## Dynamics of one-dimensional Kerr cavity solitons |

Optics Express, Vol. 21, Issue 7, pp. 9180-9191 (2013)

http://dx.doi.org/10.1364/OE.21.009180

Acrobat PDF (3696 KB)

### 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

## 1. Introduction

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. B **7**, 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,” Nature **419**, 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,” Nature **382**, 793–796 (1996) [CrossRef] .

12. A. Ustinov, “Solitons in Josephson junctions,” Physica D **123**, 315–329 (1998) [CrossRef] .

13. B. Ermentrout, X. Chen, and Z. Chen, “Transition fronts and localized structures in bistable reaction-diffusion equations,” Physica D **108**, 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. E **66**, 010901 (2002) [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,” Nature **419**, 699–702 (2002) [CrossRef] [PubMed] .

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] .

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] .

*oscillons*, have been limited to systems driven by a spatially homogeneous periodic forcing with its frequency Ω near twice the natural oscillation frequency

*ω*of the system [11

11. P. B. Umbanhowar, F. Melo, and H. L. Swinney, “Localized excitations in a vertically vibrated granular layer,” Nature **382**, 793–796 (1996) [CrossRef] .

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. A **120**, 459–463 (1987) [CrossRef] .

*ω*=

*n*: 1, with

*n*= 1, 2, 3...

*E*of the electric field in such cavities is described by the well-known mean-field Lugiato-Lefever equation (LLE), which is a particular realization of the FCGLE at the 1 : 1 resonance [20

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

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 D **227**, 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,” Nature **450**, 1214–1217 (2007) [CrossRef] .

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

## 2. Mean field model

*E*(

*t*,

*τ*) is governed by a dimensionless mean-field equation [20

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

*t*is the slow time rescaled with respect to the cavity photon lifetime,

*τ*is the fast time scale, Δ is the cavity detuning,

*S*is the continuous driving field amplitude and

*η*is the sign of the group-velocity dispersion coefficient of the fiber

*β*

_{2}. The normalization is such that where

*α*is equal to half the total cavity losses expressed in percentage of power lost per round-trip,

*t*

_{R}is the cavity round-trip time,

*L*is the cavity length, while

*γ*is the Kerr nonlinearity coefficient. The cavity detuning is defined as Δ =

*δ*/

*α*with

*δ*= 2

*mπ*−

*ϕ*

_{0}, where

*ϕ*

_{0}is the overall cavity round-trip phase shift and

*m*is the order of the closest cavity resonance.

*E*

_{in}is the amplitude of the driving field incident on the cavity input coupler with intensity transmission coefficient

*θ*and has units such that

*P*

_{in}= |

*E*

_{in}|

^{2}is the driving power. The field at the cavity output,

*E*

_{out}, which results from the interference between the field leaving the cavity and the part of the driving beam reflected on the input coupler, can then be expressed from our dimensionless variables as where

*κ*=

*α*/

*θ*. The cavity output power is given by

*P*

_{out}= |

*E*

_{out}|

^{2}.

*S*[24

24. D. Gomila, A. Scroggie, and W. Firth, “Bifurcation structure of dissipative solitons,” Physica D **227**, 70–77 (2007) [CrossRef] .

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. B **19**, 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 & Fractals **4**, 1323–1354 (1994) [CrossRef] .

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] .

## 3. Experimental demonstration of cavity soliton oscillations

### 3.1. System set-up and cavity soliton excitation scheme

**4**, 471–476 (2010) [CrossRef] .

*α*= 0.13), corresponding to a relatively high finesse of 24 and to 22-kHz-wide resonances. The largest values of normalized driving power and detuning reached in the experiment were, respectively, |

*S*|

^{2}= 8.5 and Δ = 4.1, corresponding to a CW driving power of

*P*

_{in}= 274 mW and a phase detuning

*δ*= 0.5 rad. Incoherent excitation of a CS is performed with a single 4 ps-wide writing pulse generated by a 10 MHz mode-locked laser at 1535 nm. It is launched into the cavity through the input couler and removed by a wavelength-division multiplexer (WDM) coupler after 90 m to avoid the excitation of a second CS as the 1535 nm wave travels at a different group velocity than the intra-cavity field. Note that the maximum detuning we could achieve was constrained both by the necessity to stay around the resonance, for the error signal of the PID controller to have a significant slope, and by the limited power of the writing pulses.

*π*phase-shift difference that exists between the two ports of the fiber coupler [see the minus sign in Eq. (6), necessary to satisfy energy conservation], the CSs appear as inverted peaks on a strong CW background at the output port. In our single-shot detection system, which has a bandwidth limited to 1 GHz, these inverted peaks, which are typically a few picoseconds in duration, are so severely broadened that they are in practice completely smeared out in the background and invisible on the oscilloscope. To circumvent this problem, we used a sequence of two identical narrow optical bandpass filters in front of the photodiode. These filters are dense WDM fiber couplers centered at 1552 nm, slightly above the wavelength of the CW driving beam, with a spectral width of 0.9 nm at 0.5 dB. This effectively removes the CW background and transforms the CSs into 10 ps bright peaks that can be detected with our 1 GHz measurement system.

### 3.2. Experimental results

**4**, 471–476 (2010) [CrossRef] .

**4**, 471–476 (2010) [CrossRef] .

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

## 4. Dynamical instabilities of one-dimensional cavity solitons

### 4.1. Oscillatory behavior of the 1D cavity soliton

*τ*= 50 has been taken large enough so that boundary conditions do not affect the dynamics of the CS. The time-step used is

*δt*= 5 × 10

^{−4}, much smaller than any relevant time scale of the system.

*S*|

^{2}= 8.5 and Δ = 4.1]. Figure 3(b) shows the theoretical temporal intensity profile of the CS in the fiber cavity at a time where it reaches its maximum peak power, while Figure 3(c) depicts the time evolution of the peak power of the CS. Figure 3(d) shows the corresponding contour plot of the time evolution of the central part of the CS profile shown in Figure 3(b). As in the experiment, time-periodic oscillations are observed. Figure 3(c) reveals a period of 1.418 in the normalized time of the LLE model. Using Eq. (3), this translates to an actual period of 10.6 roundtrips. As regards the modulation depth, let us notice that the modulation depth of the fluctuations shown in Figure 3(c) cannot be directly compared to the one of Figure 2(c) as the experimental measurements are performed behind two optical bandpass filters and after the CSs interfere with the reflected part of the driving beam. In order to compare the magnitude of the theoretical and experimental power oscillations, we calculate the profile of the CS at the output port using Eq. (6) and add a numerical filter mimicking the ones used in the experiment. This leads to a theoretical modulation depth of 95%.

*δ*= 0.5 rad). Higher normalized detunings should be readily accessible experimentally by simply increasing the cavity finesse. This is not easy to achieve in fiber resonators, mainly because of the lossy intracavity isolator. However, high finesse microresonators have been recently implemented [25

25. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature **450**, 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. Express **19**, 14233–14239 (2011) [CrossRef] [PubMed] .

33. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. **74**, 99–143 (2002) [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. E **84**, 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] .

11. P. B. Umbanhowar, F. Melo, and H. L. Swinney, “Localized excitations in a vertically vibrated granular layer,” Nature **382**, 793–796 (1996) [CrossRef] .

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] .

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,” Nonlinearity **12**, 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] .

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]

24. D. Gomila, A. Scroggie, and W. Firth, “Bifurcation structure of dissipative solitons,” Physica D **227**, 70–77 (2007) [CrossRef] .

42. Y.-P. Ma, J. Burke, and E. Knobloch, “Defect-mediated snaking: A new growth mechanism for localized structures,” Physica D **239**, 1867–1883 (2010) [CrossRef] .

### 4.2. Types of dynamical instabilities

*φ*,

*R*) defined by the phase

*φ*and the amplitude

*R*of the center of the CS. The middle panels show the corresponding time evolution of the intensity

*R*

^{2}of the center of the CS, while the panels on the right-hand side similarly show contour plots of the time evolution of the intensity |

*E*|

^{2}of the CS temporal intensity profile. From top to bottom the driving strength

*S*is increased from (a)

*S*= 6 to (d)

*S*= 7.1. As the CS originally emerges through a saddle-node bifurcation (

*SN*

_{CS1}), there exist two branches of CSs: one saddle solution

*S*and one possibly stable CS. In the numerical simulations presented here, we have first used a Newton-Rhapson method to detect the saddle CS

_{a}*S*. As similarly demonstrated in [43

_{a}43. D. Gomila, A. Jacobo, M. A. Matías, and P. Colet, “Phase-space structure of two-dimensional excitable localized structures,” Phys. Rev. E **75**, 026217/1–10 (2007) [CrossRef] .

*r⃗*and

_{s}*r⃗*, respectively. As an initial condition for our numerical simulations, we use the saddle solution

_{u}*S*perturbed by either ±0.005 ×

_{a}*r⃗*such that both directions of the (infinite-dimensional) unstable manifold of the saddle are followed.

_{u}*S*= 6, the dashed and solid black lines in Figure 4(a) show the time evolution in the two-dimensional sub-phase-space (

*φ*,

*R*) for both initial conditions, in each case starting from

*S*. In one case (dashed line), the system relaxes to the stable homogeneous solution

_{a}*H*

_{1}(this is the state from which the experiments are started, before writing a CS; see Section 3.2), while in the other case (solid line) a stable limit cycle

*P*

_{1}is the long-term attractor of the system. Increasing the driving

*S*to 6.5, a similar simulation shows

*H*

_{1}and a stable two-period limit cycle

*P*

_{2}as the attractors of the system [see Figure 4(b)]. Increasing

*S*leads to oscillations with higher periodicity, eventually leading to temporal chaos (the CS remains localized in the fast time-scale

*τ*though). The chaotic attractor of the CS is shown in Figure 4(c). For

*S*= 7.1 [Figure 4(d)], the homogeneous solution

*H*

_{1}is the only remaining stable attractor of the system. However, for perturbations across the stable manifold of the saddle CS

*S*, the system does not immediately relax to

_{a}*H*

_{1}. Perturbations across this threshold lead to a large excursion in phase-space before relaxing to

*H*

_{1}. This excursion corresponds to a chaotic transient where the system follows the reminiscent flow of the chaotic attractor existing for lower values of

*S*[44

44. C. Grebogi, E. Ott, and J. A. Yorke, “Crises, sudden changes in chaotic attractors, and transient chaos,” Physica D **7**, 181–200 (1983) [CrossRef] .

*excitability*[45

45. E. M. Izhikevich, “Neural excitability, spiking and bursting,” Int. J. Bifurcation Chaos **10**, 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. A **82**, 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. E **84**, 036209/1–8 (2011) [CrossRef] .

*S*further, the entire system eventually destabilizes leading not just to a localized instability, but rather to spatio-temporal chaos, see Figure 5. The white dashed line in Figure 5 shows the moment at which the system loses its perfect left-right symmetry and develops spatio-temporal chaos. The front that exists between the region with (oscillating) localized structures and the homogeneous state also loses its left-right symmetry at this point. The asymmetry in front propagation speed between the left and right side seems to be sensitive to initial conditions.

## 5. Conclusion

## Acknowledgments

## References

1. | L. A. Lugiato, “Introduction to the feature section on cavity solitons: an overview,” IEEE J. Quantum Elec. |

2. | G. S. McDonald and W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B |

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,” Nature |

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. |

5. | V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys. |

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. |

7. | M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. |

8. | J. Wu, R. Keolian, and I. Rudnick, “Observation of a nonpropagating hydrodynamic soliton,” Phys. Rev. Lett. |

9. | H. C. Kim, R. L. Stenzel, and A. Y. Wong, “Development of ‘cavitons’ and trapping of RF field,” Phys. Rev. Lett. |

10. | R. Richter and I. V. Barashenkov, “Two-dimensional solitons on the surface of magnetic fluids,” Phys. Rev. Lett. |

11. | P. B. Umbanhowar, F. Melo, and H. L. Swinney, “Localized excitations in a vertically vibrated granular layer,” Nature |

12. | A. Ustinov, “Solitons in Josephson junctions,” Physica D |

13. | B. Ermentrout, X. Chen, and Z. Chen, “Transition fronts and localized structures in bistable reaction-diffusion equations,” Physica D |

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. |

15. | O. Lejeune, M. Tlidi, and P. Couteron, “Localized vegetation patches: A self-organized response to resource scarcity,” Phys. Rev. E |

16. | B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. |

17. | S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett. |

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. |

19. | C. Elphick, G. Iooss, and E. Tirapegui, “Normal form reduction for time-periodically driven differential equations,” Phys. Lett. A |

20. | L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. |

21. | K. Nozaki and N. Bekki, “Chaotic solitons in a plasma driven by an RF field,” J. Phys. Soc. Jpn. |

22. | D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett. |

23. | A. B. Matsko, A. A. Savchenkov, and L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett. |

24. | D. Gomila, A. Scroggie, and W. Firth, “Bifurcation structure of dissipative solitons,” Physica D |

25. | P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature |

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. Express |

27. | A. Tierno, F. Gustave, and S. Barland, “Class A mode-locked semiconductor ring laser,” Opt. Lett. |

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. B |

29. | M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun. |

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31. | G. P. Agrawal, |

32. | K. Wiesenfeld, “Noisy precursors of nonlinear instabilities,” J. Stat. Phys. |

33. | I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. |

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. E |

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37. | P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photon. |

38. | L. Gelens and E. Knobloch, “Traveling waves and defects in the complex Swift-Hohenberg equation,” Phys. Rev. E |

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. |

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,” Nonlinearity |

41. | J. Burke, A. Yochelis, and E. Knobloch, “Classification of spatially localized oscillations in periodically forced dissipative systems,” SIAM J. Appl. Dyn. Syst. |

42. | Y.-P. Ma, J. Burke, and E. Knobloch, “Defect-mediated snaking: A new growth mechanism for localized structures,” Physica D |

43. | D. Gomila, A. Jacobo, M. A. Matías, and P. Colet, “Phase-space structure of two-dimensional excitable localized structures,” Phys. Rev. E |

44. | C. Grebogi, E. Ott, and J. A. Yorke, “Crises, sudden changes in chaotic attractors, and transient chaos,” Physica D |

45. | E. M. Izhikevich, “Neural excitability, spiking and bursting,” Int. J. Bifurcation Chaos |

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. A |

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. E |

48. | T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science |

**OCIS Codes**

(190.3100) Nonlinear optics : Instabilities and chaos

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 13, 2013

Revised Manuscript: March 20, 2013

Manuscript Accepted: March 21, 2013

Published: April 5, 2013

**Citation**

François Leo, Lendert Gelens, Philippe Emplit, Marc Haelterman, and Stéphane Coen, "Dynamics of one-dimensional Kerr cavity solitons," Opt. Express **21**, 9180-9191 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-9180

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