Cavity soliton inhibition of extreme events in lasers with injection.

Vortex mediated turbulence can be the key element in the generation of extreme events in spatially extended lasers with optical injection. Here, we study the interplay of vortex mediated turbulence and cavity solitons on the onset of extreme events in semiconductor lasers with injection. We first analyze and characterize these two features separately, spatiotemporal chaotic optical vortices for low values of the injection intensity and cavity solitons above the locking regime. In regimes where vortex mediated turbulence and cavity solitons coexist, localized peaks of light inhibit instead of enhancing the generation of rogue waves by locally regularizing the otherwise chaotic phase of the optical field. Cavity solitons can then be used to manipulate and control extreme events in systems displaying vortex mediated turbulence.

Observation of Mode-Locked Spatial Laser Solitons.

A stable nonlinear wave packet, self-localized in all three dimensions, is an intriguing and much sought after object in nonlinear science in general and in nonlinear photonics in particular. We report on the experimental observation of mode-locked spatial laser solitons in a vertical-cavity surface-emitting laser with frequency-selective feedback from an external cavity. These spontaneously emerging and long-term stable spatiotemporal structures have a pulse length shorter than the cavity round-trip time and may pave the way to completely independent cavity light bullets.

Quantum threshold for optomechanical self-structuring in a Bose-Einstein condensate.

Theoretical analysis of the optomechanics of degenerate bosonic atoms with a single feedback mirror shows that self-structuring occurs only above an input threshold that is quantum mechanical in origin. This threshold also implies a lower limit to the size (period) of patterns that can be produced in a condensate for a given pump intensity. These thresholds are interpreted as due to the quantum rigidity of Bose-Einstein condensates, which has no classical counterpart. Above the threshold, the condensate self-organizes into an ordered supersolid state with a spatial period self-selected by optical diffraction.

A simple but precise method for quantitative measurement of the quality of the laser focus in a scanning optical microscope.

We report a method for characterizing the focussing laser beam exiting the objective in a laser scanning microscope. This method provides the size of the optical focus, the divergence of the beam, the ellipticity and the astigmatism. We use a microscopic-scale knife edge in the form of a simple transmission electron microscopy grid attached to a glass microscope slide, and a light-collecting optical fibre and photodiode underneath the specimen. By scanning the laser spot from a reflective to a transmitting part of the grid, a beam profile in the form of an error function can be obtained and by repeating this with the knife edge at different axial positions relative to the beam waist, the divergence and astigmatism of the postobjective laser beam can be obtained. The measured divergence can be used to quantify how much of the full numerical aperture of the lens is used in practice. We present data of the beam radius, beam divergence, ellipticity and astigmatism obtained with low (0.15, 0.7) and high (1.3) numerical aperture lenses and lasers commonly used in confocal and multiphoton laser scanning microscopy. Our knife-edge method has several advantages over alternative knife-edge methods used in microscopy including that the knife edge is easy to prepare, that the beam can be characterized also directly under a cover slip, as necessary to reduce spherical aberrations for objectives designed to be used with a cover slip, and it is suitable for use with commercial laser scanning microscopes where access to the laser beam can be limited.

Kinetic theory for transverse optomechanical instabilities.

We investigate transverse symmetry-breaking instabilities emerging from the optomechanical coupling between light and the translational degrees of freedom of a collisionless, damping-free gas of cold, two-level atoms. We develop a kinetic theory that can also be mapped on to the case of an electron plasma under ponderomotive forces. A general criterion for the existence and spatial scale of transverse instabilities is identified; in particular, we demonstrate that monotonically decreasing velocity distribution functions are always unstable.

Energy shedding during nonlinear self-focusing of optical beams.

Self-focusing of intense laser beams and pulses of light in real nonlinear media is in general accompanied by material losses that require corrections to the conservative Nonlinear Schrödinger equations describing their propagation. Here we examine loss mechanisms that exist even in lossless media and are caused by shedding of energy away from the self-trapping beam making it to relax to an exact solution of lower energy. Using the conservative NLS equations with absorbing boundary conditions we show that energy shedding not only occurs during the initial reshaping process but also during oscillatory propagation induced by saturation of the nonlinear effect. For pulsed input we also show that, depending on the sign and magnitude of dispersion, pulse splitting, energy shedding, collapse or stable self-focusing may result.

Dissipative solitons in the coupled dynamics of light and cold atoms.

We investigate the coupled dynamics of light and cold atoms in a unidirectional ring cavity, in the regime of low saturation and linear single-atom response. As the dispersive opto-mechanical coupling between light and the motional degrees of freedom of the atoms makes the dynamics nonlinear, we find that localized, nonlinearity-sustained and bistable structures can be encoded in the atomic density by means of appropriate control beams.

Adler synchronization of spatial laser solitons pinned by defects.

Defects due to growth fluctuations in broad-area semiconductor lasers induce pinning and frequency shifts of spatial laser solitons. The effects of defects on the interaction of two solitons are considered in lasers with frequency-selective feedback both theoretically and experimentally. We demonstrate frequency and phase synchronization of paired laser solitons as their detuning is varied. In both theory and experiment the locking behavior is well described by the Adler model for the synchronization of coupled oscillators.

A Kerr polarization controller.

Kerr-effect-induced changes of the polarization state of light are well known in pulsed laser systems. An example is nonlinear polarization rotation, which is critical to the operation of many types of mode-locked lasers. Here, we demonstrate that the Kerr effect in a high-finesse Fabry-Pérot resonator can be utilized to control the polarization of a continuous wave laser. It is shown that a linearly-polarized input field is converted into a left- or right-circularly-polarized field, controlled via the optical power. The observations are explained by Kerr-nonlinearity induced symmetry breaking, which splits the resonance frequencies of degenerate modes with opposite polarization handedness in an otherwise symmetric resonator. The all-optical polarization control is demonstrated at threshold powers down to 7 mW. The physical principle of such Kerr effect-based polarization controllers is generic to high-Q Kerr-nonlinear resonators and could also be implemented in photonic integrated circuits. Beyond polarization control, the spontaneous symmetry breaking of polarization states could be used for polarization filters or highly sensitive polarization sensors when operating close to the symmetry-breaking point.

Reversible soliton motion.

We show that spatial solitons on either phase- or amplitude-modulated backgrounds can change their direction of motion according to the modulation frequency. A soliton may, therefore, move up or down phase gradients or remain motionless regardless of where it is in relation to the background modulation. The general theory is in good agreement with numerical results in a variety of nonlinear systems.

Introduction.

Homogeneous, periodic, quasiperiodic, irregular and disordered spatial structures have been at the heart of scientific interest and research for a long time and in many disciplines as diverse as chemistry, physics, biology and morphology. Some recent, spectacular achievements in experimental quantum optics have, however, made it possible to study quantum effects in a variety of spatial structures. The main aim of this Focus Issue is to present some of the most recent theoretical, numerical and experimental progresses in the area of Quantum Structures, i.e. spatial structures that display quantum features. A short background overview of Quantum Structures is provided at the beginning of this Focus Issue, in which individual authors' works are specifically commented. In brief, we can say that the articles can be catalogued into two main areas of research interest in Quantum Structures: quantum effects in spatial structures in nonlinear optics, and in atomic physics.

Quantum structures in nonlinear optics and atomic physics: a background overview.

A brief overview of quantum effects in spatial structures such as nonlinear optical patterns, chains of trapped ions and atoms in optical lattices is presented. Some of the main results of the contributions to this Focus Issue are also briefly described.

Controlling pattern formation and spatio-temporal disorder in nonlinear optics.

We present a feedback control method for the stabiliza- tion of unstable patterns and for the control of spatio-temporal disor- der. The control takes the form of a spatial modulation to the input pump, which is obtained via filtering in Fourier space of the output electric field. The control is powerful, exible and non-invasive: the feedback vanishes once control is achieved. We demonstrate by means of computer simulation, the effect of the control in two different optical systems.

From quantum to classical images.

We display the results of the numerical simulations of a set of Langevin equations, which describe the dynamics of a degenerate optical parametric oscillator in the Wigner representation. The scan of the threshold region shows the gradual transformation of a quantum image into a classical roll pattern. An experiment on parametric down-conversion in lithium triborate shows strikingly similar results in both the near and the far field, displaying qualitatively the classical features of quantum images. c 1997 Optical Society of America.

Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators.

Mean field models of spatially extended degenerate optical parametric oscillators possess one-dimensional stable domain wall solutions in the presence of diffraction. We characterize these structures as spiral heteroclinic connections and study the spatial frequency of the local oscillations of the signal intensity which distinguish them from diffusion kinks. Close to threshold, at resonance or with positive detunings, the dynamics of two-dimensional diffractive domain walls is ruled by curvature effects with a t(1/2) growth law, and coalescence of domains is observed. In this regime, we show how to stabilize regular and irregular distributions of two-dimensional domain walls by injection of a helical wave at the pump frequency. Further above threshold the shrinking of domains of one phase embedded in the other is stopped by the interaction of the oscillatory tails of the domain walls, leading to cavity solitons surrounded by a characteristic dark ring. We investigate the nature and stability of these localized states, provide evidence of their solitonic character, show that they correspond to spiral homoclinic orbits and find that their threshold of appearance lowers with increasing pump cavity finesse.

Computationally determined existence and stability of transverse structures. I. Periodic optical patterns.

We present a Fourier-transform based, computer-assisted, technique to find the stationary solutions of a model describing a saturable absorber in a driven optical cavity. We illustrate the method by finding essentially exact hexagonal and roll solutions as a function of wave number and of the input pump. The method, which is widely applicable, also allows the determination of the domain of stability (Busse balloon) of the pattern, and sheds light on the mechanisms responsible for any instability. To show the usefulness of our numerical technique, we describe cracking and shrinking patches of patterns in a particular region of parameter space.

Computationally determined existence and stability of transverse structures. II. Multipeaked cavity solitons.

We apply quasi-exact numerical techniques to the calculation of stationary one- and two-dimensional, bound multipeaked cavity soliton solutions of a model describing a saturable absorber in a driven optical cavity. We calculate the existence and stability domains of a wide range of such states and determine the perturbative eigenmodes that cause loss of stability. We relate the existence of N-peaked states to the locking range between patterned and homogeneous solutions, as a function of two parameters.

Polarization patterns and vectorial defects in type-II optical parametric oscillators.

Previous studies of lasers and nonlinear resonators have revealed that the polarization degree of freedom allows for the formation of polarization patterns and novel localized structures, such as vectorial defects. Type- II optical parametric oscillators are characterized by the fact that the down-converted beams are emitted in orthogonal polarizations. In this paper we show the results of the study of pattern and defect formation and dynamics in a type-II degenerate optical parametric oscillator, for which the pump field is not resonated in the cavity. We find that traveling waves are the predominant solutions and that the defects are vectorial dislocations that appear at the boundaries of the regions where traveling waves of different phase or wave-vector orientation are formed. A dislocation is defined by two topological charges, one associated with the phase and another with the wave-vector orientation. We also show how to stabilize a single defect in a realistic experimental situation. The effects of phase mismatch of nonlinear interaction are finally considered.

Perturbation theory for domain walls in the parametric Ginzburg-Landau equation.

We demonstrate that in the parametrically driven Ginzburg-Landau equation arbitrarily small nongradient corrections lead to qualitative differences in the dynamical properties of domain walls in the vicinity of the transition from rest to motion. These differences originate from singular rotation of the eigenvector governing the transition. We present analytical results on the stability of Ising walls, deriving explicit expressions for the critical eigenvalue responsible for the transition from rest to motion. We then develop a weakly nonlinear theory to characterize the singular character of the transition and analyze the dynamical effects of spatial inhomogeneities.

Self-organization in cold atomic gases: a synchronization perspective.

We study non-equilibrium spatial self-organization in cold atomic gases, where long-range spatial order spontaneously emerges from fluctuations in the plane transverse to the propagation axis of a single optical beam. The self-organization process can be interpreted as a synchronization transition in a fully connected network of fictitious oscillators, and described in terms of the Kuramoto model.