Mustapha Tlidi

Pattern formation in a passive Kerr cavity

Analytic and numerical investigations of a cavity containing a Kerr medium are reported. The mean field equation with plane-wave excitation and diffraction is assumed. Stable hexagons are dominant close to threshold for a self-focusing medium. Bistable switching frustrates pattern formation for a self-defocusing medium. Under appropriate parametric conditions that we identify, there is coexistence of a homogeneous stationary solution, of a hexagonal pattern solution and of a large (in principle infinite) number of localized structure solutions which connect the homogeneous and hexagonal state. Further above threshold, the hexagons show defects, and then break up with apparent turbulence. For Gaussian beam excitation, the different symmetry leads to polygon formation for narrow beams, but quasihexagonal structures appear for broader beams.

A model for the explanation of vegetation stripes (tiger bush)

Regular vegetation patterns appear on aerial views of plateaux in SW Niger where densely and sparsely popu- lated zones alternate with each other. This spatial organization of the vegetation is an endogenous phenomenon which is not limited to specific plants or soils; it is a characteristic land- scape of many arid regions throughout the world. The phe- nomenon is interpreted as the result of a spatial range differ- ence between two biologically distinct interactions operating at the plant population level. The proposed mechanism is independent of external heterogeneities deriving from soil geomorphology or meteorology. We present a model to simu- late the genesis of vegetation stripes. In addition, the model predicts the occurrence of vegetation hexagons corresponding to higher or lower density spots arranged in a hexagonal lattice. The distinction between the two spatial symmetries is discussed in terms of their Fourier transforms.

Transverse dynamics in cavity nonlinear optics (2000-2003)

We review publications on transverse dynamics in cavity nonlinear optics during the years 2000–2003. Topics covered are transverse pattern dynamics, localized structures, scaling of turn-on transients, properties of nonlinear PDEs, VCSELs, quadratic media, light valves with optical feedback, quantum properties of light, and quantum images.

Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials

We study a ring cavity filled with a slab of a right-handed material and a slab of a left-handed material. Both layers are assumed to be nonlinear Kerr media. First, we derive a model for the propagation of light in a left-handed material. By constructing a mean-field model, we show that the sign of diffraction can be made either positive or negative in this resonator, depending on the thicknesses of the layers. Subsequently, we demonstrate that the dynamical behavior of the modulation instability is strongly affected by the sign of the diffraction coefficient. Finally, we study the dissipative structures in this resonator and reveal the predominance of a two-dimensional up-switching process over the formation of spatially periodic structures, leading to the truncation of the homogeneous hysteresis cycle.

Localized structures in dissipative media: from optics to plant ecology

Localized structures (LSs) in dissipative media appear in various fields of natural science such as biology, chemistry, plant ecology, optics and laser physics. The proposal for this Theme Issue was to gather specialists from various fields of nonlinear science towards a cross-fertilization among active areas of research. This is a cross-disciplinary area of research dominated by nonlinear optics due to potential applications for all-optical control of light, optical storage and information processing. This Theme Issue contains contributions from 18 active groups involved in the LS field and have all made significant contributions in recent years.

High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities

Near a zero-dispersion wavelength, high-order dispersion effects play a central role in a photonic crystal fiber cavity. The study of such effects reveals the existence of stable dissipative dark solitons in the anomalous dispersion regime. Such dark localized structures are not present without high-order dispersion. They consist of dips in the profile of the intensity field. The number of dips and their temporal distribution are determined by the initial conditions. A snaking bifurcation diagram associated with these solutions is constructed.

Control and removal of modulational instabilities in low-dispersion photonic crystal fiber cavities

Taking up to fourth-order dispersion effects into account, we show that fiber resonators become stable for a large intensity regime. The range of pump intensities leading to modulational instability becomes finite and controllable. Moreover, by computing analytically the thresholds and frequencies of these instabilities, we demonstrate the existence of a new unstable frequency at the primary threshold. This frequency exists for an arbitrary small but nonzero fourth-order dispersion coefficient. Numerical simulations for a low and flattened dispersion photonic crystal fiber resonator confirm analytical predictions and open the way to experimental implementation.

Three-dimensional structures in nonlinear cavities containing left-handed materials.

We study the coupling between negative diffraction and direct dispersion in a nonlinear ring cavity containing slabs of Kerr nonlinear right-handed and left-handed materials. Within the mean field approximation, we show that a portion of the homogeneous response curve is affected by a three-dimensional modulational instability. We show numerically that the light distribution evolves through a sequence of three-dimensional dissipative structures with different lattice symmetry. These structures are unstable with respect to the upswitching process, leading to a premature transition to the upper branch in the homogeneous hysteresis cycle.

On Vegetation Clustering, Localized Bare Soil Spots and Fairy Circles

We present a model and nonlinear analysis which account for the clustering behaviors of arid vegetation ecosystems, the formation of localized bare soil spots (sometimes also called fairy circles) in these systems and the attractive or repulsive interactions governing their spatio-temporal evolution. Numerical solutions of the model closely agree with analytical predictions.

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.