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
Passive fiber ring cavities are simple devices that can exhibit extremely rich nonlinear dynamics, as first reported by Ikeda  and thoroughly investigated over the past three decades [2–6]. 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 [7–9]. Two approaches have previously been used for theoretical modeling of such cavities: the Ikeda-map formalism  (IMF) and the Lugiato-Lefever (LL) approach . 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 . 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 . 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  as well as picosecond pumping, including walk-off beyond the interferometric scale . Recently, partly as a result of some important applications in the field of supercontinuum generation , there has been a resurgence of interest in fiber-based cavities pumped with pulses . Third- as well as higher-order dispersion terms were included in the LL equation for the first time in . Inclusion of these terms may cause bistability and the emergence of temporally asymmetric pulse shapes in fiber ring cavities [16–19]. Very recently, a similar extension of the LL equation was proposed to study fundamental properties of solitons in ring cavities , as well as other schemes such as whispering gallery mode resonators .
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  to fundamental predictions like the celebrated Higgs mechanism . 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 [24–27]. 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 . 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 , traditionally used for mode-locked oscillators, and recently applied to various other types of systems such as storage rings  or free-electron lasers . In brief, this procedure allows the discrete IMF  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 , 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):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 . The ring contains 20 cm of endlessly single-mode PCF  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 , 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. 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 . Since monoatomic gases do not have any Raman response, Eq. (1) reduces to: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 . We cross-checked the solutions obtained with Auto against an implementation of the Newton relaxation algorithm  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.
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 [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)].
Furthermore, Eq. (2) does not preserve U(1) symmetry , i.e., its solutions are not invariant under global phase transformations A → Aeiφ, 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 . 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) . Similarly, we also introduce the general ansatz 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):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. .
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)].
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:
The evolution of the pulse shape between successive round-trips results from the accumulated action of gain/loss, dispersion, nonlinearity and coherent pump:Eqs. (4) and (5), this multiscale analysis yields the following single partial differential equation:
We are grateful to Serge Bielawski and Christophe Szwaj for fruitful discussions.
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