PHYSICAL REVIEW A 90, 033837 (2014) Secondary instabilities in all fiber ring cavities Zheng Liu, 1 Franc ¸ois Leo, 2, 3 Saliya Coulibaly, 1 , * and Majid Taki 1 1 PhLAM, Universit´ e de Lille 1, B ˆ atiment P5-bis, UMR CNRS/USTL 8523, F-59655 Villeneuve d’Ascq, France 2 Service OPERA-photonique, Universit´ e libre de Bruxelles (U.L.B.), 50 Avenue F. D. Roosevelt, Code Postal 194/5, B-1050 Bruxelles, Belgium 3 Photonics Research Group, Department of Information Technology, Ghent University-IMEC, Ghent B-9000, Belgium (Received 15 October 2013; revised manuscript received 18 June 2014; published 22 September 2014) We study secondary instabilities in a coherently driven passive optical fiber cavity. We show that time- modulated solutions which are generated at the onset of instability experience convective and absolute Eckhaus instabilities. The splitting of the secondary instabilities into convective and absolute instabilities drastically impacts the instability boundaries. As a consequence, the stability range of time-modulated waves is enlarged. More importantly, the threshold of absolute instability determines the transition from time-periodic wave trains to a chaotic regime. In the latter the wave trains are composed of irregular oscillations embedded in regular ones. The predictions are in excellent agreement with numerical simulations. DOI: 10.1103/PhysRevA.90.033837 PACS number(s): 42.65.Sf , 42.55.Tv, 42.55.Wd, 42.81.−i I. INTRODUCTION There is currently a considerable interest in understanding the nonlinear dynamics in optical fiber cavities that belong to the class of nonequilibrium dissipative systems. It is well known that the latter experience different generic instability characteristics of nonlinear dynamical systems, namely, the ones modeled by partial differential equations. This concerns almost all fields in science ranging from chemical reactions, biology, to nonlinear optics and fluid mechanics [1]. In optics, the formation of dissipative solutions (DSs) arises naturally in many optical devices from the coupling of dispersion (tem- poral systems) or diffraction (spatial systems), nonlinearities, and dissipation. This coupling triggers various spatiotemporal instabilities, which lead to spontaneous formation of DSs that can be stationary or not, periodic, or localized in the form of dissipative solitons [2]. Among the possible devices, coher- ently driven optical fiber ring cavities have recently appeared as one of the most promising systems, not only for the richness in their nonlinear dynamics [3] but also for their potential applications [4]. It has been shown that, in the presence of third-order dispersion, the standard theoretical approach leading to modulation instability must be extended [5,6]. More specifically, a nonlinear stationary state may be unstable with respect to localized perturbations, but the state that results will depend on the relative values of the amplification and the drift induced by the third-order dispersion term. This is the basis of the difference between convective and absolute regimes. In the former, the perturbation grows in time but decreases locally because it is advected away. In the latter it increases locally and not only in the moving frame, so it eventually extends over all the slow time domain. In this case, threshold values for primary absolute and convective instabilities were obtained [3]. However, as soon as the threshold is exceeded the system enters a nonlinear regime where secondary instabilities arise leading to a more complex dynamics characterized by transition from dissipative periodic solutions to nonperiodic and/or chaotic ones. * saliya.coulibaly@univ-lille1.fr In this paper we investigate these secondary instabilities in the presence of third-order dispersion and emphasize the crucial role of secondary convective and absolute instabilities in the dynamics of a coherently driven passive optical fiber cavity. Indeed, above threshold, dissipative time-modulated solutions are obtained. By increasing the incident pump power, these solutions destabilize and the system bifurcates either to a new periodic solution or enters a chaotic regime. At this stage a secondary instability threshold is reached. This threshold is crucial in the nonlinear dynamics of the system above threshold since it determines the stability range of the dissipative periodic solutions and subsequently the parameters range of their observation. An amplitude equation has been derived to describe the weakly nonlinear dynamics above the onset of instability that allows us to determine the threshold values for the different types of the secondary instability. An important result is that the threshold of absolute instability of modulated solutions determines the transition from modulated dissipative solutions to a regime of a temporal chaotic behavior [7]. II. THE MODEL The system under investigation depicted in Fig. 1 can be modeled by the extended nonlinear Schr ¨ odinger equation with boundary conditions. This leads to a set of two equations, usually referred to as the map equations (or mapping) that can be reduced in the mean-field approximation to obtain a single equation modeling the intracavity field dynamics. This equation, known as the Lugiato-Lefever equation (LL model) [8], has been proven relevant for describing weakly nonlinear dynamics in cavities [9]. It reads ∂ψ ∂t ′ = S − (1 + i)ψ − is ∂ 2 ψ ∂τ ′2 + B 3 ∂ 3 ψ ∂τ ′3 + i |ψ | 2 ψ, (1) where t ′ = tθ 2 /2t R is a slow normalized time variable, with t the real time and t R the round-trip time, τ ′ = Tθ/(L|β 2 |) 1/2 is a fast normalized time variable, s = sign(β 2 ), ψ = A(2γL) 1/2 /θ , S = 2A i (2γL) 1/2 /θ 2 , B 3 = β 3 θ/(3L 1/2 |β 2 | 3/2 ), and  = 2δ 0 /θ 2 . The amplitudes A i and A are, respectively, slowly varying envelopes for the incident pump electric field and the intracavity electric field, δ 0 the cavity phase detuning, L the cavity length, γ the nonlinear 1050-2947/2014/90(3)/033837(9) 033837-1 ©2014 American Physical Society