Preface

Though not often perceived in our everyday life, nonlinearity permeates our physical world. The evidence for nonlinear behavior appears in so many aspects of physics, chemistry, biology, economics, etc., that it may be nearly impossible to mention them all. Among the most striking and aesthetically appealing manifestations of nonlinearity is the propagation of solitons or, more generally, solitary waves. Strictly speaking, solitons differ from solitary waves because of the remarkable property of integrability of the governing models and its physical consequences. Such a property is, however, scarcely relevant to the physics discussed in this book. Therefore we shall make use of the term "soliton" in its broader sense, to mean a wavepacket held together by the interplay of mutually counterbalancing linear and nonlinear effects.

Historically, the first documented observation of a soliton (in a shallow-water canal in Scotland) is attributed to J.S. Russell in 1834. Since then, solitons in physics have received increasing growing interest. In the twentieth century, the soliton became one of the most fruitful concepts of nonlinear physics and a key to understanding how nonlinearity acts in nature. The main reason of this success lies in the particle-like behavior of solitons and their intrinsic nature as "modes" of nonlinear systems. Solitons have become a widespread interdisciplinary field of theoretical and experimental studies, in branches ranging from fluid dynamics, oceanography, plasma physics, mechanical vibrations, electronics, solid-state physics, cosmology, superconductivity, and acoustics, to electrodynamics. In optics, the advent of laser sources has allowed easy experimental access to the excitation of a nonlinear response in dielectric materials. It took only a few years to develop, in the early 1960s (see below in the introductory chapter by Chiao), the idea of compensating the diffractive spreading of a beam (i.e. a bell-shaped spatial wavepacket) with the self-induced lensing effect. The latter effect is associated with a graded, intensity-dependent change of refractive index, known as the optical Kerr effect, which entails a cubic (in the field) material response. This balance gives rise to self-trapping into a spatial soliton, i.e. a shape-invariant self-guided beam of light, which can be viewed as a self-induced waveguide. Diffraction is overcome because the different plane-wave components do not suffer phase delays but, rather, are forced to travel in phase, with a common propagation constant determined by the nonlinear response. Apart from its genuine self-induced nonlinear origin, the underlying guiding mechanism behind spatial solitons is identical to that which allows virtually unlimited propagation in a graded optical fiber or waveguide, that is, a properly tailored gradient of the refractive index, leading to repeated total reflections. In retrospect, it is worth noting that the idea of self-trapping came even earlier than the proposal to employ silica fibers for transmitting information (K.C. Kao, G.A. Hockham, Proc. IEE 133, 1151 (1966); A. Werts, L'Onde Electrique 46, 967 (1966)), which was the very origin of the modern revolution in optical communications.

Nowadays, the world of optical solitons has grown enormously and includes temporal solitons, such as those observed in dispersive optical fibers, stimulated scattering, and self-induced transparency. In particular, dispersion-compensating fiber solitons, although proposed more recently (A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973)), have overshadowed spatial solitons, owing to their enormous applicative importance as natural information carriers which can propagate thousands of kilometers in fiber links. In comparison, spatial solitons exploit the nonlinearity of bulk materials (or slab waveguides) and hence usually propagate only few centimeters. Moreover, diffraction is usually stronger than fiber dispersion, thereby requiring, in principle, a larger nonlinear response (higher intensities). In spite of this, the physics of spatial solitons is much richer, basically because, unlike the case for temporal solitons, trapping occurs in both one and two transverse dimensions and involves different types of nonlinear material response. Fascinating phenomena that are specifically linked to the higher number of dimensions, which have no counterpart for temporal solitons, can take place, for example self-focusing and transverse instabilities, spiraling and hybrido-dimensional soliton-soliton interaction, the existence of optical vortices, and the formation of complex patterns and localized structures in cavities. These aspects have stimulated a renewed interest in the field, as witnessed by several important achievements obtained during the last ten years. For instance, the common beliefs that spatial solitons need extremely high intensities, a self-induced refractive index-change (Kerr effect), and a coherent excitation were demolished by the observation of solitons in photorefractive materials, by second-harmonic generation, and by means of incoherent light, respectively. The recent outstanding results have motivated us to put together the material contained in this book, which is specifically dedicated to spatial solitons. Although several books on optical solitons are available, none of them give a proper description of spatial solitons. The ultimate aim of the book is twofold: (i) to give an updated overview of the field, which will serve as a reference and guide for PhD students and scientists working in the field; and (ii) to attract scientists from other areas of science, specifically in nonlinear dynamics, convincing them that optics offers unique opportunities to investigate soliton physics owing to the vast number of physical situations where solitons are observed, and the ease and reliability of experiments. This said, we shall be glad if new, unexpected results and directions make the book outdated. In particular, we hope to see spatial solitons find their definitive way into applications, such as reconfigurable waveguides, all-optical devices, beam-quality control, and many others.

The book includes invited contributions from the majority of world leading experts, often pioneers, in the field. Not all the groups who have contributed substantially to our current understanding of spatial solitons are adequately represented, for a number of reasons. We regret that it was not possible to find room for all the respected colleagues and friends who took part in the development of this exciting topical area.

The content of the book covers a wide spectrum of issues concerning the propagation of spatial solitons, ranging from the most challenging theoretical problems to the most exciting experimental achievements to date. The general topic of spatial solitons is introduced by one of its harbingers, Prof. R. Chiao, who reviews the field, sharing with us his unique historical perspective. Then, a first group of chapters deals with the most recent and important experimental results. The different materials and mechanisms commonly employed in the recent self-trapping experiments are introduced by Luther-Davies and Stegeman, who describe and compare their features. The discussion of experimental results which exploit these specific materials is deepened in the following four chapters. Silberberg and Stegeman review the activity on Kerr-like materials, with special emphasis on observation of solitons in semiconductor bulk and waveguide arrays. Then Del Re et al. focus on solitons in photorefractive materials. As far as photorefractives are concerned, their slow nonlinear response has permitted researchers to open up a fascinating new avenue, namely incoherent solitons, or self-trapping with incoherent light. This area is reviewed by Segev and Christodoulides. Multicolor trapping in frequency conversion processes due to quadratic (x⁽²⁾) nonlinearities is then described by Torruellas et al. In the following chapter, de Sterke et al. give an overview of trapping in Bragg gratings, which occurs in the form of gap solitons. These are the only objects described in this book that are not related to diffraction-compensating trapping in the transverse plane; rather, they are confined along the longitudinal direction of the grating. The reason to consider them as spatial solitons, though of a different kind, stems from the fact that they also exist in the strict stationary limit as zero-velocity envelopes.

A second group of chapters shares a more theoretical viewpoint and reveals the problems faced by the mathematical description of spatial soli- tons. The crucial problem of stability is tackled in the general chapter by Kivshar and Sukhorukov, while the aspects related to collapse (catastrophic self-focusing) are specifically addressed by Bergé. Lederer et al. acquaint us with the physics and application of discrete solitons in waveguide arrays, for which the coupling between adjacent waveguides introduces a diffraction-like spreading mechanism. The principles of optical vortex solitons, self-trapped beams with a screw dislocation in their phase, are discussed by Swartzlander.

In a third group of chapters, we have grouped together contributions dealing with spatial solitons in dissipative systems. Dissipation is first introduced by Akhmediev and Ankiewicz, who discuss the rich world of solitons of the Ginzburg-Landau equation and its application in nonlinear optics. Then, Firth and Harkness focus more specifically on cavity solitons, giving both a historical perspective and a tutorial introduction to their description. Trillo and Haelterman discuss the opportunity offered by the interplay of parametric nonlinearity and passive feedback to confine the light either in passive cavities (parametric oscillators) or in gratings. The chapter by Weiss and collaborators summarizes the main recent experimental achievements in this field.

The last two chapters show how the importance of self-trapping is widened by new areas of applications: Boardman and Xie deal with propagation in magneto-optical (anisotropic) materials, whereas Saffman and Skryabin introduce us to the basic physics of joint trapping of radiation and matter waves, which is likely to be more than a dream in view of the progress in atom cooling and Bose-Einstein condensation.

Let us conclude with a dedication. Since both of us have fully respected the tradition that requires physicists to meet very understanding companions and generate mainly daughters, let us dedicate this book to the six beautiful women in our lifes (three plus three), from whom we have subtracted the time needed to complete this book.