Systems

Equations (1)

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tu=(1+iθ)u+i|u|2u+u0+iτ2u+d3τ3u.

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