Kinetics of single stripe formation in intracavity second harmonic generation O. Lejeune a, * , M. Tlidi b a Service de Chimie Physique, Campus Plaine, Code Postal 231, Universit e Libre de Bruxelles, B-1050 Bruxelles, Belgium b Optique Nonlin eaire Th eorique, Campus Plaine, Code Postal 231, Universit e Libre de Bruxelles, B-1050 Bruxelles, Belgium Abstract The second harmonic generation of type I with diffraction is considered. When the spatial modulational instability occurs near a saddle node bifurcation associated with bistability, stable stationary single stripes may be generated spontaneously from the unstable connecting branch of homogeneous steady states. The kinetics of this localized structure formation is found to be described by power laws. Ó 2002 Elsevier Science Ltd. All rights reserved. Among other remarkable features, nonlinear cavities driven by an external field display localized structures (LS) or solitary waves [1,2]. Circular bright or dark spots are such nonlinear solutions. LS have been first predicted in bistable systems [3,4]. Later, it was shown that their existence does not require such homogeneous steady states (HSS) behavior. They arise inside the hysteresis loop formed by a branch of HSS and a branch of spatially periodic solutions which are both stable [5]. The latter originates from a spatial modulational instability [1], also called Turing instability [6], oc- curing even in monostable systems [7]. Intense theoretical research has been carried out on this subject [8]. The existence of these LS was confirmed by experiments using photorefractive material [9,10], as well as liquid crystal light valve [11], and in semiconductor microresonators [12]. Moreover, they may exhibit periodic or chaotic oscillations in time [13]. Other LS, dark and bright solitary waves, appear in the degenerate optical oscillator without any spatial modulational instability [14]. Their formation is in that case related to phase indetermination in a bistable system. In the following, we show that the interaction between the Turing and saddle node bifurcations in the hysteresis loop generated by the second harmonic generation (SHG) may stabilize single dark stripes. The dynamics of such LS is characterized by time scaling lows [15]. Experimental evidence of the solitary dark soliton in degenerate optical parametric mixing [16] has further stimulated the interest in this transverse single stripe formation. We consider an optical ring cavity filled with a quadratically nonlinear material where the SGH process takes place, i.e., two photons with frequency x are absorbed by the nonlinear medium and only one photon with frequency 2x is emitted. We limit our analysis to the phase matching of type I where the SHG process does not involve polarization degrees of freedom due to the birefringence of the v ð2Þ crystal. Due to the richness of the spectrum of its dynamical behavior, this elementary process has attracted considerable interest. Optical bistability, self-pulsing and chaos are some of the phenomena that were predicted [17–19]. The inclusion of chromatic dispersion of the cavity material in the model equations revealed that the intracavity SHG of type I presents temporal modulational instability and mixed mode with Hopf [20,21]. In those studies, the intracavity fields are spatially stabilized by using the guided-wave structure, and therefore the diffraction effects can reasonably be neglected. In contrast with that situation, we look at the coupling between diffraction and v ð2Þ nonlinearity which is the source of a pattern forming instability. Two types of stable LS may then be generated: (i) circular localized spots [22] emerge where a branch of homogeneous stable steady states * Corresponding author. Collaborateur scientifique FNRS. 0960-0779/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII:S0960-0779(02)00381-8 Chaos, Solitons and Fractals 17 (2003) 411–417 www.elsevier.com/locate/chaos