Abstract

We report theoretical and numerical study of the dynamical and spectral properties of the conservative and dissipative solitons in micro-ring resonators pumped in a proximity of the zero of the group velocity dispersion. We discuss frequency and velocity locking of the conservative solitons, when dissipation is accounted for. We present theory of the dispersive radiation emitted by such solitons, report their Hopf instability and radiation enhancement by multiple solitons.

© 2014 Optical Society of America

Full Article  |  PDF Article

Errata

C. Milián and D.V. Skryabin, "Soliton families and resonant radiation in a micro-ring resonator near zero group-velocity dispersion: erratum," Opt. Express 22, 8068-8068 (2014)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-22-7-8068

References

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  1. T. J. Kippenberg, R. Holzwarth, S. A. Diddams, “Microresonator based optical frequency combs,” Science 332, 555–559 (2011).
    [CrossRef] [PubMed]
  2. P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
    [CrossRef]
  3. Y. K. Chembo, N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
    [CrossRef]
  4. Y. K. Chembo, C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013).
    [CrossRef]
  5. A. B. Matsko, A. A. Savachenkov, W. Liang, V. S. Ilchenko, D. Seidel, L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. 36, 2845–2847 (2011).
    [CrossRef] [PubMed]
  6. A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013).
    [CrossRef]
  7. S. Coen, H. G. Randle, T. Sylvestre, M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field LugiatoLefever model,” Opt. Lett. 38, 37–39 (2013).
    [CrossRef] [PubMed]
  8. M. R. E. Lamont, Y. Okawachi, A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” Opt. Lett. 38, 3478–3481 (2013).
    [CrossRef] [PubMed]
  9. L. Zhang, C. Bao, V. Singh, J. Mu, C. Yang, A.M. Agarwal, L.C. Kimerling, J. Michel, “Generation of two-cycle pulses and octave-spanning frequency combs in a dispersion-flattened micro-resonator,” Opt. Lett. 38, 5122–5125 (2013).
    [CrossRef] [PubMed]
  10. D. V. Skryabin, A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
    [CrossRef]
  11. V. I. Karpman, “Stationary and radiating dark solitons of the third order nonlinear Schrodinger equation,” Phys. Lett. A 181, 211–215 (1993).
    [CrossRef]
  12. V. V. Afanasjev, Y. S. Kivshar, C. R. Menyuk, “Effect of third-order dispersion on dark solitons,” Opt. Lett. 21, 1975–1977 (1996).
    [CrossRef] [PubMed]
  13. C. Milián, D. V. Skryabin, A. Ferrando, “Continuum generation by dark solitons,” Opt. Lett. 34, 2096–2098 (2009).
    [CrossRef] [PubMed]
  14. M. Tlidi, L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. 35, 306–309 (2010).
    [CrossRef] [PubMed]
  15. M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A 88, 035802 (2013).
    [CrossRef]
  16. F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nature Photon. 4, 471–476 (2010).
    [CrossRef]
  17. F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, M. Taki, “Nonlinear symmetry breaking induced by third order dispersion in optical fiber cavities,” Phys. Rev. Lett. 110, 104103 (2013).
    [CrossRef]
  18. I. V. Barashenkov, E. V. Zemlyanaya, “Travelling solitons in the externally driven nonlinear Schrdinger equation,” J. Phys. A: Math. Theor. 44, 1–23 (2011).
    [CrossRef]
  19. I. V. Barashenkov, Yu. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrdinger solitons,” Phys. Rev. E 54, 5707–5725 (1996).
    [CrossRef]
  20. A. B. Matsko, A. A. Savchenkov, L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett. 37, 4856–4858 (2012).
    [CrossRef] [PubMed]

2013

Y. K. Chembo, C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013).
[CrossRef]

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013).
[CrossRef]

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

M. R. E. Lamont, Y. Okawachi, A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” Opt. Lett. 38, 3478–3481 (2013).
[CrossRef] [PubMed]

L. Zhang, C. Bao, V. Singh, J. Mu, C. Yang, A.M. Agarwal, L.C. Kimerling, J. Michel, “Generation of two-cycle pulses and octave-spanning frequency combs in a dispersion-flattened micro-resonator,” Opt. Lett. 38, 5122–5125 (2013).
[CrossRef] [PubMed]

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A 88, 035802 (2013).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, M. Taki, “Nonlinear symmetry breaking induced by third order dispersion in optical fiber cavities,” Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

2012

2011

I. V. Barashenkov, E. V. Zemlyanaya, “Travelling solitons in the externally driven nonlinear Schrdinger equation,” J. Phys. A: Math. Theor. 44, 1–23 (2011).
[CrossRef]

A. B. Matsko, A. A. Savachenkov, W. Liang, V. S. Ilchenko, D. Seidel, L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. 36, 2845–2847 (2011).
[CrossRef] [PubMed]

T. J. Kippenberg, R. Holzwarth, S. A. Diddams, “Microresonator based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef] [PubMed]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

2010

Y. K. Chembo, N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

D. V. Skryabin, A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

M. Tlidi, L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. 35, 306–309 (2010).
[CrossRef] [PubMed]

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

2009

1996

V. V. Afanasjev, Y. S. Kivshar, C. R. Menyuk, “Effect of third-order dispersion on dark solitons,” Opt. Lett. 21, 1975–1977 (1996).
[CrossRef] [PubMed]

I. V. Barashenkov, Yu. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrdinger solitons,” Phys. Rev. E 54, 5707–5725 (1996).
[CrossRef]

1993

V. I. Karpman, “Stationary and radiating dark solitons of the third order nonlinear Schrodinger equation,” Phys. Lett. A 181, 211–215 (1993).
[CrossRef]

Afanasjev, V. V.

Agarwal, A.M.

Bahloul, L.

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A 88, 035802 (2013).
[CrossRef]

Balakireva, I.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013).
[CrossRef]

Bao, C.

Barashenkov, I. V.

I. V. Barashenkov, E. V. Zemlyanaya, “Travelling solitons in the externally driven nonlinear Schrdinger equation,” J. Phys. A: Math. Theor. 44, 1–23 (2011).
[CrossRef]

I. V. Barashenkov, Yu. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrdinger solitons,” Phys. Rev. E 54, 5707–5725 (1996).
[CrossRef]

Chembo, Y. K.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013).
[CrossRef]

Y. K. Chembo, C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013).
[CrossRef]

Y. K. Chembo, N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Cherbi, L.

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A 88, 035802 (2013).
[CrossRef]

Coen, S.

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

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

Coillet, A.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013).
[CrossRef]

Coulibaly, S.

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A 88, 035802 (2013).
[CrossRef]

Del’Haye, P.

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, S. A. Diddams, “Microresonator based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef] [PubMed]

Dudley, J. M.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013).
[CrossRef]

Emplit, P.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, M. Taki, “Nonlinear symmetry breaking induced by third order dispersion in optical fiber cavities,” Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

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

Erkintalo, M.

Ferrando, A.

Gaeta, A. L.

Gavartin, E.

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Gelens, L.

Gorbach, A. V.

D. V. Skryabin, A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

Gorodetsky, M. L.

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Gorza, S.-P.

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

Haelterman, M.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, M. Taki, “Nonlinear symmetry breaking induced by third order dispersion in optical fiber cavities,” Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

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

Hariz, A.

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A 88, 035802 (2013).
[CrossRef]

Henriet, R.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013).
[CrossRef]

Herr, T.

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, S. A. Diddams, “Microresonator based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef] [PubMed]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Ilchenko, V. S.

Karpman, V. I.

V. I. Karpman, “Stationary and radiating dark solitons of the third order nonlinear Schrodinger equation,” Phys. Lett. A 181, 211–215 (1993).
[CrossRef]

Kimerling, L.C.

Kippenberg, T. J.

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, S. A. Diddams, “Microresonator based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef] [PubMed]

Kivshar, Y. S.

Kockaert, P.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, M. Taki, “Nonlinear symmetry breaking induced by third order dispersion in optical fiber cavities,” Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

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

Lamont, M. R. E.

Larger, L.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013).
[CrossRef]

Leo, F.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, M. Taki, “Nonlinear symmetry breaking induced by third order dispersion in optical fiber cavities,” Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

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

Liang, W.

Maleki, L.

Matsko, A. B.

Menyuk, C. R.

Y. K. Chembo, C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013).
[CrossRef]

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013).
[CrossRef]

V. V. Afanasjev, Y. S. Kivshar, C. R. Menyuk, “Effect of third-order dispersion on dark solitons,” Opt. Lett. 21, 1975–1977 (1996).
[CrossRef] [PubMed]

Michel, J.

Milián, C.

Mu, J.

Mussot, A.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, M. Taki, “Nonlinear symmetry breaking induced by third order dispersion in optical fiber cavities,” Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Okawachi, Y.

Randle, H. G.

Saleh, K.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013).
[CrossRef]

Savachenkov, A. A.

Savchenkov, A. A.

Seidel, D.

Singh, V.

Skryabin, D. V.

D. V. Skryabin, A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

C. Milián, D. V. Skryabin, A. Ferrando, “Continuum generation by dark solitons,” Opt. Lett. 34, 2096–2098 (2009).
[CrossRef] [PubMed]

Smirnov, Yu. S.

I. V. Barashenkov, Yu. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrdinger solitons,” Phys. Rev. E 54, 5707–5725 (1996).
[CrossRef]

Sylvestre, T.

Taki, M.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, M. Taki, “Nonlinear symmetry breaking induced by third order dispersion in optical fiber cavities,” Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Tlidi, M.

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A 88, 035802 (2013).
[CrossRef]

M. Tlidi, L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. 35, 306–309 (2010).
[CrossRef] [PubMed]

Yang, C.

Yu, N.

Y. K. Chembo, N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Zemlyanaya, E. V.

I. V. Barashenkov, E. V. Zemlyanaya, “Travelling solitons in the externally driven nonlinear Schrdinger equation,” J. Phys. A: Math. Theor. 44, 1–23 (2011).
[CrossRef]

Zhang, L.

IEEE Photonics J.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013).
[CrossRef]

J. Phys. A: Math. Theor.

I. V. Barashenkov, E. V. Zemlyanaya, “Travelling solitons in the externally driven nonlinear Schrdinger equation,” J. Phys. A: Math. Theor. 44, 1–23 (2011).
[CrossRef]

Nature Photon.

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

Opt. Lett.

Phys. Lett. A

V. I. Karpman, “Stationary and radiating dark solitons of the third order nonlinear Schrodinger equation,” Phys. Lett. A 181, 211–215 (1993).
[CrossRef]

Phys. Rev. A

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A 88, 035802 (2013).
[CrossRef]

Y. K. Chembo, N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Y. K. Chembo, C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013).
[CrossRef]

Phys. Rev. E

I. V. Barashenkov, Yu. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrdinger solitons,” Phys. Rev. E 54, 5707–5725 (1996).
[CrossRef]

Phys. Rev. Lett.

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, M. Taki, “Nonlinear symmetry breaking induced by third order dispersion in optical fiber cavities,” Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Rev. Mod. Phys.

D. V. Skryabin, A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

Science

T. J. Kippenberg, R. Holzwarth, S. A. Diddams, “Microresonator based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

(a) Bistability of the single mode Q = 0 solution. (b) Momentum M vs velocity v plots for families of conservative (no-loss) solitons, λ0 = 1570nm, δ = 0.05. Numbers in the figure show the h-values. (c) GVD and transverse profile of the ring waveguide. Red line on wavelength axis indicates the spectral interval used to plot families of dissipative solitons, Figs. 6,7.

Fig. 2
Fig. 2

(a–d) B3 = γ = 0: (a,b) Soliton profile and spectrum, small v, v = −5 × 10−4; (c,d) Soliton profile and spectrum, large v, v = −1.845×10−3. Vertical lines in (b,d) correspond to the pump wavelength λ0 = 1570 nm. (e–h) B3 = 0, γ = 10−3: (e,f) (t, z) and (t, q) evolution for (a,b); (g,h) (t, z) and (t, q) evolution for (c,d). h = 5 × 10−4 and δ = 0.05 in all the panels.

Fig. 3
Fig. 3

(a) Graphical solution of the resonance condition Ω(Q) = 0. (b) Resonance wavelength vs detuning δ for λ0 = 1570 nm and v = −1.25. Horizontal lines show cavity resonances, q. (c) Ratio of the amplitudes of the two resonant waves for several values of the reference frequency. (d) Amplitude of the strongest resonance vs δ. Dashed straight lines in (b,d) mark the cavity resonances for the resonant radiation wavelength as a function of detuning. Dashed curve in (d) corresponds to α = 〈|Ψ0|〉. Definition of α is shown in the inset of Fig. 4a. h = 5 × 10−4 in all plots.

Fig. 4
Fig. 4

Spatial (a) and spectral (b) profiles of the conservative solitons. δ = 0.05, λ0 = 1570 nm, h = 5 × 10−4. Zero GVD and resonance wavelengths are marked by the dashed and dotted-dashed vertical lines, respectively.

Fig. 5
Fig. 5

Temporal evolution of (a) intensity and (b) spectrum of the conservative soliton with δ0 = 0.0495, λ0 = 1570 nm and γ = 0 initially, and γ = 10−3 is switched on from t = 500. Slope of solid line in (a) corresponds to the input soliton velocity, v = −1.25.

Fig. 6
Fig. 6

Soliton profiles and their spectra (insets) for the different values of the reference and pump frequencies, γ = 5 × 10−3, h = 1.5 × 10−3, δ = 5 × 10−2. (a) λ0 = 1565 nm, β2 = −108.4 ps2/km, β3 = −6.6 ps3/km; (b) λ0 = 1575 nm, β2 = −56 ps2/km, β3 = −7.1 ps3/km; (c) λ0 = 1589 nm, β2 = 23 ps2/km, β3 = −7.9 ps3/km. The resonant radiation wavelengths are marked by the red dotted-dashed lines. The zero GVD wavelength is marked by the black dashed vertical lines.

Fig. 7
Fig. 7

(a) Spectral center of mass of the soliton core (h = 0.0015). (b) Velocity of the dissipative solitons (h is in the legend). (c) Growth rate of the Hopf instability of the soliton (h = 0.0015). All as functions of the reference frequency. δ = 0.05, γ = 0.005.

Fig. 8
Fig. 8

Soliton excitation by a short pulse at the same frequency than the pump. (a,b) present oscillatory instability features for λ0 = 1565, whereas (c,d) tend to a stable propagation (damping of oscillations) for λ0 = 1580 nm, in agreement with the Hopf growth analysis in Fig. 7(c). (a,c) are intensities and (b,d) are the spectra. h = 0.0015, γ = 0.005, δ = 0.05. The slopes of the black lines in the spatial evolutions correspond to the velocity of the expected soliton. The vertical lines in the spectral plots mark, form left to right, pump, zero GVD wavelengths, and predicted resonant radiation, respectively.

Fig. 9
Fig. 9

Five solitons excited in a cavity (a) and their spectrum (b): λ0 = 1580 nm, γ = 0.005, h = 0.0015, δ = 0.05. White dashed line in (b) shows spectrum of a single soliton for comparison.

Equations (6)

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i t E + i ω z E + 1 2 ! ω z 2 E i 3 ! ω z 3 E + ( i Γ ω 0 + 2 g ω 0 | E | 2 ) E + r A e i ω p t = 0.
i T Ψ + i Z Ψ + B 2 Z 2 Ψ i B 3 Z 3 Ψ + ( i γ δ + 2 | Ψ | 2 ) Ψ + h = 0 ,
i v x ψ + B 2 x 2 ψ i B 3 x 3 ψ + ( i γ δ + 2 | ψ | 2 ) ψ + h = 0 , x = Z ( v + 1 ) T .
ψ = a sech ( x a B 2 ) e i v x / ( 2 B 2 ) , a δ v 2 4 B 2 > 0.
i t [ G 1 G 2 ] = [ W ( Q ) 2 Ψ 0 2 2 Ψ 0 * 2 W * ( Q ) ] [ G 1 G 2 ] , W ^ δ i v [ x i Q ] + B 2 [ x i Q ] 2 i B 3 [ x i Q ] 3 + 4 | Ψ 0 | 2 .
Ω ( Q ) = v Q + B 3 Q 3 ± [ δ + B 2 Q 2 4 | Ψ 0 | 2 ] 2 4 | Ψ 0 | 4 .

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