Arresting soliton collapse in two-dimensional nonlinear Schrödinger systems via spatiotemporal modulation of the external potential

We predict stable, collapse-free solitonslike structures in two-dimensional nonlinear Schr\"odinger systems in subdiffractive regimes, accomplished by a spatiotemporal modulation of the external potential. We investigate the scaling laws, the stability, and the dynamical properties of these subdiffractive solitons.

Generalized complex Swift-Hohenberg equation for optical parametric oscillators

A generalized complex Swift-Hohenberg equation including diffraction and nonlinear resonance terms is derived for spatially extended nondegenerate optical parametric oscillators (OPOs) with flat end mirrors. For vanishing pump detuning this equation becomes the complex Swift-Hohenberg (SH) equation valid also for lasers. Nevertheless the similarities between OPOs and lasers are limited, since the diffractive character of OPOs is lost when the diffraction coefficients of signal and idler fields are equal. This manifests, e.g., in the absence of advection by traveling waves (TWs), a clear difference with lasers. When pump detuning is nonzero a nonlinear resonance develops, as it occurs in deg…

Transverse effects in a thin slab of material with local-field induced intrinsic optical bistability

We consider a thin slab of dense material exhibiting local-field induced intrinsic optical bistability irradiated by a transversely uniform optical field (holding beam). We study the transverse effects that can arise when local excitations are created by means of a narrow optical beam (writing beam). We show that whereas diffraction effects are negligible, diffusion effects make the excitation-domain walls to move inward or outward in the transverse direction, with a speed that depends on the holding-beam intensity and the diffusion coefficient. Conditions can be found, however, for which the wall movement is counterbalanced by the field transverse gradient so that stable narrow excitation …

Subdiffractive solitons in bose-einstein condensates

We predict the disappearance of diffraction (the increase of the mass) of Bose-Einstein condensates in counter-moving periodic potentials. We demonstrate subdiffractive solitons (stable droplets of the condensate) in the vicinity of this zero diffraction point.

Role of pump diffraction on the stability of localized structures in degenerate optical parametric oscillators.

We show that the stability range of localized structures (LS's) in the form of minimum size phase domains in degenerate optical parametric oscillators is enhanced by increasing the diffraction of the pump wave. Pump diffraction enhances spatial oscillations of decaying tails of domain boundaries, whereas spatially oscillating (weakly decaying) tails prevent the collapse of LS's, enhance their stability range, and allow the existence of more complex LS's in the form of molecules.

Dynamics of phase domains in the Swift-Hohenberg equation

Abstract We analyze analytically and numerically the dynamics of phase domains in the Swift-Hohenberg equation. For negative or small positive detuning domains contract and disappear. A large positive detuning leads to dendritic growth of the domains, and the formation of labyrinth structures. Intermediate detuning results in stable circular domains - the localized structures of the Swift-Hohenberg equation. The predicted phenomena should occur in parametrically driven chemical, hydrodynamical, and nonlinear optical systems.

Experimental demonstration of hyperbolic patterns.

We give experimental evidence of hyperbolic patterns in a nonlinear optical resonator. Such transverse patterns are a new kind of 2D dissipative structures, characterized by a distribution of the active modes along hyperbolas in the transverse wave-vector domain, in contrast with the usual (elliptic) patterns where the active modes distribute along rings. The hyperbolic character is realized by manipulating diffraction inside the optical resonator with cylindrical lenses. We also investigate theoretically hyperbolic patterns in corresponding Swift-Hohenberg models.

Cavity solitons in nondegenerate optical parametric oscillation

Abstract We find analytically cavity solitons in nondegenerate optical parametric oscillators. These solitons are exact localised solutions of a pair of coupled parametrically driven Ginzburg–Landau equations describing the system for large pump detuning. We predict the existence of a Hopf bifurcation of the soliton resulting in a periodically pulsing localised structure. We give numerical evidence of the analytical results and address the problem of cavity soliton interaction.

Faraday patterns in bose-Einstein condensates.

Temporal periodic modulation of the interatomic s-wave scattering length in Bose-Einstein condensates is shown to excite subharmonic patterns of atom density through a parametric resonance. The dominating wavelength of the spatial structures is shown to be primarily selected by the excitation frequency but also affected by the depth of the spatial modulation via a nonlinear resonance. These phenomena represent macroscopic quantum analogues of the Faraday waves excited in vertically shaken liquids.

Four-phase patterns in a forced nonlinear optical oscillator

We present preliminary theoretical and experimental results indicating that a high Fresnel number nonlinear optical oscillator with planar mirrors can display four-phase multistability, eventually leading to four-phase patterns. Such situation is similar to that emerging in extended oscillatory systems forced within a 4:1 resonance and, to the best of our knowledge, has not been predicted nor observed previously in an optical system.

Bistable phase locking in a low fresnel number nondegenerate optical oscillator with injected signal

Degenerate four-wave mixing oscillators are phase-bistable cavities. In such systems, above the oscillation threshold, two equivalent states, of equal intensities but opposite phases are generated. This phase bistability extends over the whole range of stable emission, unlike the intensity bistability (in, e.g. a saturable absorber cavity) that exits in a limited range of injection. When the cavity Fresnel number is large different patches of the beam transverse section can have different phases and a pattern forms. Basic patterns here are phase fronts (or domain walls), which are 1D structures separating regions with opposite phase that manifest as dark lines (as the phase jumps by p acros…

Diffusion stabilizes cavity solitons in bidirectional lasers

We study the influence of field diffusion on the spatial localized structures (cavity solitons) recently predicted in bidirectional lasers. We find twofold positive role of the diffusion: 1) it increases the stability range of the individual (isolated) solitons; 2) it reduces the long-range interaction between the cavity solitons. Latter allows the independent manipulation (writing and erasing) of individual cavity solitons.

Excitation of phase patterns and spatial solitons via two-frequency forcing of a 1:1 resonance.

We show that a self-oscillatory system, driven at two frequencies close to that of the unforced system (resonance 1:1), becomes phase locked and exhibits two equivalent stable states of opposite phases. For spatially extended systems this phase bistability results in patterns characteristic for real order parameter systems, such as phase domains, labyrinths, and phase spatial solitons. In variational cases, the phase-locking mechanism is interpreted as a result of the periodic "rocking" of the system potential. Rocking could be tested experimentally in lasers and in oscillatory chemical reactions.

Faraday patterns in low-dimensional Bose-Einstein condensates

We show that Faraday patterns can be excited in the weak confinement space of low-dimensional Bose-Einstein condensates by temporal modulation of the trap width, or equivalently of the trap frequency Omega_tight, in the tight confinement space. For slow modulation, as compared with Omega_tight, the low-dimensional dynamics of the condensate in the weak confinement space is described by a Gross-Pitaevskii equation with time modulated nonlinearity coefficient. For increasing modulation frequencies a noticeable reduction of the pattern formation threshold is observed close to 2*Omega_tight, which is related to the parametric excitation of the internal breathing mode in the tight confinement sp…

Phase Domains and Spatial Solitons in Degenerate Optical Parametric Oscillators with Injection

The stability of phase domains and spatial solitons in DOPO under the presence of an injected signal is investigated. The injected signal prevents the nondegenerate regime and, for a particular value of the phase, preserves the equivalence between the two homogeneous states, allowing the domain formation and, in particular, the stability of solitons. The main conclusion is that injection facilitates the experimental observation of solitons in degenerate OPOs.

Bistable phase locking in a laser with injected signal

Turing Patterns in Nonlinear Optics

The phenomenon of pattern formation in nonlinear optical resonators is commonly related to an off-resonance excitation mechanism, where patterns occur due to mismatch between the excitation and resonance frequency. In this paper we show that the patterns in nonlinear optics can also occur due to the interplay between diffractions of coupled field components. The reported mechanism is analogous to that of local activation and lateral inhibition found in reaction-diffusion systems by Turing. We study concretely the degenerate optical parametric oscillators. A local activator-lateral inhibitor mechanism is responsible for generation of Turing patterns in form of hexagons.

Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing.

Transverse pattern formation in both degenerate optical parametric oscillation and degenerate four-wave mixing is considered both theoretically and numerically. In the limit of small signal detuning both systems are shown to be described by the real Swift-Hohenberg equation. Contrarily, for small signal and large pump detunings the Swift-Hohenberg equation is modified differently in both systems, by the appearance of additional nonlinear terms, which signal the existence of nonlinear resonances that are theoretically studied through the derivation of the amplitude equation for the roll pattern in both systems. Numerical analysis supports the theoretical predictions. \textcopyright{} 1996 Th…

Stability of localized structures in the Swift-Hohenberg equation.

We show that nonmonotonic (oscillatory) decay of the boundaries of phase domains is crucial for the stability of localized structures in systems described by Swift-Hohenberg equation. The less damped (more oscillatory) are the boundaries, the larger are the existence ranges of the localized structures. For very weakly damped spatial oscillations, higher-order localized structures are possible.

Experimental demonstration of bistable phase locking in a photorefractive oscillator

We report experimental evidence of bistable phase locking in nonlinear optics, in particular, in a photorefractive oscillator emitting in few transverse modes. Bistable phase locking is a recently proposed method for converting a laserlike system, which is phase invariant, into a phase-bistable one by injecting a suitable spatially modulated monochromatic beam, resonant with the laser emission, into the optical cavity. We experimentally demonstrate that the emission on the fundamental TEM00 mode becomes phase bistable by injection of a beam with the shape of the TEM10 mode with appropriate frequency, in accordance with recent theoretical predictions [K. Staliunas et al., Phys. Rev. A 80, 02…

Diffraction management and sub-diffractive solitons in periodically driven Bose–Einstein condensates

Abstract We theoretically investigate the diffraction management in Bose–Einstein condensates (BECs) in one- (1D), two- (2D) and three-dimensional (3D) geometries. The management technique is based on the superposition of harmonic lattices’ potentials moving at a common speed but in different directions, leading to a harmonic spatio-temporal modulation of the potential. In this way a reduction in, and eventually the disappearance of usual diffraction and emergence of fourth-order diffraction are achieved. We show sub-diffractive solitons in such a diffraction managed system and demonstrate their stability in 1D, 2D and 3D. In 2D and 3D cases we investigate diffraction management by lattices…

Sub-Diffractive Band-Edge Solitons in Bose-Einstein Condensates in Periodic Potentials

A new type of matter wave diffraction management is presented that leads to sub-diffractive soliton-like structures. The proposed management technique uses two counter-moving, identical periodic potentials (e.g. optical lattices). For suitable lattice parameters a novel type of atomic band-gap structure appears in which the effective atomic mass becomes infinite at the lowest edge of an energy band. This way normal matter-wave diffraction (proportional to the square of the atomic momentum) is replaced by fourth-order diffraction, and hence the evolution of the system becomes sub-diffractive.

Phase-bistable Kerr cavity solitons and patterns

We study pattern formation in a passive nonlinear optical cavity on the basis of the classic Lugiato-Lefever model with a periodically modulated injection. When the injection amplitude sign alternates, e.g., following a sinusoidal modulation in time or in space, a phase-bistable response emerges, which is at the root of the spatial pattern formation in the system. An asymptotic description is given in terms of a damped nonlinear Schr\"odinger equation with parametric amplification, which allows gaining insight into the basic spatiotemporal dynamics of the system. One- and two-dimensional phase-bistable spatial patterns, such as bright and dark-ring cavity solitons and labyrinths, are demons…

Bistable phase locking in rocked lasers

Abstract We investigate analytically and numerically the dynamics of single mode lasers with periodic ac injection (rocked lasers). Such lasers show phase bistability as the phase of the light emitted by such lasers can lock to either of two values shifted by π. Locking regimes for different lasers are studied showing that the system response is strongly modified in class B lasers due to the influence of relaxation oscillations.

Cavity solitons in lasers with spatially modulated injected signal

The injection of a monochromatic signal into a laser is a well-known technique for locking the laser phase to that of the injection. Some years ago another type of injection, called rocking [1,2], was introduced to render the laser phase-locking bistable. Rocking consists of the modulation of the amplitude injection so that its sign changes periodically, or even randomly [3], in time. Here we present an alternative to rocking that leads to the same type of behaviour, namely the appearance of bistable phase locking and, in the case of large Fresnel number lasers, to stable (phase bistable) cavity solitons and extended patterns. The new type of injection we present here is monochromatic, unli…

Bistable phase locking of a laser via monochromatic signal injection

In free running lasers the field phase is not fixed and any value possible value is equally likely (invariant), but can be locked to an external reference by injecting a monochromatic signal field into the cavity. In this way the phase of the slave laser locks to a single value resulting in a monostable phase locking. It could be however of practical interest that the laser field be locked not to a single value but to two different possible values, hence the name bistable phase locking.

Pattern formation through phase bistability in oscillatory systems with space-modulated forcing.

We propose a novel forcing technique of spatially extended self-oscillatory systems able to excite phase bistability and the dissipative structures associated with it. The forcing is time periodic at a frequency close to the oscillators' frequency and is spatially modulated. The effects of this type of forcing are demonstrated analytically and numerically in a directly driven complex Ginzburg-Landau equation. Both spatially periodic and spatially random drives prove to be effective.

Phase-bistable pattern formation in oscillatory systems via rocking: application to nonlinear optical systems

We present a review, together with new results, of a universal forcing of oscillatory systems, termed ‘rocking’, which leads to the emergence of a phase bistability and to the kind of pattern formation associated with it, characterized by the presence of phase domains, phase spatial solitons and phase-bistable extended patterns. The effects of rocking are thus similar to those observed in the classic 2 : 1 resonance (the parametric resonance) of spatially extended systems of oscillators, which occurs under a spatially uniform, time-periodic forcing at twice the oscillations' frequency. The rocking, however, has a frequency close to that of the oscillations (it is a 1 : 1 resonant forcing) …