## Multistability and spontaneous breaking in pulse-shape symmetry in fiber ring cavities |

Optics Express, Vol. 22, Issue 3, pp. 3045-3053 (2014)

http://dx.doi.org/10.1364/OE.22.003045

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### Abstract

We describe the spatio-temporal evolution of ultrashort pulses propagating in a fiber ring cavity using an extension of the Lugiato-Lefever model. The model predicts the appearance of multistability and spontaneous symmetry breaking in temporal pulse shape. We also use a hydrodynamical approach to explain the stability of the observed regimes of asymmetry.

© 2014 Optical Society of America

## 1. Introduction

1. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. **30**(2), 257–261 (1979). [CrossRef]

2. H. U. Voss, A. Schwache, J. Kurths, and F. Mitschke, “Equations of motion from chaotic data: A driven optical fiber ring resonator,” Phys. Lett. A **256**(1), 47–54 (1999). [CrossRef]

6. R. W. Bowman, G. M. Gibson, M. J. Padgett, F. Saglimbeni, and R. Di Leonardo, “Optical trapping at gigapascal pressures,” Phys. Rev. Lett. **110**(9), 095902 (2013). [CrossRef] [PubMed]

7. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. **24**(12), 4729–4749 (2006). [CrossRef]

9. M. Schmidberger, W. Chang, P. St. J. Russell, and N. Y. Joly, “Influence of timing jitter on nonlinear dynamics of a photonic crystal fiber ring cavity,” Opt. Lett. **37**(17), 3576–3578 (2012). [CrossRef] [PubMed]

1. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. **30**(2), 257–261 (1979). [CrossRef]

10. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. **58**(21), 2209–2211 (1987). [CrossRef] [PubMed]

*τ*

_{wo}between the round-trip time and the pump pulse repetition period [8

8. N. Brauckmann, M. Kues, P. Gross, and C. Fallnich, “Noise reduction of supercontinua via optical feedback,” Opt. Express **19**(16), 14763–14778 (2011). [CrossRef] [PubMed]

11. 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**(8–9), 1323–1354 (1994). [CrossRef]

*τ*

_{wo}values of the order of an optical cycle (which we define as interferometric), it has also been successfully applied to quasi-continuous [12

12. F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics **4**(7), 471–476 (2010). [CrossRef]

13. S. Coen, M. Tlidi, P. Emplit, and M. Haelterman, “Convection versus dispersion in optical bistability,” Phys. Rev. Lett. **83**(12), 2328–2331 (1999). [CrossRef]

14. M. Kues, N. Brauckmann, P. Groß, and C. Fallnich, “Basic prerequisites for limit-cycle oscillations within a synchronously pumped passive optical nonlinear fiber-ring resonator,” Phys. Rev. A **84**(3), 033833 (2011). [CrossRef]

16. M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, “Control and removal of modulational instabilities in low-dispersion photonic crystal fiber cavities,” Opt. Lett. **32**(6), 662–664 (2007). [CrossRef] [PubMed]

16. M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, “Control and removal of modulational instabilities in low-dispersion photonic crystal fiber cavities,” Opt. Lett. **32**(6), 662–664 (2007). [CrossRef] [PubMed]

19. M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A **88**(3), 035802 (2013). [CrossRef]

20. J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Ultraweak long-range interactions of solitons observed over astronomical distances,” Nat. Photonics **7**(8), 657–663 (2013). [CrossRef]

21. Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A **87**(5), 053852 (2013). [CrossRef]

23. P. W. Higgs, “Broken symmetries and the masses of gauge bosons,” Phys. Rev. Lett. **13**(16), 508–509 (1964). [CrossRef]

24. A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, “Nonlinearly PT-symmetric systems: Spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A **84**(1), 012123 (2011). [CrossRef]

27. C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: The experimental side,” Phys. Rev. Lett. **65**(25), 3124–3127 (1990). [CrossRef] [PubMed]

9. M. Schmidberger, W. Chang, P. St. J. Russell, and N. Y. Joly, “Influence of timing jitter on nonlinear dynamics of a photonic crystal fiber ring cavity,” Opt. Lett. **37**(17), 3576–3578 (2012). [CrossRef] [PubMed]

## 2. Physical model

28. H. A. Haus, “Mode-locking of lasers,” IEEE J. Quantum Electron. **6**(6), 1173–1185 (2000). [CrossRef]

29. P. Elleaume, “Microtemporal and spectral structure of storage ring free-electron lasers,” IEEE J. Quantum Electron. **21**(7), 1012–1022 (1985). [CrossRef]

30. C. Bruni, T. Legrand, C. Szwaj, S. Bielawski, and M. E. Couprie, “Equivalence between free-electron-laser oscillators and actively-mode-locked lasers: Detailed studies of temporal, spatiotemporal, and spectrotemporal dynamics,” Phys. Rev. A **84**(6), 063804 (2011). [CrossRef]

1. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. **30**(2), 257–261 (1979). [CrossRef]

*T*and

*θ*. The dynamics of the intracavity pulse occurs on the slow time scale

*T*=

*κt*, where

*t*is physical time and the intensity decay rate is defined via

*P*describes the intracavity power of a test pulse in the absence of the pump.

*θ*=

*t*/

*t*

_{0}is a fast timescale defined in units of the characteristic pulse duration

*t*

_{0}, which resolves the pulse-shape itself.

*A*(

*T*,

*θ*) can be described by the a single partial differential equation as follows (see appendix for a derivation):where second (higher) order dispersion is given by

*β*

_{2}(

*β*

_{k}),

*δ*=

*τ*

_{wo}/(

*κT*

_{p}

*t*

_{0}) accounts for the temporal walk-off

*τ*

_{wo}between the pump pulse train and the cavity pulse,

*L*

_{f}is the physical length of the nonlinear fiber and

*T*

_{p}is the temporal separation between two pump pulses coming from the oscillator.

*F*(

*A*) models the nonlinear response, which can, e.g., include Kerr, Raman, shock and nonlinear gain effects. The influence of the coherent pump is taken into account through the last term where

*R*and

*P*

_{0}are the (intensity) reflectivity of the pump beam splitter and the pump pulse peak power.

*Â*

_{p}(

*θ*) describes the (normalized) arbitrary temporal shape of the complex pump pulse envelope and

*φ*

_{p}represents a global shift of the phase of the pump pulse. In contrast to formally similar mean-field models [10

10. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. **58**(21), 2209–2211 (1987). [CrossRef] [PubMed]

12. F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics **4**(7), 471–476 (2010). [CrossRef]

16. M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, “Control and removal of modulational instabilities in low-dispersion photonic crystal fiber cavities,” Opt. Lett. **32**(6), 662–664 (2007). [CrossRef] [PubMed]

18. G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, and A. G. Vladimirov, “Localized beating between dynamically generated frequencies,” Phys. Rev. Lett. **102**(4), 043905 (2009). [CrossRef] [PubMed]

31. 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**(1), 37–39 (2013). [CrossRef] [PubMed]

32. M. Tlidi and L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. **35**(3), 306–308 (2010). [CrossRef] [PubMed]

*κT*

_{p}< 1). Below, we shall however see that good results are obtained even in a regime where the pulse shape changes dramatically within a single loop due to both, high nonlinearity and losses.

9. M. Schmidberger, W. Chang, P. St. J. Russell, and N. Y. Joly, “Influence of timing jitter on nonlinear dynamics of a photonic crystal fiber ring cavity,” Opt. Lett. **37**(17), 3576–3578 (2012). [CrossRef] [PubMed]

33. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**(13), 961–963 (1997). [CrossRef] [PubMed]

*A*) =

*γ*|

*A*|

^{2}

*A*, where

*γ*represents the conventional nonlinear parameter, but Eq. (1) can be easily extended to include the Raman effect or any other kind of non-instantaneous nonlinear response. Figure 1 shows the asymptotic behavior of this fiber ring cavity pumped by fs-pulses using both the IMF [Fig. 1(a)] and the new multiscale approach [Fig. 1b]. For each value of

*δ*, the response of the cavity is modeled over 1200 loops for a train of Gaussian pump pulses (

*t*

_{0}= 100 fs, peak power 1.5 kW). After discarding the transient of the first 1000 loops, we plotted the normalized pulse energy

*Q*=

*P*

_{0}

*t*

_{0}

*N*, at every round trip. Three distinct regions can be identified: I and III show steady-state behavior (

*N*remains constant over all plotted round-trips) while II exhibits much more complex dynamics. The discrepancy in the position of the transition II → III can be explained by the fact that the transient regime becomes extremely long in the vicinity of

*δ*= 28. In this region the IMF has not yet fully converged to the final state after 1000 loops [Fig. 1(a)], while the multiscale model already has [Fig. 1(b)]. In contrast, both models have already converged to the same final state at

*δ*≈-5, where the transition I → II appears.

*δ*≈12, i.e., well inside the highly turbulent regime II of Fig. 1. This value of

*δ*corresponds to a walk-off value of 25 fs, i.e., much longer than an optical cycle (≈3.5 fs at a vacuum wavelength of the pump of

*λ*= 1042 nm). Both Figs. 1 and 2 show very good agreement between the commonly used discrete formalism and the multiscale model. It is also worth noticing that on the same processor, integration of the multiscale model is 5 times faster than the IMF.

## 3. Time-inversion symmetric ring cavity

34. M. Azhar, N. Y. Joly, J. C. Travers, and P. S. J. Russell, “Nonlinear optics in Xe-filled hollow-core PCF in high pressure and supercritical regimes,” Appl. Phys. B **112**(4), 457–460 (2013). [CrossRef]

*time-inversion symmetric*with respect to

*θ*. While the conventional approach to studying asymptotic dynamics in passive ring cavities is based on numerical pulse propagation, the multiscale method allows one to directly track the stationary states of Eq. (2) by imposing ∂

_{T}

*A*= 0, which can efficiently be done with e.g. the numerical tool Auto [35

35. E. Doedel, H. B. Keller, and J. P. Kernevez, “Numerical analysis and control of bifurcation problems (II): Bifurcation in infinite dimensions,” Int. J. Bifurcat. Chaos **01**(04), 745–772 (1991). [CrossRef]

36. A. Alexandrescu and J. R. Salgueiro, “Efficient numerical method for linear stability analysis of solitary waves,” Comput. Phys. Commun. **182**(12), 2479–2485 (2011). [CrossRef]

*N*of the stationary pulses circulating inside the cavity, plotted against the effective strength

*ξ*of the pulses injected into the cavity. The pump wavelength lies in the anomalous dispersion regime. Note that in the normal dispersion regime the system does not bifurcate, but reduces to one unique branch similar to the upper branch of Fig. 3(a). A standard linear stability analysis [36

36. A. Alexandrescu and J. R. Salgueiro, “Efficient numerical method for linear stability analysis of solitary waves,” Comput. Phys. Commun. **182**(12), 2479–2485 (2011). [CrossRef]

37. J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A **44**(1), 636–644 (1991). [CrossRef] [PubMed]

*ξ*= 10 corresponds to a physical peak power of 5.8 kW of the sech-shaped pump pulses.

17. F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, “Nonlinear symmetry breaking induced by third-order dispersion in optical fiber cavities,” Phys. Rev. Lett. **110**(10), 104103 (2013). [CrossRef] [PubMed]

19. M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A **88**(3), 035802 (2013). [CrossRef]

38. F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Opt. Express **21**(7), 9180–9191 (2013). [CrossRef] [PubMed]

39. Y. Xu and S. Coen, “Observation of a temporal symmetry breaking instability in a synchronously-pumped passive fibre ring cavity,” in *Proceedings of the International Quantum Electronics Conference and Conference on Lasers and Electro-Optics Pacific Rim 2011* (Optical Society of America, 2011), p. I874. [CrossRef]

*ξ*> 3.48) all states are unstable, which coincides with the onset of multistability [Fig. 3(c)]. In fact up to 5 solutions may coexist, both symmetric and AS being possible (region iii of Fig. 3(a)). To the best of our knowledge, such multistable operation in a fiber ring cavity has never been reported before. In this regime, when the system is seeded with a randomly and weakly perturbed version of one of the unstable solutions of Eq. (2), it gradually evolves into an oscillatory bound state, which does not correspond to an isolated eigenstate of the system. Figure 4 shows the simulated evolution of the circulating pulse when the system is seeded with one of the symmetric states (corresponding to the lowest branch at

*ξ*= 3.66 – see dashed line in Fig. 3(c)) in this regime. The seed abruptly transits to the closest state (after

*T*≈145), which then converges to a periodic beating between nonlinear bound states of Eq. (2) at

*T*≈450 for this value of

*ξ*. The initial change from the symmetric seed to the asymmetric steady-state solution of Eq. (2) can clearly be identified via the asymmetry parameter

*T*> 550 the asymmetry

*σ*oscillates between two values that do not correspond to isolated solutions of Eq. (2) [Fig. 4(c)].

*A*→

*A*e

*, which contrasts with other dissipative systems such as those described by Ginzburg-Landau (GL) equations [41,42*

^{iφ}42. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. **74**(1), 99–143 (2002). [CrossRef]

## 4. Hydrodynamic interpretation of stationary cavity pulse shapes

*Madelung transformation*[43]. In brief, the optical field

*A*(

*T*,

*θ*) =

*ρ*

^{1/2}e

^{i}

*is interpreted as a fluid, with density*

^{Φ}*ρ*= |

*A*|

^{2}and velocity

*v*= ∇

*Φ*(

*θ*) [44

44. D. Novoa, H. Michinel, and D. Tommasini, “Pressure, surface tension, and dripping of self-trapped laser beams,” Phys. Rev. Lett. **103**(2), 023903 (2009). [CrossRef] [PubMed]

*J*=

*i*/2(

*A*∇

*A** -

*A**∇

*A*) =

*ρv*is the energy flow and

*S*is the energy source term. As a result the pulse energy always flows from sources (regions with

*S*> 0) to sinks (regions with

*S*< 0). This means that for

*J*< 0 (

*J*> 0) energy flows towards the leading (trailing) edge. In general both

*Φ*and

*Ψ*depend on

*θ*in Eq. (3). The coherent superposition of pump and cavity pulses manifests itself as a sinusoidal modulation of

*S*, a feature that does not appear in systems (such as those governed by the GL equations) where power is supplied by internal amplification rather than a repetitive pump pulse train.

*S*and the energy flow

*J*for an AS corresponding to

*ξ*= 3.4 [Fig. 3(e)]. The redistribution of energy throughout the pulse structure, so as to compensate for losses and dispersion, is clear. Counterintuitively, the effective energy generation

*S*is almost negligible at

*θ*= 0 [Fig. 5(a)], even though the pump power [Fig. 5(c)] is peaked at that point, which is a caused by destructive interference between pump and cavity pulses. Moreover, the positive energy flow leads to a redistribution of the energy towards positive

*θ*(i.e. the pulse tail). This explains why the peak power of this stationary pulse does not overlap with the center of the pump pulse, but is shifted towards

*θ*> 0 [Fig. 3(c)].

## 5. Conclusions

## Appendix - Derivation of the continuous model

*n*+1 in terms of the one in the preceding loop

*n*, we now examine the evolution of snapshots

*E*(

_{n}*t*) of the cavity pulse at a fixed position

*z*in the cavity. This could, for example, correspond to the position of the output beam-splitter. Using a multiscale analysis, the complex field envelope can be expressed as

*E*(

_{n}*t*) =

*A*(

*T*,

*θ*). Within this framework:where the advection term describes a possible mismatch between the lengths of the passive cavity and the pump oscillator and

*κT*

_{p}< 1.

*E*

_{p}(

*t*) is the coherent pump field. By combining Eqs. (4) and (5), this multiscale analysis yields the following single partial differential equation:where

*effective pump strength*and the peak power

*P*

_{0}of the pump pulses and the normalized pump shape

## Acknowledgments

## References and links

1. | K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. |

2. | H. U. Voss, A. Schwache, J. Kurths, and F. Mitschke, “Equations of motion from chaotic data: A driven optical fiber ring resonator,” Phys. Lett. A |

3. | R. Vallée, “Temporal instabilities in the output of an all-fiber ring cavity,” Opt. Commun. |

4. | G. Steinmeyer and F. Mitschke, “Longitudinal structure formation in a nonlinear resonator,” Appl. Phys. B |

5. | J. García-Mateos, F. C. Bienzobas, and M. Haelterman, “Optical bistability and temporal symmetry-breaking instability in nonlinear fiber resonators,” Fiber Integr. Opt. |

6. | R. W. Bowman, G. M. Gibson, M. J. Padgett, F. Saglimbeni, and R. Di Leonardo, “Optical trapping at gigapascal pressures,” Phys. Rev. Lett. |

7. | P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. |

8. | N. Brauckmann, M. Kues, P. Gross, and C. Fallnich, “Noise reduction of supercontinua via optical feedback,” Opt. Express |

9. | M. Schmidberger, W. Chang, P. St. J. Russell, and N. Y. Joly, “Influence of timing jitter on nonlinear dynamics of a photonic crystal fiber ring cavity,” Opt. Lett. |

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

11. | 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 |

12. | F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics |

13. | S. Coen, M. Tlidi, P. Emplit, and M. Haelterman, “Convection versus dispersion in optical bistability,” Phys. Rev. Lett. |

14. | M. Kues, N. Brauckmann, P. Groß, and C. Fallnich, “Basic prerequisites for limit-cycle oscillations within a synchronously pumped passive optical nonlinear fiber-ring resonator,” Phys. Rev. A |

15. | M. J. Schmidberger, F. Biancalana, P. St. J. Russell, and N. Y. Joly, “Semi-analytical model for the evolution of femtosecond pulses during supercontinuum generation in synchronously pumped ring cavities,” in The European Conference on Lasers and Electro-Optics (OSA, 2013). |

16. | M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, “Control and removal of modulational instabilities in low-dispersion photonic crystal fiber cavities,” Opt. Lett. |

17. | F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, “Nonlinear symmetry breaking induced by third-order dispersion in optical fiber cavities,” Phys. Rev. Lett. |

18. | G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, and A. G. Vladimirov, “Localized beating between dynamically generated frequencies,” Phys. Rev. Lett. |

19. | M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A |

20. | J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Ultraweak long-range interactions of solitons observed over astronomical distances,” Nat. Photonics |

21. | Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A |

22. | L. D. Landau and E. M. Lifshitz, |

23. | P. W. Higgs, “Broken symmetries and the masses of gauge bosons,” Phys. Rev. Lett. |

24. | A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, “Nonlinearly PT-symmetric systems: Spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A |

25. | Y. Li, J. Liu, W. Pang, and B. A. Malomed, “Symmetry breaking in dipolar matter-wave solitons in dual-core couplers,” Phys. Rev. A |

26. | J. R. Salgueiro and Y. S. Kivshar, “Nonlinear dual-core photonic crystal fiber couplers,” Opt. Lett. |

27. | C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: The experimental side,” Phys. Rev. Lett. |

28. | H. A. Haus, “Mode-locking of lasers,” IEEE J. Quantum Electron. |

29. | P. Elleaume, “Microtemporal and spectral structure of storage ring free-electron lasers,” IEEE J. Quantum Electron. |

30. | C. Bruni, T. Legrand, C. Szwaj, S. Bielawski, and M. E. Couprie, “Equivalence between free-electron-laser oscillators and actively-mode-locked lasers: Detailed studies of temporal, spatiotemporal, and spectrotemporal dynamics,” Phys. Rev. A |

31. | 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. |

32. | M. Tlidi and L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. |

33. | T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. |

34. | M. Azhar, N. Y. Joly, J. C. Travers, and P. S. J. Russell, “Nonlinear optics in Xe-filled hollow-core PCF in high pressure and supercritical regimes,” Appl. Phys. B |

35. | E. Doedel, H. B. Keller, and J. P. Kernevez, “Numerical analysis and control of bifurcation problems (II): Bifurcation in infinite dimensions,” Int. J. Bifurcat. Chaos |

36. | A. Alexandrescu and J. R. Salgueiro, “Efficient numerical method for linear stability analysis of solitary waves,” Comput. Phys. Commun. |

37. | J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A |

38. | F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Opt. Express |

39. | Y. Xu and S. Coen, “Observation of a temporal symmetry breaking instability in a synchronously-pumped passive fibre ring cavity,” in |

40. | C. Sulem and P.-L. Sulem, |

41. | N. Akhmediev and A. Ankiewicz, “Dissipative Solitons in the Complex Ginzburg-Landau and Swift-Hohenberg Equations,” in |

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

43. | E. Madelung, |

44. | D. Novoa, H. Michinel, and D. Tommasini, “Pressure, surface tension, and dripping of self-trapped laser beams,” Phys. Rev. Lett. |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 25, 2013

Revised Manuscript: January 16, 2014

Manuscript Accepted: January 22, 2014

Published: February 3, 2014

**Citation**

M. J. Schmidberger, D. Novoa, F. Biancalana, P. St.J. Russell, and N. Y. Joly, "Multistability and spontaneous breaking in pulse-shape symmetry in fiber ring cavities," Opt. Express **22**, 3045-3053 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3045

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### References

- K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30(2), 257–261 (1979). [CrossRef]
- H. U. Voss, A. Schwache, J. Kurths, F. Mitschke, “Equations of motion from chaotic data: A driven optical fiber ring resonator,” Phys. Lett. A 256(1), 47–54 (1999). [CrossRef]
- R. Vallée, “Temporal instabilities in the output of an all-fiber ring cavity,” Opt. Commun. 81(6), 419–426 (1991). [CrossRef]
- G. Steinmeyer, F. Mitschke, “Longitudinal structure formation in a nonlinear resonator,” Appl. Phys. B 62(4), 367–374 (1996). [CrossRef]
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