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We report on the stable self-focusing of light for various degrees of spatial coherence. Since the diffraction of a partially spatial coherent light beam is mainly governed by the transverse correlation width, its self-focusing is generally non-trivial. Here, we systematically demonstrate that partially coherent solitons can be generated in a non-instantaneous nonlinear material when the external parameters are properly adapted to the degree of coherence of the light beam.

We study the existence and stability of multi-component spatial optical solitons in anisotropic nonlocal photorefractive media. For the case of three components, we find numerically the whole family of composite solitons with two perpendicular dipole components, and show their stability for a wide range of parameters. We confirm our theoretical results by an experimental observation of these novel types of composite optical solitons in a photorefractive strontium barium niobate crystal.

Stable dipole-mode vector solitons in photorefractive (PR) crystals are studied, both in numerical simulations and in experimental investigation. Numerically exact multi-humped solitary solutions to the propagation equations of two incoherently coupled beams in PR media with an anisotropic nonlocal material response are found. A continuous set of dipole-mode vector solitons, ranging from the solitons with negligible dipole-mode contribution to the solitons with negligible ground mode contribution, is identified. The results are compared with experimental data.

Self-bending of photorefractive solitons is caused by diffusion in photorefractive crystals and becomes an important effect when the beam size is in the range of the charge carriers diffusion length. In this paper we present an experimental and numerical examination of the beam bending dependence on relevant parameters such as the applied electric field and the beam intensity. We demonstrate that the bending dependence on the electric field in the low saturation regime has the form of a square function at low values of the field and becomes linear for higher values. For stronger saturation the curve gets the form of a square root function. The bending dependence on the beam intensity has a maximum at defined intensity. The experimental data are compared with numerical simulations, giving a good qualitative agreement.

We review the propagation and interaction behavior of spatial screening solitons generated in a photorefractive SBN crystal in order to exploit them for new perspectives in soliton-driven photonics. We report on the successful experimental generation of an array of 5 x 5 spatial screening solitons. The waveguide properties of each channel are probed by a beam of different wavelength, and each channel is found to guide the probe beam independently. By combining the interaction and waveguide properties, we achieved channel combination and separation, thereby realizing complex waveguide couplers and dividers. Furthermore we show that image processing is possible using parallel propagating photorefractive solitons, allowing to create new features in the frame of the concept of "light is guiding light."

Chapter 8, on self-organized pattern formation in single-feedback systems, describes how the complexity of the material-light interaction is enhanced by allowing the system to feed the beam back into the nonlinear material. A single-mirror feedback scheme with a photorefractive nonlinearity constitutes an excellent model system for studying pattern formation effects. An extensive description of the main properties of the free-running system is the topic of this chapter.

In Chapter 9, multiple patterns and complex pattern competition, experimental results which reveal parameter regions of simultaneously stable patterns and of pattern competition are presented. The most surprising feature of a photorefractive single-feedback system is the existence of multistability which provides a wealth of new patterns far beyond the threshold. In order to understand the complex dynamics of pattern interaction, detailed investigations of the uncontrolled system are necessary, and provide an appropriate and essential starting point for the implementation of manipulation and control schemes for active pattern selection.

Chapter 7 treats the case when a multimode waveguide is induced not by a single but several optical beams and vector solitons in a photorefractive medium can be constructed. In this case the participating modes are not guided in a passive way but interact with the medium and contribute equally to the formation of the required refractive-index profile. This is a substantially different approach from that described in the preceding chapters since two or more beams copropagate in a joint refractive-index profile generated by themselves. Vector solitons can therefore be considered as molecules of light, and their stability is one of the most crucial questions in complex spatial-soliton optics. We show that combination of higher-order modes with the fundamental Gaussian mode can lead to stabilization of multipole vector solitons, whereas vortex beams, as a general rule, decay into multipole vector solitons. Moreover, we demonstrate the stabilization of multicomponent solitons and soliton clusters.

Chapter 10 which covers manipulation and control of self-organized patterns by spatio-temporal techniques, the desire of researchers in this field to adjust and stabilize defined pattern states instead of passively observing spontaneous pattern formation is approached. An additional system parameter, a frequency detuning of the pump beams, is introduced and allows us to choose a particular pattern state when two or more patterns are accessible for one given set of parameters. The unique real-time access to the spatial frequency domain that is available in optics is exploited for the realization of a spatial control scheme based on the manipulation of wave numbers in the Fourier plane. Direct filtering in the feedback arm is shown to be an excellent tool for forcing the system to a desired solution. The feedback system and, thereby, its solutions are changed by any intervention in the system, which motivates us to consider an extension of this invasive scheme.

Chapter 2 on light propagation in nonlinear optical media describes the origin of self-focusing of beams in nonlinear optical media in general. It shows how these focusing effects can be exploited to create optical spatial solitons. Starting with one-dimensional transverse soliton phenomena the general features of soliton formation which lead to a description by the nonlinear Schrodinger equation are shown. On the basis of these introductory investigations the existing conditions for two-dimensional spatial optical solitons are considered which lead to novel solitary structures with exciting interaction features.

Chapter 5 treats the interaction of spatial solitons in saturable photorefractive media which may lead to novel, surprising scenarios. We distinguish between coherent and incoherent soliton interaction. For coherent interaction, we show that the relative phase of the interacting solitons is essential to the outcome of the interaction, leading to exciting new effects such as attraction and repulsion, soliton birth, annihilation and phase-dependent energy exchange. For the case of incoherent soliton interaction, the anisotropy of the photorefractive effect leads to a refractive index that displays regions of both attraction and repulsion, a previously unknown and surprising feature of the interaction between incoherent spatial solitons. In the direction of the applied electric field, solitons may attract or repel, depending on their relative separation, whereas they always attract when oriented along the Ydirection. This anomalous interaction gives rise to a wealth of new phenoma that were not expected from the general theory of spatial solitons in saturable media. Among these phenomena, rotation, oscillation and complex spiraling behavior show the complexity of photorefractive soliton interaction.

Chapter 2 on light propagation in nonlinear optical media describes the origin of self-focusing of beams in nonlinear optical media in general. It shows how these focusing effects can be exploited to create optical spatial solitons. Starting with one-dimensional transverse soliton phenomena the general features of soliton formation which lead to a description by the nonlinear Schrodinger equation are shown. On the basis of these introductory investigations the existing conditions for two-dimensional spatial optical solitons are considered which lead to novel solitary structures with exciting interaction features.

The photorefractive nonlinearity is described in Chap. 3 beginning with the origins of the space charge field arising in these materials from their interaction with light. The band transport model equations are used to formulate the basic equations for the photorefractive nonlinearity. The complexity is gradually increased, starting with one beam interacting with the nonlinear material and then taking a second beam into account to show the beamcoupling properties of photorefractive crystals. An important feature of a diffusion-dominated photorefractive material is the possibility of performing an energy exchange between interacting beams in a wave-mixing process. The two-beam coupling properties are thoroughly discussed for the special case of a counterpropagating geometry, which is the configuration of choice for the investigations of pattern formation effects described in Chaps. 8 to 10. Special care has been taken to introduce the main variables and equations that represent the starting point for subsequent considerations.

Chapter 6, waveguiding in photorefractive solitons for soliton-driven photonics, uses the investigations of the preceding chapters to exploit the refractive-index modulation that is built up during soliton formation for waveguiding purposes. Owing to the strong wavelength selectivity of the photorefractive effect, it is possible to guide light of different wavelength regions. Thereby, several different scenarios of the interaction of coherent and incoherent solitons can be demonstrated, leading to waveguide couplers, dividers and splitters. Furthermore, have we been able to use the inherent parallelism of optical beam propagation to create two-dimensional lattices of photorefractive solitons that allow simultaneous waveguiding of a large number of information channels. By combining the interaction of photorefractive solitons with this feature, highly parallel and adaptive solitonic crystal structures can be formed, representing the basis of soliton-driven photonic applications.

Chapter 1 on nonlinear waves and transverse patterns is an introduction dedicated to the basic description of modulation instabilities in nonlinear systems. This chapter classiffies the different situations that may arise in nonlinear systems and provide the physical background necessary to understand the following chapters.

Chapter 4 is dedicated to the physics of soliton formation in a photorefractive material and shows that the nonlinear formation of a space charge field due to the incident light beam can be balanced by screening an externally applied electric field, thus leading to a self-guided, stable beam. From a mathematical as well as a physical point of view, this solitary wave is very similar to spatial solitons in saturable media. The anisotropy of the photorefractive effect due to the application of the external electric field leads to an asymmetry of the refractive index created by the beams that strongly influences the formation of spatial photorefractive solitons. The most peculiar features are elliptically shaped solitary solutions, beam bending and diameter oscillations during soliton formation.

We study (2+1)-dimensional multicomponent spatial vector solitons with a nontrivial topological structure of their constituents, and demonstrate that these solitary waves exhibit a symmetry-breaking instability provided their total topological charge is nonzero. We describe a novel type of stable multicomponent dipole-mode solitons with intriguing swinging dynamics., 3 pages, 4 figures, accepted for publication in Optics Letters

We study, theoretically and experimentally, multicomponent spatial solitons in nonlinear saturable (isotropic and anisotropic photorefractive) bulk media. We find numerically a family of the three-component dipole-mode solitons and demonstrate their stability in a wide range of the input parameters. We also observe the formation and stability of these spatial solitons in experiment with photorefractive strontium barium niobate (SBN) crystals.

We investigate the specific influence of anisotropy on generation, stability and dynamic properties of spatial dipole-mode vector solitons in a photorefractive medium, experimentally and numerically. Further, we demonstrate collision-induced transformation of transverse to angular momentum.

We investigate the generation of higher-order optical vector solitons in two transverse dimensions in anisotropic nonlinear media consisting of an incoherent superposition of a Gaussian beam and a higher-order laser mode with a complex internal modal structure. We demonstrate both numerically and experimentally various examples of these stable self-trapped light structures and show that vortex modes carrying topological charge always decay into multiple-humped structures that remain self trapped during propagation. Furthermore, we demonstrate the mutual stabilization of a triple- and a double-humped transverse light structure leading to the formation of a two-dimensional vector soliton without a stabilizing fundamental Gaussian mode.

We discuss properties of multicomponent spatial formed by simultaneous co-propagation of few mutually incoherent beams in a slow nonlinear medium. W present experimental results on formation and interaction of these solitons in photorefractive nonlinear crystal.

We investigate the generation of higher-order optical vector solitons in two transverse dimensions in anisotropic nonlinear media consisting of an incoherent superposition of a Gaussian beam and a higher-order laser mode with a complex internal modal structure. We demonstrate both, numerically and experimentally various examples of these stable self-trapped light structures and show that vortex modes carrying topological charge always decay into multiple-humped structures which remain self-trapped during propagation. Furthermore we combine two dipole mode beams in the absence of a fundamental Gaussian component and demonstrate their mutual attraction that displays an unstable propagation behaviour and leads to a splitting of one constituent beam.

We predict theoretically a novel type of an extremely robust optical vector soliton that originates from trapping of a dipole (HG/sub 01/-type) mode by a soliton-induced waveguide. As a specific example, we consider two incoherently interacting optical beams propagating along the direction z in a bulk saturable medium. The model describes, in the isotropic approximation, solitons in photorefractive materials.

An experimental investigation of the dynamical, time-dependent effects during the formation and interaction of incoherent two-dimensional spatial screening solitons in a photorefraetive SBN crystal is performed. The scenarios observed include initial diffraction, collapse to the soliton shape, the oscillation of beam diameters, beam bending and the rotation and aligning of soliton pairs. The dynamics of nonplanar interaction and spiraling of two incoherent solitons is considered in more detail, showing that power exchange between the interacting beams is responsible for complex three-dimensional trajectories in the nonlinear crystal.

An experimental and numerical investigation of the dynamical, time-dependent effects accompanying the formation and interaction of two-dimensional spatial screening solitons in a photorefractive strontium barium niobate crystal is performed. These effects include initial diffraction, collapse to the soliton shape, the oscillation of beam diameters, beam bending, and the rotation, twisting, and turning of soliton pairs. The dynamics of complex spiraling of two incoherent solitons is considered in more detail.

We report on the first experimental observation of a novel type of optical vector soliton, a {\em dipole-mode soliton}, recently predicted theoretically. We show that these vector solitons can be generated in a photorefractive medium employing two different processes: a phase imprinting, and a symmetry-breaking instability of a vortex-mode vector soliton. The experimental results display remarkable agreement with the theory, and confirm the robust nature of these radially asymmetric two-component solitary waves., 4 pages, 8 figures; pictures in the PRL version are better quality

We study the mutual trapping of two dipole-mode light beams in a medium with a nonlocal anisotropic saturable nonlinearity. It is shown that the incoherent attraction of perpendicularly aligned dipole-mode beams leads to the stabilization of their composite structure. Our numerical analysis gives a stationary solution that is stable only with respect to small amplitude modulations. The coupled solitary structure disintegrates in a well-defined way when the numerical perturbations or the experimental propagation length exceeds a certain limit. In this case a new and more robust, multicomponent solitary transverse light structure consisting of two dipole-mode vector solitons will be generated.

We study, both theoretically and experimentally, the scattering properties of optical dipole-mode vector solitons - radially asymmetric composite self-trapped optical beams. First, we analyze the soliton collisions in an isotropic two-component model with a saturable nonlinearity and demonstrate that in many cases the scattering dynamics of the dipole-mode solitons allows us to classify them as ``molecules of light'' - extremely robust spatially localized objects which survive a wide range of interactions and display many properties of composite states with a rotational degree of freedom. Next, we study the composite solitons in an anisotropic nonlinear model that describes photorefractive nonlinearities, and also present a number of experimental verifications of our analysis., Comment: 8 pages + 4 pages of figures

We review the generation of the recently predicted multi-component spatial optical solitons in a saturable nonlinear bulk medium. We present numerical simulations for an effectively isotropic model and experimental results for a set of different combinations of a Gaussian beam co-propagating incoherently with a beam of a more complex internal structure, such as a higher order transverse laser mode. We discuss the different formation processes and the general properties of a variety of different dipole-mode composite solitons and expand our investigations to the generation of a quadrupole-mode composite soliton.

(2+1)-dimensional optical spatial solitons have become a major field of research in nonlinear physics throughout the last decade due to their potential in adaptive optical communication technologies. With the help of photorefractive crystals that supply the required type of nonlinearity for soliton generation, we are able to demonstrate experimentally the formation, the dynamic properties, and especially the interaction of solitary waves, which were so far only known from general soliton theory. Among the complex interaction scenarios of scalar solitons, we reveal a distinct behavior denoted as anomalous interaction, which is unique in soliton-supporting systems. Further on, we realize highly parallel, light-induced waveguide configurations based on photorefractive screening solitons that give rise to technical applications towards waveguide couplers and dividers as well as all-optical information processing devices where light is controlled by light itself. Finally, we demonstrate the generation, stability and propagation dynamics of multi-component or vector solitons, multipole transverse optical structures bearing a complex geometry. In analogy to the particle-light dualism of scalar solitons, various types of vector solitons can - in a broader sense - be interpreted as molecules of light.