Soliton-driven Photonics

The word gyrotropy turns up quite often in physics and it comes from the Greek word gyros, meaning circle [1]. It is used not only in science but in engineering too, as a generic description of an event involving some rotation of the plane of linear polarisation of light. In fact, following Fresnel’s proposition that linearly polarised light is a superposition of two forms of light called left and right-circularly polarised light it is clear that a gyrotropic material is associated with the appearance of elliptically, or circularly, polarised light. It is a very important area that embraces many complex materials which display a wealth of fascinating properties, like optical activity. In general then, a complex relationship exists between the field vectors E and H and the induction vectors D and B, where these quantities have their usual meanings. This relationship can be adjusted to take into account that gyrotropy can be free, natural or forced [1]. Free gyrotropy and forced gyrotropy are in the same category, because ‘forced’ means that it is created by an external magnetic field, for example, and ‘free’ is associated with internal fields. The best known example of natural gyrotropy is optical activity that is exhibited by sugar solutions and this is immediately distinguishable from the forced case by the following signature. Suppose a plane linearly polarised light wave passes once through a natural gyrotropic material causing the plane of polarisation to be rotated. If the same beam is reflected back through the material then, because it has natural gyrotropy, the rotation on the first pass is undone and no final rotation results i.e. no reversal of handedness occurs in this case [2]. This is a very important distinction from forced gyrotropy, which is the property of magnetooptic materials where the rotation of the plane of polarisation would have been doubled. Faraday discovered this and the Faraday effect, as it is now called, involves propagation parallel or antiparallel to an applied magnetic field. Other well-known magnetooptic effects are Voigt and Cotton-Mouton after their discoverers, which occur when the wave propagation is perpendicular to an applied field. Either name can be used but, historically, Voigt dealt with vapours while the second name-pair used liquids. Voigt will be the term adopted here to denote this type of birefringence, which is also revealed by uniaxial crystals, when a wave propagates perpendicular to the optic axis. As will be shown later, for bulk media, the Faraday effect is a non-reciprocal phenomenon and the bulk Voigt [Cotton-Mouton] [1, 2] effect is reciprocal. The really interesting outcome, however, is that even the Voigt effect is non-reciprocal in an asymmetric waveguide.

Spatial solitons are beams that do not diffract by virtue of a strong nonlinear interaction with the medium in which they propagate.[1] Quadratic solitons are a rather special member of the spatial soliton family because their existence is not linked to a self- induced refractive index change and subsequent guiding of the beam by the induced waveguide. Instead they exist by virtue of the strong coupling via the second order nonlinearity χ(2) between beams of different frequencies.[2] For example, for Type I second harmonic generation (SHG) in which there is a single fundamental and harmonic beam, the beams exchange photons with propagation distance leading to mutual self- trapping. Although such solitons were predicted back in the 1970s, they were not observed experimentally until the mid 1990s. [2–4] Over the last five years there has been a great deal of experimental progress in this field and the purpose of this chapter is to review some aspects of this work. (A separate chapter by Sukhorukov in this book deals with the details of the theory of quadratic solitons and related effects.)

Reorientational phenomenon in liquid crystals is unique for nonlinear optics applications. Recently there have been shown that the reorientational nonlinearity in nematic liquid crystals can govern spatial solitons in both waveguide and bulk geometry. Such solitons require a few milliwats of light power and they can be controlled by the state of light polarisation or an external electric field. In this paper, the theoretical analysis and experimental results on optical solitons in nematic liquid crystals due to the reorientational nonlinearity are reviewed. The discussion of their properties is preceded by a brief introduction to nonlinear optics of liquid crystals.

It is known that there is strong association between the basic equation of nonlinear optics, the nonlinear Schrödinger equation (NSE), and the fundamental equation of magnetodynamics, the Landau-Lifshitz equation (LLE), that, in the case of isotropic and uniaxially anisotropic medium, just takes the form of equivalence [1, 2]. This gives grounds to expect close similarity between nonlinear soliton phenomena in optics and magnetism. The traditional objects of study in magnetodynamics are so-called domain walls (DWs) (narrow, moving or standing, transition regions in magnets separating the regions of different uniform magnetization) and small-amplitude oscillating soliton-like packets of magnetostatic waves (MSW), which are clearly observable and reliably reproducible in experiments. However, the class of known exact soliton solutions is more wide. It contains the solutions correspondent to DWs as a particular case, so- called topological solitons, and includes the family of localized solutions (“dynamical solitons”). These solitons may be considered as promising information carriers for devices of functional magnetoelectronics, because their maximal velocities are much higher than the velocities of bubble magnetic domains, which were studied intensively earlier. Unfortunately, it should be noted that there are no fool-proof physical experiments in observation and generation of them.

The bit error rate (BER) in soliton amplification through a cascade of Erbium-doped fiber amplifiers (EDFA) is determined accounting for the signal-induced gain saturation. Steady state analysis under pulsed conditions is used since the level population densities respond to the time-averaged power of the soliton stream. Since shorter soliton duration is required for higher bit rates and the average soliton power increases as the duration decreases, EDFAs are operated under saturation at high bit rates. The EDFA gain and amplified spontaneous emission (ASE) noise change with the time-averaged input power and attain different values when operated under saturation than when not. In this work, the span distance and gain and noise levels are accurately determined accounting for the saturation effect in order to meet a bit error rate criteria in a given soliton communication system using a cascade of EDFAs.

In article, three problems of interaction of spatial solitons in the non-uniform environment are considered. Basic feature of these problems consists in the fact that nonlineariry and heterogeneity are of the same order. The approach to the decision of such type of problems is offered. Efficiency and singularity of the soliton interaction is shown.

Nowadays a great attention is spared between various nonlinear optical effects to the spatio-temporal transformation of the laser beams such as: self-focusing, self- diffraction, bistability, optical turbulence and other transversal instabilities [1, 2]. The main part of experiments was accomplished in Kerr media placed in the resonator with a positive feedback. Comparatively a new object of the study of such effects is the thin lightsensitive films where the resonance interaction with the incident light is determined by Rayleigh scattering in the waveguide modes [3]. As a result the periodical structure on account of the interference between the incident beam and the scattering modes is formed in this film. This structure is developed within the beam action and in general manifest itself as the nonperfect diffraction grating. Its nonperfection is connected with that the periodical structure consists of the set of separate microgratings (domains). These microgratings are different from each other by the orientation of grating vectors K in regard to the principal direction K₀ determined by the polarization vector E of the incident light beam. This work is devoted to the effects connected with the formation of the spontaneous periodical structures (SPS) of the waveguide nature in the films under the action of one single continuous Gaussian laser beam (P=5 mW, γ=633 nm) with arbitrary polarization.

Although nearly a quarter of a century has elapsed from the first theoretical prediction of quadratic optical solitons [1], a large interest has been conveyed to the topic only during the last decade [2], partially due to a mature understanding of χ(2) cascaded processes in second harmonic generation (SHG) [3]. After the first experimental demonstration of quadratic spatial solitons (QSS’s) in KTP by Torruellas et al. [4], a considerable amount of theoretical as well as experimental work has been performed, as apparent in this Volume.

Soliton transmission through a silica based single-mode fiber having a cross sectional area of 50 μm2, core refractive index of 1.46, nonlinear coefficient 1.22 x 10-12 m2/V2 and dispersion of 16 psec/nm/km is realized. The shape of the input pulse is hyperbolic secant and pulse width is 25 psec.

All-optical devices based on cascading effects in second order nonlinear materials seem to be very promising for the next generation of optical telecommunication systems [1]. A beam propagation method (BPM) based on paraxial approach [2] is a most popular technique employed to investigate light propagation in such devices. The drawback of this technique is necessity to accurately guess reference indices to satisfy slowly varying envelope approximation. This limitation of the conventional BPM is removed by using nonparaxial, wide-angle formulation of BPM [3, 4]. The nonparaxial BPM can simulate fields with rapidly changing envelopes and there is no need to accurately guess reference indices. In implementation of nonparaxial BPM for simulation of boundless space propagation boundary conditions play a very important role, because popular technique, transparent boundary conditions (TBC) [5], have very limited effectiveness [4]. Recently, novel boundary conditions, perfectly matched layer (PML) boundary conditions, were proposed by Bérenger [6]. In this paper, an investigation of effectiveness of nonlinear PML (NL-PML) boundary conditions terminated by different realisations of TBC (adaptive, controlled and uniform TBC) is reported. NL-PML boundary conditions terminated by controlled TBC were proposed that extremely improve out-going wave absorbing for small tilted propagation angles of narrow parametric spatial solitons and are most suitable for nonparaxial simulations.

Interaction and mutual control of the propagation behaviour of optical beams in Kerrtype nonlinear planar waveguides are analyzed based on numerical solutions of the modified nonlinear Schrodinger equation derived within nonparaxial vector approach. It is shown that after the interaction the angle of propagation of the solitons changes and this results in transversal shift of their output position. The trajectories of the beams are obtained and analyzed in terms of variation of the initial conditions.

We present experiments and related numerical simulations on spatial induced modulational instability, which is used to study periodic arrays of bright soliton-like beams in a planar CS₂ waveguide. In the picosecond regime the finite molecular relaxation time of the nonlinear refraction contributes to inhibiting the periodic recurrence, thus stabilizing the propagation of the soliton-like array.

This paper presents polarization properties of light in liquid crystal (LC) optical fiber waveguides with elliptical cores.

The properties of solitary waves in materials exhibiting both second-order and photorefractive nonlinearities are presented. Interactions of two solitary waves in such materials are presented and possible devices are proposed. Split-field perturbation method is used to confirm numerical results.

We study quadratic solitons supported by two-wave parametric interaction in χ(2)nonlinear media. We obtain very accurate explicit solutions for bright solitons with the help of a specially developed analytical approach.

There has been a lot of interest recently in the study of self-guided optical beams, also known as optical spatial solitons, which propagate in nonlinear optical media in conditions where the natural tendency of the beam to diffract is exactly compensated by the optical nonlinearity [1, 2]. As a consequence the beam profile is invariant in the propagation direction — a property shared with the modes of a fabricated optical waveguide and one which gives rise to the notion of self-guidance. One of the nonlinearities most studied that supports the creation of spatial solitons is the so-called Kerr nonlinearity where the refractive index of the material is linearly dependent on the local intensity of the light beam. A Gaussian laser beam, for example, propagating in a Kerr material would induce an increase in the material’s refractive index on-axis relative to off-axis regions [2]. This creates a focusing action in the medium that provides the basic condition for compensating diffraction.

This course of lectures is meant to give an idea of what physical effects can be obtained beyond the paraxial approximation in a study of the dynamics of spatial solitons, soliton- like waveguide channels and pulse signals. In description of narrow wave packets featuring a broad space-time spectrum it is often not sufficient to use the approximate parabolic equation which accounts only for a slight diffraction and dispersion spreading. At the same time, analysis of a problem in the nonabridged equations limit is generally quite complicated no matter which method — analytical or numerical -is used, so any new approach is highly valuable. In this work we present some analytical methods for description of the wave fields in inhomogeneous and nonstationary nonlinear media, which allow to design the dynamics of nonparaxial quasi-localized soliton-like wave structures. Using these methods one can largely simplify the initial problem reducing it to the form ready for numerical and, in some cases, analytical solving. Specifically, we focus on two problems here: a) the features of the processes of self-action of soliton-like wave beams and pulse signals in smoothly inhomogeneous slightly nonstationary media, that are related with the nonlinear distortions of propagation paths and the carrier frequency [1, 2], and b) the dynamics of low radiation loss nonlinear wave structures quasi-localized in space (nonlinear leaky modes) [3]. These problems are analyzed for self-consistent soliton-like waveguide channels propagating near the linear-nonlinear media interface (nonlinear quasi-surface waves). To better understand which nonparaxial effects are meant here we need to recall what a paraxial (or quasi-optical) approximation is in itself.

We discuss solitons in nonlinear resonators. In particular we discuss the cases of “laser” resonator, laser resonator containing a nonlinear absorber, parametric mixing and semiconductor microresonator. In these resonators the following localized structures (or spatial solitons) exist: vortices, bright solitons, phase solitons, and bright and dark solitons, respectively. We discuss the types of equations to which these systems correspond and report the experimental observations and properties of these different solitons, with extension to three-dimensional structures.

It is well recognized that two-step second-order (orχ(2): χ(2) cascading provides an efficient way to generate an effective nonlinear phase shift for the purposes of all-optical nonlinear switching [1]. Cascading of χ(2) processes allows all-optical switching to be achieved at pump levels substantially lower than those usually available in centrosymmetric media with highest known cubic nonlinearity [2]. A further search for the methods allowing to reduce the switching power is crucial for the future applications of all-optical devices based on cascading of χ(2) processes. The switching intensity is usually connected with the input power density necessary for achieving a nonlinear phase shift (NPS) of the amount of π or π/2, depending of the type of the device. The larger is the efficiency of the NPS generation, the lower is the switching intensity. Some methods for an enhancement of the NPS in quadratic nonlinear media with two-step second-order cascading processes have been recently suggested in [3, 4].

The (1+1)-dimensional theory of envelope soliton propagations rests on the nonlinear Schrödinger (NLS) equation, which reads: 1$$ iAu_z + Bu_{xx} + Cu|u|^2 = 0 $$x is the pulse shape variable, either transverse or longitudinal (time variable t). A well- known, but essential property of the NLS equation is to be completely integrable by means of the inverse scattering transform (IST) method for any value of the real coefficients A, B, and C. The most simple generalization of the NLS equation to (2+1) dimensions is the so-called two-dimensional nonlinear Schrödinger equation (2D NLS), that reads: 2$$ iAu_z + Bu_{xx} + Cu_{yy} + Du|u|^2 = 0 $$ Equation (2) is not integrable, whatever the value of the coefficients is, and does not admit any stable localized solution.

We report the experimental study of seeded and noise initiated modulational instability (MI) during second harmonic generation in a LiNbO₃ slab waveguide. In the non-seeded case, two MI initiating mechanisms were identified, namely random noise on the input beam and waveguide imperfections. We also measured the MI gain coefficient as a function of periodicity and beam intensity employing a small, well-defined periodic perturbation (seeding). Excellent agreement with theory was obtained.

There is a critical magnetic field, which separates two different zones for soliton signals propagation in magnetic chains: the sin-Gordon zone and the Heisenberg zone. We investigate the fine structure of these signals in the neighborhood of the critical field also nonzero soliton velocity. Explicit formulae both for azimuthal kink and meridional soliton are obtained, they take into account the nonlinear interaction of the soliton structures.

We report on the experimental observation of two-dimensional dipole-mode vector solitons. We demonstrate a decay of an unstable vortex-mode soliton into a robust dipole-mode soliton in a SBN crystal.

Spatial quadratic optical soliton formation has been studied extensively in the past [1] – [4] and recently Li et al [5] demonstrated quadratic spatio-temporal soliton formation in LilO₃ using tilted pulses. In this article we report self-focusing and intensity dependent walk-off compensation in LilO₃ using tilted pulses. Tilted pulses are space time coupled pulses which allow control of dispersive propagation in normally dispersive materials. In this study we focus a short tilted pulse in the dimension transverse to the tilt and observe intensity dependent spatial shifts. We explore this intensity dependent spatial shift for thresholding and limiting application using ultrashort tilted pulses.

In recent years a considerable interest has been focused on theoretical [1–10] and experimental [12, 13] studies of spatial dissipative structures in nonlinear optical resonators. Cavity soliton (CS) can be created and destroyed by localised pulses of light, and can thus be used to store images or information. The ability to control and manipulate CS has been proposed in the pioneering paper [2] for the case of two-level medium. The problem of stationary soliton solution has been numerically and analytically investigated mostly in the case of the semiconductor optical resonators [5] and cubic nonlinear cavity [6]. Note that the quadratic non-linearity provides a wide range of new opportunities. For instance, the threshold of pattern formation in a degenerate OPO was considered [7]. Some authors concentrate their attention on the CS interaction [8]. The problem of soliton stability in OPO is discussed in [9]. The most complicated case of SHG configuration with driving field corresponding to the fundamental frequency (FF) was studied in [10]. It was found that the quadratic CS can appear due to modulation instability of the steady-state solution. It should be emphasized that trapping dynamics of CS is not investigated sufficiently.

This paper presents an overview of a new interpretation of nonlinear solitary waves based on the momentum space, or k-space, which allows non-paraxial effects to be readily included, and reduces to NLS under certain circumstances. In this interpretation, solitary waves are described by the absence of curvature of the nonlinear k-space.

Simultaneous propagation of two optical pulses in Kerr type planar waveguides is considered. It is assumed that pulses propagate in different dispersion regimes: anomalous and normal. Such effects as catastrophic self-focusing, spatio-temporal splitting, and a possibility of a formation of soliton-like solutions axe examined. It is discussed whether such a configuration could be used for compression or switching of pulses.

The process of four-wave mixing is of a great importance in guiding systems, since some leaky modes can appear through the possible emission of fast combinative waves. There is an example in papers [1, 2], where the mechanism of radiative losses of electromagnetic energy at a nonlinear interaction of forward and backward surface waves in third order nonlinear plasma films was theoretically investigated. One should note that this effect also takes place at an interaction of volume modes in nonlinear waveguides. It is interesting to seek a similar effect in strongly nonlinear systems, such as spatial solitons, for example. As it is shown in given paper, emission of combinative wave leads to dramatic changes in solitons dynamics, when the usual repulsion or attraction is completely equilibrated by reaction of radiation.

We observe induced modulation instability (MI) of a partially spatially incoherent beam by seeding noise through cross-phase-modulation. Experiments revealed the existence of a threshold for such induced incoherent MI, which depends on the degree of spatial coherence as well as the strength of the nonlinearity. Above the threshold, the MI leads to formation of ordered and disordered patterns of incoherent light.

It is generally understood that the interplay between a nonlinear response and the feedback provided by a periodic perturbation can give rise to light localization and slowly-propagating envelope waves within the stopband of an otherwise reflecting structure, i. e. a structure where running linear waves are damped. Such concept was widely investigated in the pioneering numerical/theoretical studies by Mills et al. [1- 3] and by Sipe and Winful [4] concerning Kerr materials, until a number of experimental findings in Silicon-on-Oxide planar waveguides [5], silica fibers [6–8] and AlGaAs channels [9] have confirmed the most important results: the possibility of temporal solitary waves stemming from the balance between the nonlinearity (optical Kerr effect) and the group-velocity-dispersion (GVD), the latter mainly due to the presence of a Bragg periodic grating [10].

The chapter presents a new scalar model of optical beam propagation in nonlinear media, as it is developed in [1–4]. The model addresses narrow beams and stresses on nonlinearly induced diffraction, an effect of medium inhomogeneity introduced by the spatial variation of the nonlinear polarization. Strarting from the vector nonparaxial model of beam propagation in nonlinear media, it is shown that not the vectorial nature of the carrier wave field, but a scalar effect which comes out from the (div/E)-term in the wave equation and has the meaning of nonlinear diffraction, controls predominating over the nonparaxiality, the balance between diffraction and nonlinearity in the formation of the spatial solitons. The conclusion is based on analytical and numerical solutiuons of the nonlinear equations for the beam envelopes and on analysis of the wave power conservation laws derived. Both third (Kerr-type)- and second- order nonlinearities are treated as well as both planar waveguides and bulk media are covered. Single beam propagation and beam interaction and coupling are described. New solitary-wave solutions are presented.

We observed experimentally a power-dependent polarization instability of the fast spatial soliton in AIGaAs slab waveguides. The slow soliton remains stable. The instability occurs at power levels below which the nonlinear index change becomes comparable to the birefringence and is caused by coupling to radiation fields via four- wave mixing.

The theory of sub-wavelength quadratic solitons has been developed in the frame of Maxwell’s equations for type I nonlinear interaction. The fundamental limitation of soliton width has been obtained by numerical and analytical methods.

Wave beam propagation in the smoothly nonuniform medium can be described in the framework of geometrical optics when the wave length is small in comparison with characteristic scales of the problem, namely, size of the beam, size of inliomogeneities etc. The propagation trajectories of linear and nonlinear wave beams do not differ when the nonlinearity coefficient does not depend on coordinates. However, in a case of sharp variations of the medium properties, especially on the boundary of two media, when the dielectric permittivity of the medium is described by discontinuous function, the behavior of the nonlinear wave beam considerably differs from the linear description of wave fields [1–5]. In the linear case the field on the boundary is splited on the reflected field and the transmitted field. The nonlinear beam practically either completely reflects, or completely transits into other medium, since the nonlinear focusing tries to conserve the concentration of the wave field. This feature of nonlinear fields allows to use the geometrooptical approach for the description of the wave packet propagation in the strongly nonuniform medium with sharp gradients of the dielectric permittivity.

In the past few years, photorefractive spatial solitons that exist at low power level have been predicted [1] and experimentally confirmed in transient regime [2]. Later, by addition of a background illumination, steady-state screening solitons have been demonstrated [3]. Photorefractive solitons have since been the subject of an intensive work. In addition to SBN crystals in which photorefractive solitons were first observed, different materials (BTO, KNbO₃, LiNbO₃ or InP) can support photorefractive solitons. Spatial solitons relying on the photovoltaic effect have also been studied and observed in bulk crystals [4–8]. In the widely used material LiNbO₃ light induces a photovoltaic space charge field effect that diminishes the index of refraction. This allows dark soliton observation. Dark photovoltaic 1 -D soliton were first obtained in Bulk LiNbO₃ crystals [6] and have been used to create Y-junction [7]. Recently, 2-D photovoltaic solitons have also been reported in a KNSBN crystal [8].

The theory of stationary propagation of laser radiation in nonlinear directional couples (NDCs) usually assumes that the propagation constants β depend on the wave intensity J. At present time properties of two-core NDCs whose propagation constants contain the Kerr correction [1–3] have been studied in detail. A system of nonlinear differential equation describing the propagation of light in coupled parallel fibres has been constructed and analytic solutions of these equations in terms of elliptic functions have been obtained.

In this paper, we discuss the use of a generalized nonlinear Schrödinger equation 1$$ i\psi _x + \psi _{tt} + 2|\psi |^2 \psi - i\beta \psi _{ttt} + i\gamma \left( {|\psi |^2 \psi } \right), = 0, \beta > 0, \gamma > 0 $$ for the complex amplitude ψ(x,t) of the light guide pulse envelope. The use of the subpicosecond and femtosecond pulses gives one many additional opportunities for light guides devices and, in particular, opportunity to increase transmitted powers. However in the mentioned ranges it is necessary more accurate to take into account nonlinear and dispersion effects. There are serious theoretical arguments [1–3], that the additional terms with β and γ in (1) permit one to describe transition to the sub- picosecond range. It is necessary to note, that these terms naturally appear under consecutive asymptotic derivation of the equation of type (1) in [4]. The highest nonlinear term was taken into account in [1, 5–7], and term Ψ _m is considered by [6–8].

In the past decade spatial solitary waves in materials exhibiting non-Kerr nonlinearities have attracted a lot of attention. The dynamics of solitons in quadratic media was investigated both theoretically and experimentally [1, 2, 3]. Solitons in photorefractive crystals have also been observed and investigated [4, 5, 6, 7, 8]. In this paper, optical solitons in materials exhibiting both photorefractive and quadratic nonlinearity are investigated. These two nonlinearities are of completely different origin but as shown, in a recent, pioneering paper [9], they can both influence the propagation of a beam of light (it is shown that the efficiency of second harmonic generation increases due to creation of a photorefractive channel). Here we show that relatively weak second- harmonic wave, can guide a powerful fundamental wave.

It is well-known that interest to spatial quadratic solitons has arisen in many respects due to their unique switching properties [1–7]. Change of a direction of light beam in birefrigent media due to the mutual trapping of fundamental and harmonic beams was studied earlier [3, 4]. Except for that the change of optical soliton direction is shown to be possible colliding the soliton with other identical one [5, 6]. However, the description of soliton trapping and interaction usually does not accounted for the radiative waves as the model of solitons like particles does [7]. In the given work such model appears insolvent to describe radiative attenuation of spatial beam fluctuations. We show the radiative model of misalign beam interaction which predicts the direction of soliton trapped.

The stabilizing role of the nonlinearly induced diffraction on the critical balance between nonlinearity and linear diffraction at (2+l)D soliton-like beam propagation is studied. It is shown that regarding the static properties of such beams, the nonlinearly induced diffraction predominates over the effects of nonparaxiality and longitudinal field component. The dynamics of these solitary-wave beams is obtained and analyzed numerically in terms of variation of the input power.

We report for the first experimental generation of complete (3+1) dimensional spatio-temporal soliton, or “light bullet”. This is achieved by propagation of high- intensity femtosecond laser pulses in atomic and molecular gases.

Liquid crystals are commonly applied anisotropic materials in modern optoelectronics and determination of material parameters characterized their properties is very important. In the paper the method of measurement the elastic constant of nematic liquid crystal based on nonlinear reorientation effect is presented. Obtained results are in qualitative compatibility with the results obtained by the classical methods.

About a history of the question. There are two contrast situations in optics: classical diffraction of light, e.g. Bragg-Ewald’s diffraction on space lattice of point dipoles, and laser trapping of microscopic particles or atoms. At the former light fields are subjected to the material distribution, at the latter particles are obeyed to light [1]. Substance and fields are equal in strength, to some extent, when we deal with a stimulated light scattering. As early as 35 years it’s known particular, two-dimensional kind of that phenomenon. Formation of ripples [2], spontaneous gratings (SG) with stimulated Wood’s anomalies [4], laser-induced periodical surface structures [5] is connected with an instability development of a substance in the interference fields, arising due to superposition of the single incident pumping beam with scattered surface modes. SG spatio-temporal structure has been investigated early in the 80-th by Dr. Fritz Keilmann from Max-Plank-Institute in Shtuttgart [2, 3], who, for the first time, observed a dispersive behavior of the ripple period and connected it with an excitement of surface polaritons. Dr. Keilmann pointed out also that “the situation in our case is somewhat different” comparing to Bragg reflection. Detailed theoretical treatments are based on “surface-scattered waves” [4, 6], “radiation remnants” [5], “analytical solution of the diffraction problem under Wood’s anomalies conditions” [7] and others. All models are sufficiently complicated for physical understanding. Despite the broad range of theories, SG display some bright and universal properties, which testify in favour of a possibility to treat a simple and universal mathematical model of the phenomenon.

A soliton is a concept which describes various physical phenomena ranging from solitary waves on a water surface to ultra-short optical pulses in an optical fiber. The main feature of solitons is that they can propagate long distances without visible changes. From a mathematical point of view, a soliton is a localized solution of a partial differential equation describing the evolution of a nonlinear system with an infinite number of degrees of freedom. Solitons are usually attributed to integrable systems. In this instance, solitons remain unchanged during interactions, apart from a phase shift. They can be viewed as ‘modes’ of the system, and, along with radiation modes, they can be used to solve initial-value problems using a nonlinear superposition of the modes [1]. However, in the recent years, the notion of solitons has been extended to various systems which are not necessarily integrable. Following this new trend, we extend the notion of solitons and include a wider range of systems in our treatment. These include dissipative systems, Hamiltonian systems and a particular case of them, viz. integrable systems.

The switching operation is one of the crucial functions of all information processing or information transmission systems. Some of them, like a communication network or a computer, are generally composed of connected switches. The vital simplification for optical systems would be a possibility to avoid multiple conversions of information from a photonic to an electronic form to provide switching at subsequent nodes of the net. All- optical switching devices can give this chance. Generally an all-optical switching operation occurs when output characteristics of the device can be determined either by the parameters of the input signal or by a separate control beam [1]. A number of devices to switch optical signals have been proposed. The switching operation in the first group of them relies on optically controlled power exchange between two modes guided in an integrated optics or fibre system. Functioning of the second group relies on optically induced changes in phase difference between two pulses.

In the present paper the fundamental theory of parametric solitons, trapping and interacting in bulk media, cavities and gratings with quadratic nonlinearity are considered. We discuss the mechanism of parametric self-action due to which quadratic solitons are formed. Nonlinear dispersion of both plane waves and solitons is investigated. Change of quadratic soliton properties in the process of narrowing of its width is traced with the help of numerical and analytical solutions of Maxwell equations. The criteria of soliton stability and nature of modulation instability are analyzed. The dynamics of soliton trapping is demonstrated. Advantages and disadvantages of effective particle model of quadratic solitons are presented. The main features of soliton generation in resonators are discussed as well.

Within the proposed approach we derive a nonperturbative closed form solution for the semiclassical dressed states, explicitly accounting for the fast, non- adiabatic effects due to the amplitude and phase variations of the electromagnetic field.

We present a new type of composite soliton; a rotating propeller soliton. This soliton is made of a rotating dipole component jointly trapped with a bell-shaped component.

The existence and dynamical and other properties of cavity solitons are reviewed. These are bright, stable, non-diffracting spots of light in a driven optical cavity. The cavity must contain a nonlinear medium, but cavity solitons are supported by many media which do not support ordinary (propagating) spatial solitons. We use the Kerr cavity as a first example to describe methods to find them and analyse their stability. We demonstrate a sizeable domain of stability of two-dimensional cavity solitons in a Kerr cavity. Some other cavity soliton systems are briefly described. We show that cavity solitons have properties interesting for applications to optical information processing. Semiconductor microresonators are particularly promising, and we outline some results from models of such systems.

We overview our recent results on the discrete spatial solitons — nonlinear localized modes — in two-dimensional (2D) photonic crystals and photonic-crystal waveguides. Employing the technique based on the Green function, we describe the existence domains for nonlinear guided modes in photonic crystal waveguides and study their unique properties including bistability. We also show that low-amplitude nonlinear modes near the band edge of a reduced-symmetry 2D square-lattice photonic crystals, which are usually unstable, can be stabilized due to effective long-range linear and nonlinear interactions.

The basic mathematical apparatus of nonlinear optics consists of an array of nonlinear PDEs for the complex amplitudes of an envelope of interacting wave trains. In the general case, these equations include linear and nonlinear dissipative terms. However, in many important cases, they are small and can be neglected: therefore the equations are conservative, and the medium is transparent. According to the Kramers-Kronig relations, stemming from the principle of causality, the transparency can be realized at most in a limited spectral band, and even in this case some dissipation inevitably exists. Nevertheless, such fundamental nonlinear effects as the generation of high harmonics, induced Raman scattering, and self-focusing can be described by the conservative equations, preserving energy.