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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 3 — Feb. 10, 2014
  • pp: 3045–3053
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Multistability and spontaneous breaking in pulse-shape symmetry in fiber ring cavities

M. J. Schmidberger, D. Novoa, F. Biancalana, P. St.J. Russell, and N. Y. Joly  »View Author Affiliations


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

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

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

] to fundamental predictions like the celebrated Higgs mechanism [23

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

]. 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 [24

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

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]

]. 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

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]

]. 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

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

], traditionally used for mode-locked oscillators, and recently applied to various other types of systems such as storage rings [29

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

] or free-electron lasers [30

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]

]. In brief, this procedure allows the discrete IMF [1

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]

] 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
(1)
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

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

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

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

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

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

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]

], 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

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]

]. The ring contains 20 cm of endlessly single-mode PCF [33

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]

] 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].
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”.
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.

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.
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.
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

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

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]

]. Since monoatomic gases do not have any Raman response, Eq. (1) reduces to:
TA(T,θ)=AiLfβ22κTpt02θ2A+iLfκTpF(A)+RP0κTpAp^(θ)
(2)
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

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]

]. We cross-checked the solutions obtained with Auto against an implementation of the Newton relaxation algorithm [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]

] 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.
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.
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

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]

] 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.

Furthermore, Eq. (2) does not preserve U(1) symmetry [40

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

], 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

41. 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.

,42

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

]. 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

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

]. 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

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]

]. 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
(3)
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)].
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.
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

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)
(4)
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
(5)
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
(6)
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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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]

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]

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]

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]

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]

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]

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]

40.

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

41.

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.

42.

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

43.

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

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]

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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