Bouncing localized structures in a liquid-crystal light-valve experiment

Abstract

Experimental evidence of bouncing localized structures in a nonlinear optical system is reported. Oscillations in the position of the localized states are described by a consistent amplitude equation, which we call the Lifshitz normal form equation, in analogy with phase transitions. Localized structures are shown to arise close to the Lifshtiz point, where non-variational terms drive the dynamics into complex and oscillatory behaviors. [11,12,13] and cavity solitons in lasers [14]. Localized states are patterns which extend only over a small portion of a spatially extended and homogeneous system [15]. Different mechanisms leading to stable localization have been proposed [16]. Among these, two main classes of localized structures have to be distinguished, namely those localized structures arising as solutions of a quintic Swift-Hohenberg like equation [16] and those that are stabilized by nonvariational terms in the subcritical Ginzburg-Landau equation [17]. The main difference between the two cases is that the first-type localized structures have a characteristic size which is fixed by the pinning mechanism over the underlying pattern or by spatial damped oscillations between homogenous states [16,18], whereas the second-type ones have no intrinsic spatial length, their size being selected by non-variational effects and going to infinity when dissipation goes to zero. In both cases, non-variational effects may lead to dynamical behaviors of localized structures [19]. Variational models based on a generalized Swift-Hohenberg equation have been proposed to describe the appearance of localized structures in nonlinear optics [20]. However, a generalization including non-variational terms is generically expected to apply even in optics, as happens, for instance, in semiconductor laser instabilities [21], giving rise to dynamical behaviors of localized structures, such as propagation and oscillations of their positions [22]. PacsWe report here an experimental evidence of localized structures dynamics in a Liquid-Crystal-Light-Valve (LCLV) with optical feedback. It is already known that, in the simultaneous presence of bistability and pattern forming diffractive feedback, the LCLV system shows localized structures [12,23,24,25]. Recently, rotation of localized structures along concentric rings have been reported in the case of a rotation angle introduced in the feedback loop [26]. Here, we fix a zero rotation angle and we show a new dynamical behavior, the bouncing of two adjacent localized structures, that is not related to imposed boundary conditions but is instead a direct consequence of the non-variational character of the system under study. Theoretically, we show that the LCLV system has several branches of bistability connecting an homogeneous state to a patterned one and we derive an amplitude equation accounting for the appearance of localized structures. This is a one-dimensional model, that we call the Lifshitz normal form equation [22], characterizing the dynamics of localized structures close to each po...

Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems

“…Some consequences of these have already been identified: ͑1͒ Localized structures can move with a constant speed. 25 This behavior is clearly attributed to the absence of any Lyapunov or potential to minimize for Eq. ͑1͒; ͑2͒ Two modulational instabilities can exist with different wavelengths.…”

“…The above equation is one of the simplest possible nonlinear models of spatial dynamics; it has been derived first for the coherently pumped semiconductor cavity, 21,22 and soon later for a liquid crystal light valve with optical feedback. [23][24][25] More recently, it was derived for an acoustic resonator containing a viscous medium. 26 A distinctive feature of this equation is the presence of the nonlinear diffusion term ٌ 2 and ٌ͉ ͉ 2 , which breaks the → − and Y → −Y symmetry and renders it nonvariational, i.e., it does not possess a Lyapunov functional.…”

Spontaneous motion of localized structures and localized patterns induced by delayed feedback

“…In two-dimensional (2D) settings, theoretical prediction of high degree of multistability between structures having different number of peaks was established for driven nonlinear planar cavities [6][7][8][9][10][11]. This prediction has been confirmed by experimental evidence of DLS in various nonlinear optical systems [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Transverse optical structures can be either stationary or not.…”

Localized States in Bi‐Pattern Systems

“…By increasing the input light intensity I in we observe a sequence of transitions, as shown by the experimental snapshots of Figure 2. First, the homogeneous steady-state looses stability and develops a pattern of hexagons (Figure 2 The theoretical model for the LCLV feedback system was previously derived in [37] and consists in two coupled equations, one for the average director tilt θ(…”