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

Passive fiber ring cavities are simple devices that can exhibit extremely rich nonlinear dynamics, as first reported by Ikeda [1] and thoroughly investigated over the past three decades [26]. From the experimental point of view, the advent of photonic crystal fiber (PCF) as a nonlinear element has sparked new interest in this established field due to the possibility of adjusting dispersion and nonlinearity into previously inaccessible parameter ranges [79]. Two approaches have previously been used for theoretical modeling of such cavities: the Ikeda-map formalism [1] (IMF) and the Lugiato-Lefever (LL) approach [10]. In the IMF, the evolution of the pulse during each trip around the cavity is modeled numerically in discrete steps. In this frame, all physical phenomena such as dispersion, nonlinearity and losses are applied sequentially to the field. Commonly used with pulsed sources, this approach allows for arbitrary pump pulse durations and shapes as well as large values of temporal walk-off τwo between the round-trip time and the pump pulse repetition period [8]. In contrast, the LL model relies on a single partial differential equation for the mean field, and is hence computationally much less demanding and more amenable to analysis. In particular, the LL-equation has been shown to give rise to stable pulsed solutions [11]. Although it is, in principle, restricted to continuous pumping and τwo values of the order of an optical cycle (which we define as interferometric), it has also been successfully applied to quasi-continuous [12] as well as picosecond pumping, including walk-off beyond the interferometric scale [13]. Recently, partly as a result of some important applications in the field of supercontinuum generation [14], there has been a resurgence of interest in fiber-based cavities pumped with pulses [15]. Third- as well as higher-order dispersion terms were included in the LL equation for the first time in [16]. Inclusion of these terms may cause bistability and the emergence of temporally asymmetric pulse shapes in fiber ring cavities [1619]. Very recently, a similar extension of the LL equation was proposed to study fundamental properties of solitons in ring cavities [20], as well as other schemes such as whispering gallery mode resonators [21].

On the other hand, the presence of both linear and nonlinear features in waveguide-ring cavities makes them ideal in the study of fundamental effects such as spontaneous symmetry breaking (SSB) – a phenomenon that relates to the existence of solutions explicitly violating one or more continuous symmetries exhibited by an equation. SSB effects are known to be ubiquitous in nature, being responsible for a plethora of paradigmatic physical phenomena, ranging from ferromagnetism and superconductivity [22] to fundamental predictions like the celebrated Higgs mechanism [23]. In the fields of nonlinear photonics and matter-wave dynamics, SSB is usually attributed to the combined effects of linear waveguiding structures and nonlinear interactions (see e.g [2427]. and references therein), generating asymmetry in the profiles of stationary states of the system.

In this paper we extend the very general LL equation to make it suitable for handling ultrashort pulses as well as walk-off values of the order of, or even larger than, the pulse duration. We verify this equation by first modeling the dynamics of an asynchronously pumped solid-core PCF ring cavity and comparing it to the established Ikeda-map simulations as well as experiments [9]. We then apply it to a gas-filled hollow-core PCF ring cavity. In this case the model reduces to a fully temporally inversion-symmetric equation, which remarkably predicts the spontaneous emergence of temporally asymmetric pulse shapes, thus exhibiting a kind of SSB. In fact, for certain parameter ranges the asymmetric states (ASs) are the only stable solutions of the system, which can be explained using a hydrodynamical representation of the internal energy flow of the pulse within the cavity. We could furthermore observe regimes with multistable operation.

2. Physical model

This proposal relies on a multiscale analysis similar to the well-known Haus master equation [28], traditionally used for mode-locked oscillators, and recently applied to various other types of systems such as storage rings [29] or free-electron lasers [30]. In brief, this procedure allows the discrete IMF [1] to be transformed into a simple differential equation using two independent timescales 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(T0+T)=P(T0)exp(κT), where P describes the intracavity power of a test pulse in the absence of the pump. θ = t/t0 is a fast timescale defined in units of the characteristic pulse duration t0, which resolves the pulse-shape itself.

With this approach, the effects of dispersion, nonlinearity and gain/loss are modeled continuously around the ring cavity. One can show that the evolution of the envelope of the circulating electric field A(T,θ) can be described by the a single partial differential equation as follows (see appendix for a derivation):

[T+δθ]A(T,θ)=A+LfκTpk=2ik+1k!t0kβkθkA+iLfκTpF(A)+ξAp^(θ)eiφp
where second (higher) order dispersion is given by β2 (βk), δ = τwo/(κTpt0) accounts for the temporal walk-off τwo between the pump pulse train and the cavity pulse, Lf is the physical length of the nonlinear fiber and Tp 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 ξ=RP0/(κTp) represents the effective pump strength and R and P0 are the (intensity) reflectivity of the pump beam splitter and the pump pulse peak power. Âp(θ) describes the (normalized) arbitrary temporal shape of the complex pump pulse envelope and φp represents a global shift of the phase of the pump pulse. In contrast to formally similar mean-field models [10,12,16,18,31,32], Eq. (1) allows for pump pulses with arbitrary temporal shapes and durations, as well as non-interferometric walk-off values. This multiscale approach in principle demands that the pulse shape is only weakly modified within a single loop (see appendix) and that the decay rate is small compared to a single round trip (i.e. κTp < 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.

We tested this model in a solid-core PCF ring cavity with characteristics similar to those used in recent experiments, which includes the full wavelength-dependent dispersion [9]. The ring contains 20 cm of endlessly single-mode PCF [33] pumped in the anomalous dispersion region, close to the zero dispersion wavelength. For simplicity, the analysis is limited to instantaneous Kerr nonlinearity, i.e. F(A) = γ|A|2A, 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 (t0 = 100 fs, peak power 1.5 kW). After discarding the transient of the first 1000 loops, we plotted the normalized pulse energy N=|A|2dθ, which is proportional to the physical pulse energy Q = P0t0N, 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.

 

Fig. 1 Bifurcation diagrams of the normalized pulse energy N with respect to the normalized temporal walk-off δ corresponding to a delay range from −200 to 200 fs, simulated with (a) the IMF and (b) Eq. (1). Each value of δ contains 200 values of N. Vertical dashed lines represent dynamical transitions. The evolution diagrams in Fig. 2 correspond to the slice highlighted in green, which is labeled “Fig. 2”.

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As a second test, the evolution of the temporal and spectral pulse shape inside the cavity is shown for 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.

 

Fig. 2 Comparison of temporal (left column) and spectral (right column) pulse evolution simulated with the IMF (top) and Eq. (1) (bottom) for δ = 12. The simulation corresponds to the green region labeled “Fig. 2” in Fig. 1. The time window for θ corresponds to 6 ps in physical units and k = 1/θ.

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3. Time-inversion symmetric ring cavity

In the following, we will focus on a reduced version of Eq. (1) in which only second-order dispersion is considered. Experimentally, this could be realized using a kagomé-lattice hollow-core PCF filled with a high-pressure monoatomic gas such as Xe [34]. Since monoatomic gases do not have any Raman response, Eq. (1) reduces to:

TA(T,θ)=AiLfβ22κTpt02θ2A+iLfκTpF(A)+RP0κTpAp^(θ)
Note that Eq. (2) is 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 ∂TA = 0, which can efficiently be done with e.g. the numerical tool Auto [35]. We cross-checked the solutions obtained with Auto against an implementation of the Newton relaxation algorithm [36] and direct numerical pulse propagation.

Figure 3(a) shows the complete bifurcation diagram of the normalized pulse energy 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 [36,37] was used to test the stability of the stationary states; stable states are drawn with solid lines in Fig. 3, while unstable states are dashed. For the parameters of the ring cavity, the highest displayed pump strength of ξ = 10 corresponds to a physical peak power of 5.8 kW of the sech-shaped pump pulses.

 

Fig. 3 (a) Bifurcation diagram of the normalized pulse energy Ν for the stationary solutions of Eq. (2) with respect to the pump strength ξ, where solid branches are linearly stable and dashed branches unstable. i, ii, and iii indicate regions where the system is monostable, bistable and multistable. (b,c) Close-ups of the bistable and multistable regions. In (b), the symmetric and asymmetric sections are labeled “s” and “as”, while the states outside the dark blue area are symmetric. (d,e) Power and phase of the two solutions in the inset of (c). The vertical grey dashed line in (c) indicates the states discussed in Fig. 4.

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Globally, Fig. 3a presents three distinct regions exhibiting different dynamics. As the effective pump strength is increased, the system bifurcates from monostable regions (i) to regions displaying bistability (ii) and even multistability (iii). At the same time the symmetry of the states changes in a complex way [Fig. 3]. The spontaneous emergence of AS is remarkable since Eq. (2) is time-inversion symmetric. Previous reports of AS in fiber ring cavities required the presence of odd-order dispersion [17,19], the Raman-effect or interferometric walk-off values [38,39]. Even more remarkable is that in the regions labeled ii the AS is stable against perturbations, whereas the symmetric solution is unstable. In region iii (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 σ=0|A(T,θ)A(T,θ)|2dθ/N [Fig. 4(c)]. Note that for T > 550 the asymmetry σ oscillates between two values that do not correspond to isolated solutions of Eq. (2) [Fig. 4(c)].

 

Fig. 4 Evolution of the (a) normalized pulse energy, (b) temporal shape and (c) asymmetry of the cavity pulse when seeding with an unstable symmetric solution at ξ = 3.66 as indicated by the vertical grey dashed line in Fig. 3(c).

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Furthermore, Eq. (2) does not preserve U(1) symmetry [40], i.e., its solutions are not invariant under global phase transformations AAe, which contrasts with other dissipative systems such as those described by Ginzburg-Landau (GL) equations [41,42]. In particular, U(1) symmetry is readily broken in Eq. (2) as a result of the interferometric superposition of the pump and the intracavity pulses. Among other implications, this remarkable feature of Eq. (2) compels any complex steady-state to be sensitive to global phase transformations and hence to display a unique phase portrait [Figs. 3(d) and 3(e)]. In particular, the phase must be “engineered” and preserved by the ring cavity if the pulse shape is to be sustained.

4. Hydrodynamic interpretation of stationary cavity pulse shapes

In order to explain how the AS can be sustained within this temporally inversion-symmetric system, we employ a hydrodynamical analogy, based on the so-called Madelung transformation [43]. In brief, the optical field A(T,θ) = ρ1/2eiΦ is interpreted as a fluid, with density ρ = |A|2 and velocity v = ∇Φ(θ) [44]. Similarly, we also introduce the general ansatz P0Ap^=f(θ)eiΨ for the coherent pump in Eq. (2). Solving for the real and imaginary parts yields an Euler-like equation (real part) that accounts for conservation of momentum, and the following continuity-equation (imaginary part):

Tρ+J=2ξf(θ)ρcos(ΨΦ)ρ=S
where J = i/2(AA* - 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.

It is worth noticing that any stationary state of the system, either temporally symmetric or asymmetric, fulfills energy conservation, i.e. Sdt=0.

Figure 5 plots the energy generation 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)].

 

Fig. 5 Hydrodynamical representation of the (a) energy generation S and (b) internal energy flow J for the AS presented in Fig. 3(e) at ξ ≈3.4. (c) shows the temporal shape of the complex envelope of the pump pulses |Âp(θ)|2.

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5. Conclusions

We have applied an extension of the LL model to describe the dynamics of a passive fiber ring cavity pumped by ultrashort pulses. Although multiscale models, such as the one employed here, in principle demand that the pulse shape is only weakly modified within a single loop, good results are obtained even in a high-nonlinearity regime where the pulse shape dramatically changes from one round-trip to the next.

A Raman-free, synchronously pumped version of the model shows (somewhat surprisingly) that fiber ring cavities described by time-inversion symmetric equations can exhibit spontaneous symmetry breaking as well as multistability. In the case of multistable operation, where both symmetric and asymmetric states coexist, we find that the asymmetric one is stable. We also show that a hydrodynamic approach can be used to explain how the ASs are self-sustained. Experimentally, systems with very weak higher-order dispersion are realizable in a hollow core kagomé-style PCF filled with high-pressure noble gas. The generality of the approach and the governing equation may also be useful to a broad range of different scientific communities, for example assisting in the design of new laser cavities or in the study of nonlinear dynamics in ring configurations.

Appendix - Derivation of the continuous model

In order to simplify the iterative map that describes the cavity pulse shape in any loop n+1 in terms of the one in the preceding loop n, we now examine the evolution of snapshots En(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 En(t) = A(T,θ). Within this framework:

En+1(t)=A(T+κTp,θ+τwot0)A(T,θ)+κTpTA(T,θ)+τwot0θA(T,θ)
where the advection term describes a possible mismatch between the lengths of the passive cavity and the pump oscillator and κTp < 1.

The evolution of the pulse shape between successive round-trips results from the accumulated action of gain/loss, dispersion, nonlinearity and coherent pump:

En+1(t)En(t)=κτwoEn(t)+Lfk=2ik+1k!βktkEn(t)+iLfF(En(t))+REp(t)eiφp
where Ep(t) is the coherent pump field. By combining Eqs. (4) and (5), this multiscale analysis yields the following single partial differential equation:
[T+δθ]A(T,θ)=A+LfκTpk=2ik+1k!t0kβkθkA+iLfκTpF(A)+ξAp^(θ)eiφp
where ξ=RP0/(κTp) has been introduced as the effective pump strength and the peak power P0 of the pump pulses and the normalized pump shape Ap^(θ) have been defined such that Ep(t)=P0Ap^(θ) . Note that this derivation a priori implies that the pulse shape only slightly changes within one round-trip.

Acknowledgments

We are grateful to Serge Bielawski and Christophe Szwaj for fruitful discussions.

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References

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  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, F. Mitschke, “Equations of motion from chaotic data: A driven optical fiber ring resonator,” Phys. Lett. A 256(1), 47–54 (1999).
    [CrossRef]
  3. R. Vallée, “Temporal instabilities in the output of an all-fiber ring cavity,” Opt. Commun. 81(6), 419–426 (1991).
    [CrossRef]
  4. G. Steinmeyer, F. Mitschke, “Longitudinal structure formation in a nonlinear resonator,” Appl. Phys. B 62(4), 367–374 (1996).
    [CrossRef]
  5. J. García-Mateos, F. C. Bienzobas, M. Haelterman, “Optical bistability and temporal symmetry-breaking instability in nonlinear fiber resonators,” Fiber Integr. Opt. 14(4), 337–346 (1995).
    [CrossRef]
  6. R. W. Bowman, G. M. Gibson, M. J. Padgett, F. Saglimbeni, 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).
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2013

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(3), 035802 (2013).
[CrossRef]

J. K. Jang, M. Erkintalo, S. G. Murdoch, S. Coen, “Ultraweak long-range interactions of solitons observed over astronomical distances,” Nat. Photonics 7(8), 657–663 (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(5), 053852 (2013).
[CrossRef]

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

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(10), 104103 (2013).
[CrossRef] [PubMed]

Y. Li, J. Liu, W. Pang, B. A. Malomed, “Symmetry breaking in dipolar matter-wave solitons in dual-core couplers,” Phys. Rev. A 87(1), 013604 (2013).
[CrossRef]

M. Azhar, N. Y. Joly, J. C. Travers, 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]

S. Coen, H. G. Randle, T. Sylvestre, 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]

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

2012

2011

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

C. Bruni, T. Legrand, C. Szwaj, S. Bielawski, 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]

M. Kues, N. Brauckmann, P. Groß, 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]

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

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

2010

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,” Nat. Photonics 4(7), 471–476 (2010).
[CrossRef]

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

2009

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

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

2007

2006

2005

2002

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

2000

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

1999

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

1997

1996

G. Steinmeyer, F. Mitschke, “Longitudinal structure formation in a nonlinear resonator,” Appl. Phys. B 62(4), 367–374 (1996).
[CrossRef]

1995

J. García-Mateos, F. C. Bienzobas, M. Haelterman, “Optical bistability and temporal symmetry-breaking instability in nonlinear fiber resonators,” Fiber Integr. Opt. 14(4), 337–346 (1995).
[CrossRef]

1994

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos Solitons Fractals 4(8–9), 1323–1354 (1994).
[CrossRef]

1991

E. Doedel, H. B. Keller, 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]

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, 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]

R. Vallée, “Temporal instabilities in the output of an all-fiber ring cavity,” Opt. Commun. 81(6), 419–426 (1991).
[CrossRef]

1990

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

1987

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

1985

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

1979

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]

1964

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

Akhmediev, N. N.

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, 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]

Alexandrescu, A.

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

Aranson, I. S.

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

Azhar, M.

M. Azhar, N. Y. Joly, J. C. Travers, 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]

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(3), 035802 (2013).
[CrossRef]

Biancalana, F.

M. J. Schmidberger, F. Biancalana, P. St. J. Russell, 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).

Bielawski, S.

C. Bruni, T. Legrand, C. Szwaj, S. Bielawski, 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]

Bienzobas, F. C.

J. García-Mateos, F. C. Bienzobas, M. Haelterman, “Optical bistability and temporal symmetry-breaking instability in nonlinear fiber resonators,” Fiber Integr. Opt. 14(4), 337–346 (1995).
[CrossRef]

Birks, T. A.

Bowman, R. W.

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

Brauckmann, N.

M. Kues, N. Brauckmann, P. Groß, 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]

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

Bruni, C.

C. Bruni, T. Legrand, C. Szwaj, S. Bielawski, 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]

Chang, W.

Chembo, Y. K.

Y. K. Chembo, C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87(5), 053852 (2013).
[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(3), 035802 (2013).
[CrossRef]

Coen, S.

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

S. Coen, H. G. Randle, T. Sylvestre, 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]

F. Leo, L. Gelens, P. Emplit, M. Haelterman, S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Opt. Express 21(7), 9180–9191 (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,” Nat. Photonics 4(7), 471–476 (2010).
[CrossRef]

S. Coen, M. Tlidi, P. Emplit, M. Haelterman, “Convection versus dispersion in optical bistability,” Phys. Rev. Lett. 83(12), 2328–2331 (1999).
[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(3), 035802 (2013).
[CrossRef]

Couprie, M. E.

C. Bruni, T. Legrand, C. Szwaj, S. Bielawski, 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]

D’Angelo, E. J.

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

Di Leonardo, R.

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

Doedel, E.

E. Doedel, H. B. Keller, 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]

Elleaume, P.

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

Emplit, P.

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

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(10), 104103 (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,” Nat. Photonics 4(7), 471–476 (2010).
[CrossRef]

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

Erkintalo, M.

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

S. Coen, H. G. Randle, T. Sylvestre, 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]

Fallnich, C.

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

M. Kues, N. Brauckmann, P. Groß, 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]

Firth, W. J.

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos Solitons Fractals 4(8–9), 1323–1354 (1994).
[CrossRef]

García-Mateos, J.

J. García-Mateos, F. C. Bienzobas, M. Haelterman, “Optical bistability and temporal symmetry-breaking instability in nonlinear fiber resonators,” Fiber Integr. Opt. 14(4), 337–346 (1995).
[CrossRef]

Gelens, L.

Gibson, G. M.

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

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,” Nat. Photonics 4(7), 471–476 (2010).
[CrossRef]

Green, C.

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

Gross, P.

Groß, P.

M. Kues, N. Brauckmann, P. Groß, 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]

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(10), 104103 (2013).
[CrossRef] [PubMed]

F. Leo, L. Gelens, P. Emplit, M. Haelterman, S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Opt. Express 21(7), 9180–9191 (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,” Nat. Photonics 4(7), 471–476 (2010).
[CrossRef]

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

J. García-Mateos, F. C. Bienzobas, M. Haelterman, “Optical bistability and temporal symmetry-breaking instability in nonlinear fiber resonators,” Fiber Integr. Opt. 14(4), 337–346 (1995).
[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(3), 035802 (2013).
[CrossRef]

Haus, H. A.

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

Heatley, D. R.

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, 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]

Higgs, P. W.

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

Ikeda, K.

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]

Jang, J. K.

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

Joly, N. Y.

M. Azhar, N. Y. Joly, J. C. Travers, 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]

M. Schmidberger, W. Chang, P. St. J. Russell, 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]

M. J. Schmidberger, F. Biancalana, P. St. J. Russell, 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).

Keller, H. B.

E. Doedel, H. B. Keller, 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]

Kernevez, J. P.

E. Doedel, H. B. Keller, 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]

Kivshar, Y. S.

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

J. R. Salgueiro, Y. S. Kivshar, “Nonlinear dual-core photonic crystal fiber couplers,” Opt. Lett. 30(14), 1858–1860 (2005).
[CrossRef] [PubMed]

Knight, J. C.

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(10), 104103 (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,” Nat. Photonics 4(7), 471–476 (2010).
[CrossRef]

Kozyreff, G.

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

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

Kramer, L.

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

Kues, M.

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

M. Kues, N. Brauckmann, P. Groß, 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]

Kurths, J.

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]

Lefever, R.

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos Solitons Fractals 4(8–9), 1323–1354 (1994).
[CrossRef]

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

Legrand, T.

C. Bruni, T. Legrand, C. Szwaj, S. Bielawski, 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]

Leo, F.

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

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(10), 104103 (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,” Nat. Photonics 4(7), 471–476 (2010).
[CrossRef]

Li, Y.

Y. Li, J. Liu, W. Pang, B. A. Malomed, “Symmetry breaking in dipolar matter-wave solitons in dual-core couplers,” Phys. Rev. A 87(1), 013604 (2013).
[CrossRef]

Liu, J.

Y. Li, J. Liu, W. Pang, B. A. Malomed, “Symmetry breaking in dipolar matter-wave solitons in dual-core couplers,” Phys. Rev. A 87(1), 013604 (2013).
[CrossRef]

Louvergneaux, E.

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

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

Lugiato, L. A.

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos Solitons Fractals 4(8–9), 1323–1354 (1994).
[CrossRef]

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

Malomed, B. A.

Y. Li, J. Liu, W. Pang, B. A. Malomed, “Symmetry breaking in dipolar matter-wave solitons in dual-core couplers,” Phys. Rev. A 87(1), 013604 (2013).
[CrossRef]

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

McDonald, G. S.

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos Solitons Fractals 4(8–9), 1323–1354 (1994).
[CrossRef]

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(5), 053852 (2013).
[CrossRef]

Michinel, H.

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

Mindlin, G. B.

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

Miroshnichenko, A. E.

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

Mitschke, F.

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]

G. Steinmeyer, F. Mitschke, “Longitudinal structure formation in a nonlinear resonator,” Appl. Phys. B 62(4), 367–374 (1996).
[CrossRef]

Murdoch, S. G.

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

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(10), 104103 (2013).
[CrossRef] [PubMed]

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

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

Novoa, D.

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

Padgett, M. J.

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

Pang, W.

Y. Li, J. Liu, W. Pang, B. A. Malomed, “Symmetry breaking in dipolar matter-wave solitons in dual-core couplers,” Phys. Rev. A 87(1), 013604 (2013).
[CrossRef]

Randle, H. G.

Russell, P. S. J.

M. Azhar, N. Y. Joly, J. C. Travers, 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]

Russell, P. St. J.

Saglimbeni, F.

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

Salgueiro, J. R.

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

J. R. Salgueiro, Y. S. Kivshar, “Nonlinear dual-core photonic crystal fiber couplers,” Opt. Lett. 30(14), 1858–1860 (2005).
[CrossRef] [PubMed]

Schmidberger, M.

Schmidberger, M. J.

M. J. Schmidberger, F. Biancalana, P. St. J. Russell, 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).

Schwache, A.

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]

Scroggie, A. J.

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos Solitons Fractals 4(8–9), 1323–1354 (1994).
[CrossRef]

Solari, H. G.

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

Soto-Crespo, J. M.

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, 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]

Steinmeyer, G.

G. Steinmeyer, F. Mitschke, “Longitudinal structure formation in a nonlinear resonator,” Appl. Phys. B 62(4), 367–374 (1996).
[CrossRef]

Sylvestre, T.

Szwaj, C.

C. Bruni, T. Legrand, C. Szwaj, S. Bielawski, 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]

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(10), 104103 (2013).
[CrossRef] [PubMed]

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

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

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(3), 035802 (2013).
[CrossRef]

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

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

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

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

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos Solitons Fractals 4(8–9), 1323–1354 (1994).
[CrossRef]

Tommasini, D.

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

Travers, J. C.

M. Azhar, N. Y. Joly, J. C. Travers, 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]

Tredicce, J. R.

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

Vallée, R.

R. Vallée, “Temporal instabilities in the output of an all-fiber ring cavity,” Opt. Commun. 81(6), 419–426 (1991).
[CrossRef]

Vladimirov, A. G.

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

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

Voss, H. U.

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]

Wright, E. M.

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, 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]

Appl. Phys. B

G. Steinmeyer, F. Mitschke, “Longitudinal structure formation in a nonlinear resonator,” Appl. Phys. B 62(4), 367–374 (1996).
[CrossRef]

M. Azhar, N. Y. Joly, J. C. Travers, 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]

Chaos Solitons Fractals

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos Solitons Fractals 4(8–9), 1323–1354 (1994).
[CrossRef]

Comput. Phys. Commun.

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

Fiber Integr. Opt.

J. García-Mateos, F. C. Bienzobas, M. Haelterman, “Optical bistability and temporal symmetry-breaking instability in nonlinear fiber resonators,” Fiber Integr. Opt. 14(4), 337–346 (1995).
[CrossRef]

IEEE J. Quantum Electron.

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

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

Int. J. Bifurcat. Chaos

E. Doedel, H. B. Keller, 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]

J. Lightwave Technol.

Nat. Photonics

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,” Nat. Photonics 4(7), 471–476 (2010).
[CrossRef]

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

Opt. Commun.

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]

R. Vallée, “Temporal instabilities in the output of an all-fiber ring cavity,” Opt. Commun. 81(6), 419–426 (1991).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

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]

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(3), 035802 (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(5), 053852 (2013).
[CrossRef]

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

Y. Li, J. Liu, W. Pang, B. A. Malomed, “Symmetry breaking in dipolar matter-wave solitons in dual-core couplers,” Phys. Rev. A 87(1), 013604 (2013).
[CrossRef]

M. Kues, N. Brauckmann, P. Groß, 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]

C. Bruni, T. Legrand, C. Szwaj, S. Bielawski, 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]

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, 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]

Phys. Rev. Lett.

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

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

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

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

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

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(10), 104103 (2013).
[CrossRef] [PubMed]

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

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

Rev. Mod. Phys.

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

Other

E. Madelung, Die mathematischen Hilfsmittel des Physikers (Springer, 1964).

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]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer, 1999).

N. Akhmediev and A. Ankiewicz, “Dissipative Solitons in the Complex Ginzburg-Landau and Swift-Hohenberg Equations,” in Dissipative Solitons, N. Akhmediev and A. Ankiewicz, eds., Vol. 661 in Lecture Notes in Physics (Springer, 2005), pp. 1–17.

L. D. Landau and E. M. Lifshitz, Statistical Physics (Elsevier, 1996).

M. J. Schmidberger, F. Biancalana, P. St. J. Russell, 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).

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Figures (5)

Fig. 1
Fig. 1

Bifurcation diagrams of the normalized pulse energy N with respect to the normalized temporal walk-off δ corresponding to a delay range from −200 to 200 fs, simulated with (a) the IMF and (b) Eq. (1). Each value of δ contains 200 values of N. Vertical dashed lines represent dynamical transitions. The evolution diagrams in Fig. 2 correspond to the slice highlighted in green, which is labeled “Fig. 2”.

Fig. 2
Fig. 2

Comparison of temporal (left column) and spectral (right column) pulse evolution simulated with the IMF (top) and Eq. (1) (bottom) for δ = 12. The simulation corresponds to the green region labeled “Fig. 2” in Fig. 1. The time window for θ corresponds to 6 ps in physical units and k = 1/θ.

Fig. 3
Fig. 3

(a) Bifurcation diagram of the normalized pulse energy Ν for the stationary solutions of Eq. (2) with respect to the pump strength ξ, where solid branches are linearly stable and dashed branches unstable. i, ii, and iii indicate regions where the system is monostable, bistable and multistable. (b,c) Close-ups of the bistable and multistable regions. In (b), the symmetric and asymmetric sections are labeled “s” and “as”, while the states outside the dark blue area are symmetric. (d,e) Power and phase of the two solutions in the inset of (c). The vertical grey dashed line in (c) indicates the states discussed in Fig. 4.

Fig. 4
Fig. 4

Evolution of the (a) normalized pulse energy, (b) temporal shape and (c) asymmetry of the cavity pulse when seeding with an unstable symmetric solution at ξ = 3.66 as indicated by the vertical grey dashed line in Fig. 3(c).

Fig. 5
Fig. 5

Hydrodynamical representation of the (a) energy generation S and (b) internal energy flow J for the AS presented in Fig. 3(e) at ξ ≈3.4. (c) shows the temporal shape of the complex envelope of the pump pulses |Âp(θ)|2.

Equations (3)

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[ T +δ θ ]A(T,θ)=A+ L f κ T p k=2 i k+1 k! t 0 k β k θ k A+i L f κ T p F(A)+ξ A p ^ ( θ ) e i φ p
E n+1 ( t ) E n ( t )=κ τ wo E n ( t )+ L f k=2 i k+1 k! β k t k E n ( t ) +i L f F( E n ( t ) )+ R E p ( t ) e i φ p
[ T +δ θ ]A(T,θ)=A+ L f κ T p k=2 i k+1 k! t 0 k β k θ k A+i L f κ T p F(A)+ξ A p ^ ( θ ) e i φ p

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