Daniel Walgraef

Walk-off and pattern selection in optical parametric oscillators.

The effect of walk-off in pattern selection in optical parametric oscillators is theoretically examined. We show that a dynamic mechanism also allows us to observe the formation of structures for positive signal detunings. In this regime the pattern that is generated is a periodic array of kinks that separate regions in which one of two stable steady states is alternately selected. This structure can be regarded as a train of dark soliton stripes because the two solutions have opposite signs. The wavelength of the selected pattern is theoretically predicted, and the prediction agrees with the results of the numerical solutions of the equations governing the device.

Growth dynamics of noise-sustained structures in nonlinear optical resonators

The existence of macroscopic noise-sustained structures in nonlinear optics is theoretically predicted and numerically observed, in the regime of convective instability. The advection-like term, necessary to turn the instability to convective for the parameter region where advection overwhelms the growth, can stem from pump beam tilting or birefringence induced walk-off. The growth dynamics of both noise-sustained and deterministic patterns is exemplified by means of movies. This allows to observe the process of formation of these structures and to confirm the analytical predictions. The amplification of quantum noise by several orders of magnitude is predicted. The qualitative analysis of the near- and far-field is given. It suffices to distinguish noise-sustained from deterministic structures; quantitative informations can be obtained in terms of the statistical properties of the spectra.

Wave-unlocking transition in resonantly coupled complex Ginzburg-Landau equations

We study the effect of spatial frequency forcing on standing-wave solutions of coupled complex Ginzburg-Landau equations. The model considered describes several situations of nonlinear counterpropagating waves and also of the dynamics of polarized light waves. We show that forcing introduces spatial modulations on standing waves which remain frequency locked with a forcing-independent frequency. For forcing above a threshold the modulated standing waves unlock, bifurcating into a temporally periodic state. Below the threshold the system presents a kind of excitability.

Self-organization and nanostructure formation in chemical vapor deposition.

When thin films are grown on a substrate by chemical vapor deposition, the evolution of the first deposited layers may be described, on mesoscopic scales, by dynamical models of the reaction-diffusion type. For monatomic layers, such models describe the evolution of atomic coverage due to the combined effect of reaction terms representing adsorption-desorption and chemical processes and nonlinear diffusion terms that are of the Cahn-Hilliard type. This combination may lead, below a critical temperature, to the instability of uniform deposited layers. This instability triggers the formation of nanostructures corresponding to regular spatial variations of substrate coverage. Patterns wavelengths and symmetries are selected by dynamical variables and not by variational arguments. According to the balance between reaction- and diffusion-induced nonlinearities, a succession of nanostructures including hexagonal arrays of dots, stripes, and localized structures of various types may be obtained. These structures may initiate different growth mechanisms, including Volmer-Weber and Frank-Van der Merwe types of growth. The relevance of this approach to the study of deposited layers of different species is discussed.

SPACE INVERSION SYMMETRY BREAKING AND PATTERN SELECTION IN NONLINEAR OPTICS

Pattern formation in nonlinear optical cavities, when an advection-like term is present, is analysed. This term breaks the space inversion symmetry causing the existence of a regime of convective instabilities, where noise-sustained structures can be found, and changing the pattern orientation and the selected wavevector. The concepts of convective and absolute instability, noise-sustained structures and the selection mechanisms in two dimensions are discussed in the case of optical parametric oscillators and a Kerr resonator. In the latter case, in which hexagons are the selected structure, we predict and observe that stripes are the most unstable structures in the initial linear transient. In the nonlinear regime of the absolute instability these stripes destabilize and hexagons form. Their orientation is dictated by that of the transient stripes and therefore by the advection term. In the convective regime we predict and observe disordered noise-sustained hexagons, preceded in space by noise-sustained stripes.

Stability and symmetry of ion-induced surface patterning

We present a continuum model of ion-induced surface patterning. The model incorporates the atomic processes of sputtering, re-deposition and surface diffusion, and is shown to display the generic features of the damped Kuramoto-Sivashinsky (KS) equation of non-linear dynamics. Linear and non-linear stability analyses of the evolution equation give estimates of the emerging pattern wavelength and spatial symmetry. The analytical theory is confirmed by numerical simulations of the evolution equation with the Fast Fourier Transform method, where we show the influence of the incident ion angle, flux, and substrate surface temperature. It is shown that large local geometry variations resulting in quadratic non-linearities in the evolution equation dominate pattern selection and stability at long time scales.

Noise-Sustained Transverse Structures in Nonlinear Optical Cavities

We theoretically demonstrate the existence of noise-sustained patterns in passive nonlinear quadratic or cubic (Kerr) optical cavitia [I], in the regime of convective instability. This regime refers to the c m of system in which advection overwhelms the growth dynamics due to the instability. Thus the formed pattern leaves the system and a structure can be permanently obaerved only if noise continuously regenerates it. Henceforth. in this condition patterns arise BS macroscopic manifestations of amplified noise; in particular, in optical cavities this noise can be of quantum origin [Z]. For optical parametric oscillators (OPOs) an advection-like term is given by the transversal walk-off between the pump and the signal. This effect is due to the crystal birefringence, vhich is exploited to phasematch the nonlinear interaction. More generally. any misalignement between the pump and the cavity field can produce the same effect. In our presentation we show hou, to determine the range in the parameter space where the instability is of convective nature. Two specific examples of calculation are given. The first refers to a Kerr resouator with a pump misalignemenl: a simple on?-dimensional transversal modcl allows us to introduce the main concepts of convective instabilities. In the semnd example. u~e will analyae a two-dimensional transversal model of an OPO. In this ease the walk-off term breaks the spatial rotational symmetry of the equations. The analysis indicates that this causes a preferential stripe pattern orientation both in the convectively and ahsalutely unstable regimes. In the latter rqime, it also predicts the co-existence of absolutely and convectively unstable spatial modes BS B result of the symmetry breaking. In both c e s . the numerical solutions of the equations governing the pattern evolution confirm all the predictions and noisesustained structures are found and characterized. The differences between noisesustained and deterministic patterns are shown. qualitatively in terms of the nearand far field observation and quantitatiwly through the statistical properties of the field spectral intensity. The numericai solutions also provide valuable informations about the growth dynamics of these structureb. Although txo particular kind of devices were investigated the concipts explained are relevant to other similar optical systems where walk-off or misalignment are present. Noisesustained patterns have been probably found but not recognized and charaeterized in previous experiments. This hypothesis is mainly based on the fact that for system with an advection-like term the instabilities just abwe threshold are always convective.

Polarization Patterns in Kerr Media

We study spatiotemporal pattern formation associated with the polarization degree of freedom of the electric field amplitude in a mean field model describing a Kerr medium in a cavity with flat mirrors and driven by a coherent plane-wave field. We consider linearly as well as elliptically polarized driving fields, and situations of self-focusing and self-defocusing. For the case of self-defocusing and a linearly polarized driving field, there is a stripe pattern orthogonally polarized to the driving field. Such a pattern changes into a hexagonal pattern for an elliptically polarized driving field. The range of driving intensities for which the pattern is formed shrinks to zero with increasing ellipticity. For the case of self-focusing, changing the driving field ellipticity leads from a linearly polarized hexagonal pattern ~for linearly polarized driving! to a circularly polarized hexagonal pattern ~for circularly polarized driving!. Intermediate situations include a modified Hopf bifurcation at a finite wave number, leading to a time dependent pattern of deformed hexagons and a codimension 2 Turing-Hopf instability resulting in an elliptically polarized stationary hexagonal pattern. Our numerical observations of different spatiotemporal structures are described by appropriate model and amplitude equations. @S1063-651X~98!12608-9#