Mediated by Localized Structures in Kerr Cavities

We characterize a scenario where localized structures in nonlinear optical cavities display an oscillatory behavior which becomes unstable leading to an excitable regime. Excitability emerges from spatial dependence since the system locally is not excitable. We show the existence of different mechanisms leading to excitability depending on the profile of the pump field. DOI: 10.2529/PIERS060907134206 Localized structures (LS) in dissipative optical cavities arise as a consequence of the interplay between diffraction, nonlinearity, driving, and dissipation [1]. These structures, also known as cavity solitons, are unique once the parameters of the system have been fixed. This fact makes this structures potentially useful in optical storage and processing of information [2, 3]. LS may develop a number of instabilities, for instance their amplitude can oscillate in time while remaining static in space. Here we report on a novel regime of excitability associated to the existence of localized structures in a nonlinear optical system [4, 5]. Excitability has been found in a variety of systems [6], including optical systems [7], and is characterized by a nonlinear response under applied external perturbation. Perturbations exceeding a certain threshold are able to elicit in the system a full and well defined response. Furthermore after one perturbation the system cannot be excited again within a refractory period of time. Excitability is behind excitation waves in heart tissue and the existence of action potentials in neurons, and, so, may confer new computational capabilities to optical systems beyond information storage. In this paper we show the existence of different mechanisms leading to excitability depending on the profile of the pump field. For a homogeneous pump the mechanism leading to excitable behavior is a saddle-loop bifurcation through which an stable oscillating LS collides with an unstable LS [4]. For a system pumped by a localized Gaussian beam on top of homogeneous background the scenario is richer and one finds two different mechanisms leading to excitability. One is based on a saddleloop bifurcation as above while the other takes place through a saddle-node in the invariant circle (SNIC) bifurcation. This second mechanism has excitability threshold which can be much lower. We consider a ring cavity filled with a nonlinear self-focusing Kerr medium pumped by an external field. In the mean field approximation, the dynamics of the electric field inside the cavity can be described by a single partial differential equation for the scaled slowly varying amplitude E(~x, t) [8] ∂tE = −(1 + iθ)E + i∇2E + E0 + i |E| E, (1) where E0 is the homogeneous driving field and θ is the cavity detuning. The homogeneous steady state solution of Eq. (1) is implicitly given by Es = E0/(1 + i(θ − Is)), where Is = |Es|. We use the intra-cavity background intensity Is together with θ as convenient control parameters. The homogeneous solution has a modulational instability (MI) at Is = 1.0 leading to the formation of subcritical hexagonal patterns. The existence and dynamical properties of localized structures in this system, the so called Kerr cavity solitons (KCS), have been studied in [9] and references therein. Fig. 1 shows the region of existence of KCS in the Is − θ parameter space. The KCS branch starts subcritically at the MI point. The middle-branch KCS is unstable and it has only a single unstable mode. There is a turning point (saddle-node) where an stable upper-branch KCS starts. The bifurcation diagram as function of Is for fixed θ is shown in Fig. 1. A similar diagram can be obtained for fixed Is as function of θ. Increasing Is or θ the upper-branch KCS undergoes a Hopf bifurcation resulting in a periodically oscillating localized structure. The oscillation is such that it approaches the stable manifold of the middle-branch KCS and then escapes along the unstable manifold. As the detuning is increased the limit cycle goes closer and closer to the middle-branch KCS as illustrated in Fig. 2. At a critical value θc a global bifurcation takes place: the cycle touches the middle-branch KCS and PIERS ONLINE, VOL. 3, NO. 3, 2007 350 No Lalized Suctur es E xitabi l i t y (a)