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The concept of coherence is of fundamental importance for describing the physical characteristics of light and for evaluating the suitability for experimental application. In the case of pulsed laser sources, the pulse-to-pulse coherence is usually considered for a judgment of the compressibility of the pulse train. This type of coherence is often lost during propagation through a highly nonlinear medium, and pulses prove incompressible despite multioctave spectral coverage. Notwithstanding the apparent loss of interpulse coherence, however, supercontinua enable applications in precision frequency metrology that rely on coherence between different spectral components within a laser pulse. To judge the suitability of a light source for the latter application, we define an alternative criterion, which we term intrapulse coherence. This definition plays a limiting role in the carrier-envelope phase measurement and stabilization of ultrashort pulses. It is shown by numerical simulation and further corroborated by experimental data that filamentation-based supercontinuum generation may lead to a loss of intrapulse coherence despite near-perfect compressibility of the pulse train. This loss of coherence may severely limit active and passive carrier-envelope phase stabilization schemes and applications in optical high-field physics.

The electronic, rotational, and vibrational components of the ultrafast optical nonlinearity in ${\mathrm{H}}_{2}$ and ${\mathrm{D}}_{2}$ are measured directly and absolutely at intensities up to the ionization threshold of $\ensuremath{\sim}{10}^{14}\phantom{\rule{0.16em}{0ex}}{\mathrm{W}/\mathrm{cm}}^{2}$. As the most basic nonlinear interactions of the simplest molecules exposed to high fields, these results constitute a benchmark for high-field laser-matter theory and simulation.

In light filamentation induced by axicon-generated, powerful Bessel beams, the spatial propagation dynamics in the nonlinear medium determines the geometry of the filament channel and hence its potential applications. We show that the observed steady and unsteady Bessel beam propagation regimes can be understood in a unified way from the existence of an attractor and its stability properties. The attractor is identified as the nonlinear unbalanced Bessel beam (NL-UBB) whose inward H\"ankel beam amplitude equals the amplitude of the linear Bessel beam that the axicon would generate in linear propagation. A simple analytical formula that determines de NL-UBB attractor is given. Steady or unsteady propagation depends on whether the attracting NL-UBB has a small, exponentially growing, unstable mode. In case of unsteady propagation, periodic, quasi-periodic or chaotic dynamics after the axicon reproduces similar dynamics after the development of the small unstable mode into the large perturbation regime., Comment: 9 pages, 7 figures

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by purely nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are surveyed too, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation (BEC). The solitons are considered in one, two, and three dimensions (1D, 2D, and 3D). Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of 1D solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for the theoretical and experimental studies alike. In both the 1D and 2D cases, the mechanism which creates solitons in NLs is principally different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices., 169 pages, 35 figures, a comprehensive survey of results on solitons in purely nonlinear and mixed lattices, to appear in Reviews of Modern Physics

Filamentation and supercontinuum generation with 1.3-ps, 1055-nm laser pulses in YAG crystal is investigated numerically and experimentally. Numerical simulations based on solving the unidirectional nonparaxial propagation equation uncover that the self-focusing dynamics of a picosecond laser pulse markedly differs from that observed in a femtosecond filamentation regime. We show that spatiotemporal transformation of the picosecond pulse is governed by the free electron plasma, which defocuses and absorbs its rear part, resulting in the formation of several subpulses of femtosecond duration, which thereafter undergo peculiar spatiotemporal dynamics and have different contributions to spectral superbroadening. The numerical findings are confirmed experimentally by measuring the spatiotemporal intensity profiles of the wave packet at various stages of propagation where relevant events of the spectral broadening occur.

A regime of extreme self-compression of optical solitons to single-cycle duration with further shortening along with superbroad spectrum up-shifting is revealed when the Kerr nonlinearity and ionization process are independently controlled. This results in efficient optical-pulse compression as a whole towards extremely short single-cycle pulses at essentially shorter wavelengths, which may open a new way to generate optical pulses with durations of hundreds of attoseconds in the ultraviolet domain.

We analyze the propagation of self-trapped optical beams close to the Fr\'eedericksz threshold in nematic liquid crystals. Accounting for power-dependent changes in walk-off due to the all-optical response, we demonstrate that light beams can switch from positive to negative refraction according to the excitation.

We present results of an investigation of the spatial-temporal structure of the delayed Kerr nonlinearity excited during filamentation of a femtosecond laser pulse inside a crystalline target. It is found that the pulse splitting, which is an inherent property of the filamentation process, constrains the length of the spatial coherence of the Raman wake wave which follows the pump pulse. Each subpulse induces lattice vibrations, which interfere either constructively or destructively depending on the delay between the subpulses. This results in abrupt ``switching'' of the phase of the wake wave along and across the filament volume. For the same reason, the maximal amplitude of the Raman response is observed not at the point of the maximal laser intensity and electron concentration\char22{}it is shifted forward to the end of the filament. This effect should be observed if the period of Raman oscillations is close to the duration of the pump pulse.

Conditions for stable propagation of one-dimensional bright spatial solitons in media exhibiting optical nonlinearities up to the seventh-order are investigated. The results show well-defined stability regions even when all the nonlinear terms are focusing. Conditions for onset of the supercritical collapse of the optical beam are identified too. A variational approximation is used to predict dependence of the soliton propagation constant on the norm, and respective stability regions are identified using the Vakhitov-Kolokolov criterion. Analytical results obtained by means of the variational approximation are corroborated by numerical simulations of the cubic-quintic-septimal nonlinear Schroedinger equation., Physical Review A, in press

We develop a simple analytical model for calculating the vibrational contribution to the nonlinear refractive index ${n}_{2}$ (Kerr coefficient) of a crystal in terms of known crystalline parameters such as the linear refractive index, the coefficient of thermal expansion, the atomic density, and the reduced mass and the natural oscillation frequency of the vibrational modes of the crystal lattice. We show that the value of this contribution in the terahertz spectral region can exceed the value of the nonlinear refractive index ${n}_{2}$ in the visible and near-IR spectral ranges (which is largely electronic in origin) by several orders of magnitude. For example, for crystal quartz the value of the Kerr coefficient in the low-frequency limit is ${n}_{2}=2.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$ esu or, equivalently, $4.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}\phantom{\rule{0.28em}{0ex}}{\mathrm{m}}^{2}$/W, which is very much larger than its value of $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}\phantom{\rule{0.28em}{0ex}}{\mathrm{m}}^{2}$/W in the visible range. Furthermore, we present an analysis of the dispersion of ${n}_{2}$ in the terahertz spectral range and show that even larger values of ${n}_{2}$ occur at frequencies close to the vibrational resonances.

Generally, optical diffraction is only weakly dependent on the refractive index of a medium in which light propagates. In this paper, diffraction in a nearly isotropic Kerr photonic crystal (PhC) made of silicon pillars embedded in nonlinear carbon disulfide ambient was reversed and enhanced by its linear refractive index, which is negative and much less than unity. The effective nonlinear refractive index coefficient ${n}_{2}$ of the PhC was found by fitting spectral broadening induced by self-phase modulation. The enhanced inverse diffraction, attributed to positive ${n}_{2}$, allows self-defocusing in one single PhC. More interestingly, the same PhC can selectively exhibit dual functionalities, i.e., self-defocusing and self-focusing, based on the wavefront property of a given input beam. Our results may pave the way for protecting nanostructured photonic devices from laser damage and provide a method for controlling wavefronts.

We have theoretically and experimentally investigated surface dispersive shock waves (SDSWs) at the interface between a self-defocusing medium and a linear medium. We demonstrate that SDSWs can form when the linear refractive index of the self-defocusing medium is much greater than that of the linear medium, and the initial nonlinearity far outweighs diffraction. SDSWs have been observed at the interface between air and a weakly absorbing liquid when the power of the input beam far exceeds that needed to trap a surface dark soliton. We also observed the formation of SDSWs when an input beam was projected away from the interface, and observed these patterns at the curved surface.

We report generation of a broadband supercontinuum (SC) in a birefringent crystal upon irradiation by intense, near-infrared, femtosecond laser pulses and investigate its dependence on incident laser power and the initial laser polarization. We choose as our sample $\ensuremath{\alpha}\ensuremath{-}{\mathrm{BaB}}_{2}{\mathrm{O}}_{4}$ ($\ensuremath{\alpha}$-BBO), a material in which the third order is the lowest order of nonvanishing nonlinear optical susceptibility. Three different SC generation mechanisms are identified under our experimental conditions; these are found to depend upon the pump power ${P}_{pu}$. Close to the critical power for self-focusing (${P}_{pu}\ensuremath{\sim}5$ MW), the spectrum is dominated by Kerr nonlinearity or self-phase modulation, whereas at intermediate pumping ${P}_{pu}\ensuremath{\sim}125$ MW, Raman amplification on the Stokes side is a dominating mechanism. At higher pump power $\ensuremath{\sim}300$ MW, significant asymmetric broadening extending up to $\ensuremath{\sim}570$ nm is observed due to the coherent anti-Stokes Raman scattering. Our study provides an intuitive explanation for the observed power- and polarization-dependent SC generation in $\ensuremath{\alpha}$-BBO crystals.

We address the existence and stability properties of families of gap solitons in a lower refractive-index core, sandwiched between two optical lattices with a parity-time ($\mathcal{PT}$) symmetry imprinted in a defocusing medium. The scale of flat-topped complex solitons can be controlled arbitrarily by varying the embedded index core. Multipeaked solitons are found to exhibit equal peak values in the region of the index core, and they have no analog in other lattice-modulated or bulk media. We demonstrate that, in sharp contrast to solitons in regular $\mathcal{PT}$ lattices, flat-topped and multipeaked solitons are either stable or suffer a negligibly weak instability, even when the gain-loss coefficient approaches the $\mathcal{PT}$-symmetry-breaking point. Our results, thus, build a bridge between the $\mathcal{PT}$ defect solitons in a narrow index core and the $\mathcal{PT}$ kink pairs in a broad index core. We also suggest an effective way for the observation of ``surface solitons'' in $\mathcal{PT}$-symmetric lattices.

We numerically investigate harmonic generation in Ar gas under high ionization conditions and demonstrate that a lighthouse effect is present. We examine the structure of the driving field during propagation in temporal, spectral, and spatial domains, and conclude that the complete depletion of neutral Ar on axis gives rise to additional wavelets at off-axis regions. We show that these wavelets propagate with increasing divergence as the radial distances from the axis increase, generating the rotation of the wave front, thus fulfilling a necessary condition for the lighthouse effect. We obtain attosecond bursts of light emitted with different divergences in successive optical half-cycles so that in the far field these bursts arrive at different distances from the beam axis.

We theoretically study the central wavelength tuning of a laser pulse propagating in a hollow-core fiber filled with prealigned nitrogen gas. The simulation results show that the double pump pulses, suitable pump pulse energy, gas pressure, and temperature can extend the wavelength shift range, and the spectral blueshift or redshift can be controlled by tuning the time delay between the pump and probe pulses. It is also shown that several hundred nanometers wavelength tuning is obtained by the double pump pulses with appropriate parameters.

Extending the longitudinal range of plasma channels produced by femtosecond annular (ring) Gaussian beams in atmosphere is numerically investigated. We find that the length of a stable plasma channel induced by the annular Gaussian beam, comparing with a Gaussian beam under the same initial condition, has a significant improvement, which can be well understood from two aspects. The dynamic complementation caused by the optical system composing of an axicon and a diverging lens can extend the plasma channel greatly, and the special initial transverse distribution of ring-Gaussian pulse leads to pulse splitting and redistribution of the pulse energy before the formation of optical filament. The ultrashort ring-Gaussian beam perhaps offers a new and an efficient route towards the generation of extended optical filament.

We report an observation of spatial solitons in a medium managed to present fifth-seventh (focusing-defocusing) refractive nonlinearities with suppressed third-order nonlinearity. Propagation of two-dimensional bright spatial solitons for \ensuremath{\sim}10 Rayleigh lengths was observed and characterized in a suspension of silver nanoparticles in acetone using the scattered light imaging method. Numerical calculations based on a nonlinear Schr\"odinger-type equation, including contributions up to the seventh-order susceptibility, were performed showing good agreement with the experimental results.

The impact of nonadiabatic laser-induced molecular alignment on filamentation is numerically studied. Weak and strong field model of impulsive molecular alignment are compared in the context of nonlinear pulse propagation. It is shown that the widely used weak field model describing the refractive index modification induced by impulsive molecular alignment accurately reproduces the propagation dynamics providing that only a single pulse is involved during the experiment. On the contrary, it fails at reproducing the nonlinear propagation experienced by an intense laser pulse traveling in the wake of a second strong laser pulse. The discrepancy depends on the relative delay between the two pulses and is maximal for delays corresponding to half the rotational period of the molecule.

We report that nonlocal nonlinear media allow a controlled interaction among coherent solitons with a relative tilt previously imposed, resulting in the generation of rotary self-trapped beams. We demonstrate that two initially separated fundamental solitons can interact, generating stable rotating dipoles for a continuum interval of relative tilt values. Surprisingly, we find that for a higher number of initial solitons launched and after some emission of radiation waves, the initial self-trapped structures can decay into rotating dipole solitons. The normalized orbital angular momentum of these rotating dipoles can be controlled by adjusting the initial tilt.

A detailed theoretical investigation is performed for the self-compression of relativistically strong laser pulses under plasma wake wave excitation conditions. An analytical expression is obtained for the characteristic length of laser pulse self-compression in the one-dimensional case. A good agreement between the theoretical and numerical calculation of the dependence of the wave-packet compression length on both amplitude and plasma concentration is demonstrated. The influence of the wave beam self-focusing radiation process on the rate of shortening of the laser pulse duration is considered.

The dynamics of truncated Airy beams in optical media with Kerr nonlinearity is studied in the framework of the nonlinear Schr\"odinger (NLS) model. It is demonstrated that an intense Airy beam with zero total momentum can generate static solitons, as well as moving solitons. The parameters of these solitons are calculated using the Zakharov-Shabat scattering problem associated with the NLS equation. It is found that solitons take the main part of the initial power, while only a small fraction of the power is transformed into a self-accelerating linear packet. The threshold parameters of the Airy beam for the soliton formation are obtained. It is shown that the threshold for the formation of the first static soliton is also a threshold of the solitonless regime.

Traveling unidirectional localized edge states in optical honeycomb lattices are analytically constructed. They are found in honeycomb arrays of helical waveguides designed to induce a periodic pseudomagnetic field varying in the direction of propagation. Conditions on whether a given pseudofield supports a traveling edge mode are discussed; a special case of the pseudofields studied agrees with recent experiments. Interesting classes of dispersion relations are obtained. Envelopes of nonlinear edge modes are described by the classical one-dimensional nonlinear Schr\"odinger equation along the edge. Nonlinear states termed edge solitons are predicted analytically and are found numerically.

We present a procedure for nonlinearity management of metal-dielectric composites. Varying the volume fraction occupied by silver nanoparticles suspended in acetone we could cancel the refractive index related to the third-order susceptibility, $\chi_{eff}^{(3)}$, and the nonlinear refraction behavior was due to the fifth-order susceptibility, $\chi_{eff}^{(5)}$. Hence, in a cross-phase modulation experiment, we demonstrated for the first time the effect of spatial-modulation- instability due to $\chi_{eff}^{(5)}$. The results are corroborated with numerical calculations based on a generalized Maxwell-Garnet model., Comment: 20 pages, 5 figures

We show that when a single vortex convected by a flowing background collides with an oblique dark soliton in defocusing Kerr-type medium, it passes through the oblique soliton, but this process is accompanied by a considerable vertical shift of the vortex that grows with decrease of the inclination angle of the dark soliton with respect to the direction of the flow and decreases with increase of the flow (or collision) velocity. In the parameter range where the interaction with a vortex does not lead to drastic deformations of the shape of the oblique dark soliton, this shift can be estimated analytically under the assumption that vortex soliton moves along the "stream lines" that exhibit vertical shift in the vicinity of oblique dark soliton. Similar effect was observed upon collision of dark solitons with moving vortex dipoles or rotating pairs of vortices with equal topological charges., Comment: 5 pages, 3 figures, to appear in Physical Review A

The filamentation of ultraviolet and infrared nanosecond light pulses in fused silica is investigated theoretically and numerically. Emphasis is put on the action of a dynamical plasma response on two counterpropagating waves, amplified by Kerr self-focusing and stimulated Brillouin scattering (SBS). For a single unperturbed wave, laser filamentation takes place through a quasistationary balance between Kerr self-focusing and plasma defocusing, for which a variational approach reproduces global propagation features. However, such a quasistationary balance cannot hold as temporal modulational instability breaks up the pulse over electron recombination times. For two counterpropagating waves, we report similar instabilities which are justified through a plane-wave stability analysis. These instabilities originate from an intense backscattered wave that induces strong plasma contribution and thus destabilizes the pump near the entrance face of the material. Rapid phase modulations are then applied to suppress backscattering. We show that pump waves with broad enough bandwidths can inhibit SBS and thus prevent instability. The robustness of phase modulations against random fluctuations in the input pump pulse is finally addressed.

We theoretically investigate the free-carrier induced spatio-temporal dynamics of continuous waves in silicon waveguides embedded in an amplifying medium. Optical propagation is governed by a cubic Ginzburg-Landau equation coupled with an ordinary differential equation accounting for the free-carrier dynamics. We find that, owing to free-carrier dispersion, countinuous waves are modulationally unstable in both anomalous and normal dispersion regimes and chaotically generate unstable accelerating pulses., Comment: 5 pages, 4 figures

In this Letter, it is experimentally and theoretically shown that weak odd harmonics generated during the propagation of an infrared ultrashort ultraintense pulse unexpectedly modify the nonlinear properties of the medium and lead to a strong modification of the propagation dynamics. This result is in contrast with all current state-of-the-art propagation model predictions, in which secondary radiations, such as third harmonic, are expected to have a negligible action upon the fundamental pulse propagation. By analyzing full three-dimensional ab initio quantum calculations describing the microscopic atomic optical response, we have identified a fundamental mechanism resulting from interferences between a direct ionization channel and a channel involving one single ultraviolet photon. This mechanism is responsible for wide refractive index modifications in relation with significant variation of the ionization rate. This Letter paves the way to the full physical understanding of the filamentation mechanism and could lead to unexplored phenomena, such as coherent control of the filamentation by harmonic seeding.

Akhmediev and Kuznetsov-Ma breathers are rogue wave solutions of the nonlinear Schr\"odinger equation (NLSE). Talbot effect (TE) is an image recurrence phenomenon in the diffraction of light waves. We report the nonlinear TE of rogue waves in a cubic medium. It is different from the linear TE, in that the wave propagates in a NL medium and is an eigenmode of NLSE. Periodic rogue waves impinging on a NL medium exhibit recurrent behavior, but only at the TE length and at the half-TE length with a \pi-phase shift; the fractional TE is absent. The NL TE is the result of the NL interference of the lobes of rogue wave breathers. This interaction is related to the transverse period and intensity of breathers, in that the bigger the period and the higher the intensity, the shorter the TE length., Comment: 4 pages, 4 figures

We demonstrate that femtosecond filaments can set up an extended and robust thermal waveguide structure in air with a lifetime of several milliseconds, making possible the very-long-range guiding and distant projection of high-energy laser pulses and high-average power beams. As a proof of principle, we demonstrate guiding of 110-mJ, 7-ns, 532-nm pulses with 90% throughput over ∼15 Rayleigh lengths in a 70-cm-long air waveguide generated by the long time-scale thermal relaxation of an array of femtosecond filaments. The guided pulse was limited only by our available laser energy. In general, these waveguides should be robust against the effects of thermal blooming of extremely high-average-power laser beams.

International audience; We demonstrate spatiotemporal cleaning of a femtosecond laser pulse impinging on two counterpropagating filaments in air. The retroreflected signal essentially has a perfect beam profile. Prepulses present in the incident pulse are also efficiently removed. The performance of the “filament mirror” is explained by a plasma-mediated wave mixing process that can be viewed as a transient three-dimensional hologram.

In contrast to an infinite degree of nonlocality, we demonstrate that vortex solitons in nonlinear media under competing self-focusing cubic and self-defocusing quintic nonlocal nonlinearities can be stabilized with a finite degree of nonlocality. Stable vortex solitons in the upper branch, bifurcated from the competing cubic-quintic nonlinearities, are found to be supported when the original double-ring refractive index change is transferred into a single-ring configuration due to the balance between diffusive nonlocality and defocusing quintic nonlinearity. The dynamics and stabilities of the vortex solitons are studied analytically and numerically.

We study the wave breaking mechanism for the weakly dispersive defocusing nonlinear Schrödinger equation with a constant phase dark initial datum that contains a vacuum point at the origin. We prove by means of the exact solution to the initial value problem that, in the dispersionless limit, the vacuum point is preserved by the dynamics until breaking occurs at a finite critical time. In particular, both Riemann invariants experience a simultaneous breaking at the origin. Although the initial vacuum point is no longer preserved in the presence of a finite dispersion, the critical behavior manifests itself through an abrupt transition occurring around the breaking time.

The influence of group-velocity-dispersion (GVD) effects on the propagation of femtosecond laser pulses in air at different pressures is investigated by numerically solving the extended nonlinear Schr\"odinger equation. It is observed that GVD has a great influence on the collapse distance ${L}_{\mathrm{c}}$ of the self-focusing short laser pulses when the pressure is relatively high (e.g., 10 atm), in which case the semiempirical formula describing the collapse distance of self-focusing laser pulse [Dawes and Marburger, Phys. Rev. 179, 862 (1969)] is no longer applicable, while GVD has little influence on the self-focusing process of longer laser pulses. Through the results of numerical simulations as well as analytical analysis, we find that the initial duration of the laser pulse is the main factor that determines the GVD effect on the propagation and the increase of the pressure and pulse compression during the propagation enhance this effect. The spectral data show that GVD also influences the spectrum broadening in high-pressure cases.

We investigate beam diffraction and spatial modulation instability of coherent light beams propagating in the non-paraxial regime in a nonlinear Kerr medium. We study the instability of plane wave solutions in terms of the degree of non-paraxiality, beyond the regime of validity of the nonlinear Schroedinger equation. We also numerically analyze the way non-paraxial terms breaks the integrability and affect the periodical evolution of higher order soliton solutions. We discuss non-locality and effective dissipation introduced by the non-linear coupling with evanescent waves., 8 pages, 6 figures

In this paper, we address the propagation of a near resonant laser inside an atomic vapor in the case when, due to the Kerr effect, the laser beam would either self-focus or self-defocus. We show, both theoretically and experimentally, how to get rid of such an alteration in the transverse beam profile without changing any of the characteristics of the laser light under consideration (wavelength, intensity, etc.), nor of the atomic vapor. Moreover, our proposed method offers a lot of control on the beam profile, whose transverse size after propagation may be chosen at will by making use of a second, copropagating laser, whose required wavelength and intensity may be derived analytically.

A unified criterion of self-focusing, applicable to both freely propagating beams and waveguide modes, is proposed, with the self-focusing threshold defined as the critical peak power needed to induce a refractive-index step capable of trapping one more spatial mode. For freely propagating beams and high-order waveguide modes, this definition is shown to recover the canonical expression for the critical power of self-focusing routinely used in nonlinear optics. In the limiting case of low-order guided modes, the proposed definition is found to be fully consistent with the earlier criteria of self-focusing in a waveguide.

We experimentally investigate the two-dimensional condensate (optical dropletlike) soliton formation and dynamics of the generated signal and probe beams in four-wave mixing (FWM) process with atomic coherence, under competition between the third- and fifth-order nonlinear susceptibilities. With such competing nonlinearities, mutual transformations among dropletlike fundamental, dipole, and azimuthally modulated vortex FWM solitons are observed. The influence of nonlinear competition on the photonic band gap is also investigated. All the results are obtained under low powers.

We demonstrate both theoretically and experimentally that a finite Airy beam changes its trajectory and shape in optically induced waveguide arrays consisting of different kinds of defects. The propagation dynamics and beam acceleration are controlled with positive and negative defects, and appropriate refractive index change. An additional class of discrete beams and Airy defect modes are demonstrated.

We investigate the propagation of a partially coherent beam in a nonlinear medium with logarithmic nonlinearity. We show that all information about the properties of the beam, as well as the condition for formation of incoherent solitons, can be obtained from the evolution equation for the mutual coherence function. The key parameter is the detuning $\ensuremath{\Delta}$ between the effective diffraction radius and the strength of the nonlinearity. Stationary partially coherent solitons exist when $\ensuremath{\Delta}=0$ and the nonlinearity exactly compensates for the spreading due to both diffraction and incoherence. For nonzero detunings the solitons are oscillating in nature, and we find approximate solutions in terms of elliptic functions. Our results establish an elegant equivalence among several different approaches to partially coherent beams in nonlinear media.

The nonlinear propagation of ultrashort laser pulses launched into the air is investigated. The formation of optical light "bullets," or spatially localized structures, has been experimentally observed recently. Their stability is shown as due to the occurrence of a dynamical balance between two opposite nonlinear effects: an optical focusing Kerr effect balanced by a defocusing self-induced multiphoton partial ionization of the neutral gas. Characteristics of the "bullets" are predicted analytically and confirmed numerically. They are found to be in agreement with observations.

We report the observation of optical Bloch oscillations in waveguide arrays. The required linear variation of the propagation constant across the array was obtained by using the thermo-optic effect and can be controlled by temperature. Bloch oscillations manifesting themselves m transverse oscillations of the light beam upon propagation can be attributed to the existence of localized states (Wannier-Stark states) with equidistant eigenvalue spacing (Wannier-Stark ladder). Selfphase modulation by a Kerr nonlinearity will disintegrate the Wannier-Stark ladder as well as the Bloch oscillations.

Collapse of (1 + 1)-dimensional beams in the inhomogeneous one-dimensional quintic nonlinear Schrodinger equation is analyzed both numerically and analytically. It is shown that in the vicinity of a narrow attractive inhomogeneity, the collapse of beams in which the homogeneous medium would blow up may be delayed and even arrested. [S1063-651X(99)03610-7].

We study the interaction between two localized waves that propagate in a bulk ~two transverse dimensions! Kerr medium, while being incoherently coupled through cross-phase modulation. The different types of stationary solitary wave solutions are found and their stability is discussed. The results of numerical simulations suggest that the solitary waves are unstable. We derive sufficient conditions for when the wave function is bound to collapse or spread out, and we develop a theory to describe the regions of different dynamical behavior. For localized waves with the same center we confirm these sufficient conditions numerically and show that only when the equations and the initial conditions are symmetric are they also close to being necessary conditions. Using Gaussian initial conditions we predict and confirm numerically the powerdependent characteristic initial separations that divide the phase space into collapsing and diffracting solutions, and further divide each of these regions into subregions of coupled ~fusion! and uncoupled dynamics. Finally we illustrate how, close to the threshold of collapse, the waves can cross several times before eventually collapsing or diffracting. @S1063-651X~99!00604-2#

We present experimental results related to collisional properties of partially coherent solitons (PCSs). In experiments with a photorefractive crystal we show that collisions in nonlinear media cause the PCS comprised of few mutually incoherent soliton components to change its shape.

The self-focusing of femtosecond optical pulses in dielectric solids is studied under the influence of the combined effects of diffraction, Kerr nonlinearity, normal group velocity dispersion, and self-induced multiphoton ionization. The numerical simulations show a significant pulse shortening with a compression factor of 5.5 and self-channeling of the beam when the critical power is exceeded.