Damià Gomila

Dynamics of localized and patterned structures in the Lugiato-Lefever equation determine the stability and shape of optical frequency combs

This research was supported by the Research Foundation-Flanders (FWO), by the Spanish MINECO, and FEDER under Grants FISICOS (Grant No. FIS2007-60327) and INTENSE@COSYP (Grant No. FIS2012-30634), by Comunitat Autonoma de les Illes Balears, by the Research Council of the Vrije Universiteit Brussel (VUB), and by the Belgian Science Policy Office (BelSPO) under Grant No. IAP 7-35. S. Coen also acknowledges the support of the Marsden Fund of the Royal Society of New Zealand.

Dynamical properties of two-dimensional Kerr cavity solitons

We present the results of our study of the dynamics of two-dimensional Kerr cavity solitons. The solitons are absolutely stable over a substantial parameter domain. We analyze their dynamics beyond the instability boundary, finding regions of stable oscillation and of fivefold or sixfold azimuthal instability. The Hopf oscillation is surprisingly robust, owing to the influence of a lower-amplitude unstable soliton.

Excitability mediated by localized structures in a dissipative nonlinear optical cavity

This work reports on a new regime of excitability associated to the existence of localized structures in a nonlinear optical system. Findings emphasize that, in absence of spatial degreed of freedom, the system described by the partial differential equation is not an excitable system. The system exhibits excitability only after a localized structure has undergone a Hopf and a saddle-loop bifurcation. Finally, this study shows that all this scenario is organized by a co-dimension two Takens-Bogdanov bifurcation point.

Impact of nonlocal interactions in dissipative systems: Towards minimal-sized localized structures

In order to investigate the size limit of spatial localized structures in a nonlinear system, we explore the impact of linear nonlocality on their domains of existence and stability. Our system of choice is an optical microresonator containing an additional metamaterial layer in the cavity, allowing the nonlocal response of the material to become the dominating spatial process. In that case, our bifurcation analysis shows that this nonlocality imposes another limit on the width of localized structures going beyond the traditional diffraction limit.

Third-order chromatic dispersion stabilizes Kerr frequency combs.

Using numerical simulations of an extended Lugiato-Lefever equation we analyze the stability and nonlinear dynamics of Kerr frequency combs generated in microresonators and fiber resonators, taking into account third-order dispersion effects. We show that cavity solitons underlying Kerr frequency combs, normally sensitive to oscillatory and chaotic instabilities, are stabilized in a wide range of parameter space by third-order dispersion. Moreover, we demonstrate how the snaking structure organizing compound states of multiple cavity solitons is qualitatively changed by third-order dispersion, promoting an increased stability of Kerr combs underlined by a single cavity soliton.

Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials

In this work we characterize the dynamical instabilities of localized structures exhibited by a recently introduced [Gelens et al., Phys. Rev. A 75, 063812 (2007)] generalization of the Lugiato-Lefever model that includes a weakly nonlocal response of an intracavity metamaterial. A rich scenario, in which the localized structures exhibit different types of oscillatory instabilities, tristability, and excitability, including a regime of conditional excitability in which the system is bistable, is presented and discussed. Finally, it is shown that the scenario is organized by a pair of Takens-Bogdanov codimension-2 points.

Phase-space structure of two-dimensional excitable localized structures

In this work we characterize in detail the bifurcation leading to an excitable regime mediated by localized structures in a dissipative nonlinear Kerr cavity with a homogeneous pump. Here we show how the route can be understood through a planar dynamical system in which a limit cycle becomes the homoclinic orbit of a saddle point (saddle-loop bifurcation). The whole picture is unveiled, and the mechanism by which this reduction occurs from the full infinite-dimensional dynamical system is studied. Finally, it is shown that the bifurcation leads to an excitability regime, under the application of suitable perturbations. Excitability is an emergent property for this system, as it emerges from the spatial dependence since the system does not exhibit any excitable behavior locally.

Vortex solitons in lasers with feedback

We report on the existence, stability and dynamical properties of two-dimensional self-localized vortices with azimuthal numbers up to 4 in a simple model for lasers with frequency-selective feedback.We build the full bifurcation diagram for vortex solutions and characterize the different dynamical regimes. The mathematical model used, which consists of a laser rate equation coupled to a linear equation for the feedback field, can describe the spatiotemporal dynamics of broad area vertical cavity surface emitting lasers with external frequency selective feedback in the limit of zero delay.

Dark solitons in the Lugiato-Lefever equation with normal dispersion

This research was supported by the Research Foundation–Flanders (FWO-Vlaanderen) (P.P. and L.G.), by the Junior Mobility Programme (JuMo) of the KU Leuven (L.G.), by the Belgian Science Policy Office (BelSPO) under Grant No. IAP 7-35 (P.P. and L.G.), by the Research Council of the Vrije Universiteit Brussel (P.P. and L.G.), by the Spanish MINECO and FEDER under Grant Intense@Cosyp (FIS2012-30634) (D.G.), and by the National Science Foundation under Grant No. DMS-1211953 (E.K.).

From one- to two-dimensional solitons in the Ginzburg-Landau model of lasers with frequency-selective feedback

We use the cubic complex Ginzburg-Landau equation linearly coupled to a dissipative linear equation as a model for lasers with an external frequency-selective feedback. This system may also serve as a general pattern-formation model in media driven by an intrinsic gain and selective feedback. While, strictly speaking, the approximation of the laser nonlinearity by a cubic term is only valid for small field intensities, it qualitatively reproduces results for dissipative solitons obtained in models with a more complex nonlinearity in the whole parameter region where the solitons exist. The analysis is focused on two-dimensional stripe-shaped and vortex solitons. An analytical expression for the stripe solitons is obtained from the known one-dimensional soliton solution, and its relation with vortex solitons is highlighted. The radius of the vortices increases linearly with their topological charge m, therefore the stripe-shaped soliton may be interpreted as the vortex with m=∞, and, conversely, vortex solitons can be realized as unstable stripes bent into stable rings. The results for the vortices are applicable for a broad class of physical systems.