A periodic array of optical waveguides creates a novel kind of device in which new types of spatial solitons can be generated and studied experimentally. The properties of spatially localized modes in a waveguide array are usually analyzed in the framework of a set of coupled-mode equations, each equation representing the soliton amplitude in a specific waveguide, but coupled to the neighboring waveguides. Such a coupled set of equations is referred to as the discrete nonlinear Schrödinger (NLS) equation, and its localized solutions are known as discrete solitons. A similar approach for studying discrete spatial solitons in solid-state physics is known as the tight-binding approximation. In the context of an optical waveguide array, this approximation corresponds to the assumption that the fundamental modes in all waveguides are only weakly coupled. The solitoncentered inside a single waveguide is stable because it corresponds to the minimum of the Hamiltonian. If the discrete soliton is forced to move sideways, it has to jumpfrom one waveguide to the next, passing from a stable to an unstable configuration. Thedifference between the Hamiltonians in the two cases—the so-called Peierls-Nabarropotential—accounts for the resistance that the soliton has to overcome during transverse propagation. This potential increases as the input power level increases. As a result, the soliton becomes localized to a single waveguide and is effectively decoupled from the rest of the array.

This chapter discusses the features of spatial solitons using a non-integrable generalized nonlinear SchröSdinger (NLS) equation. Spatial solitons in non-Kerr nonlinear materials exhibit many features that differ dramatically from those associated with the exactly integrable NLS equation. The solitary waves associated with non-Kerr nonlinear media preserve their shape, but their stability is not guaranteed, because of the non-integrable nature of the underlying generalized NLS equation. In fact, their stability against small perturbations is a crucial issue because only stable (or weakly unstable) self-trapped beams can be observed experimentally. Spatial solitons can propagate at an angle to the propagation direction z. Such solitons are of considerable practical interest because they allow steering of light by changing the angle that the soliton makes with the z axis. However, by changing the magnitude of velocity V electrically or optically, such a soliton can be steered in any direction. This shows the feasibility of a dynamically reconfigurable optical interconnect based on spatial solitons.

Optical vortex solitons represent a generalization of the concept of spatial dark solitons to two transverse dimensions. A vortex soliton is associated with the self-trapping of a phase singularity embedded within a broad optical beam in a self-defocusing medium. Experimentally, vortex solitons appear in a natural way through the transverse instability of dark-soliton stripes in a nonlinear bulk medium. Unlike the spinning solitons, which carry an angular momentum and require a self-focusing medium, the vortex solitons exist only in self-defocusing nonlinear media (n 2 < 0). Optical vortex solitons are sometimes described as the stable, black, and self-guided beams of circular symmetry. Vortex solitons can be generalized to include the polarization properties of light, and two such generalizations are possible. Firstly, two optical fields of different polarizations can have a vortex-like structure. This kind of double-vortex soliton is known to exist in other fields. Secondly, a vortex of one polarization can guide the optical field of another polarization, giving rise to the two-dimensional generalization of the bright-dark soliton pairs. Vortex solitons can also be generalized to include the temporal dimension. Such an extension leads to the concept of optical bullets in the context of bright solitons.

This chapter provides an overview of optical solitons by emphasizing on the physical principles and the simplest theoretical model based on the cubic nonlinear SchröSdinger (NLS) equation and its generalizations. The term “solitary wave” or “solitons” is used to reflect the particle-like nature of solitary waves that remained intact even after mutual collisions. In the context of nonlinear optics, solitons are classified as being either temporal or spatial, depending on whether the confinement of light occurs in time or space during wave propagation. Temporal solitons represent optical pulses that maintain their shape, whereas spatial solitons represent self-guided beams that remain confined in the transverse directions orthogonal to the direction of propagation. Both types of solitons evolve from a nonlinear change in the refractive index of an optical material induced by the light intensity — a phenomenon known as the optical Kerr effect in the field of nonlinear optics. The intensity dependence of the refractive index leads to spatial self-focusing (or self-defocusing) and temporal self-phase modulation (SPM)- the two major nonlinear effects that are responsible for the formation of optical solitons. The scalar NLS equation applies for both temporal and spatial aspects of wave Propagation It is derived on the basis of quite general assumptions about the dispersive (and diffractive) effects and the nonlinear properties of physical systems.

This chapter focuses on the properties of temporal solitons. Temporal solitons exhibit features that are identical to those for spatial Kerr solitons. The feature of maintaining its shape and remaining chirp-free during propagation inside optical fibers makes temporal solitons an ideal candidate for optical communications. A simple way to understand this behavior is to think of optical solitons as the temporal modes of a nonlinear waveguide. Higher intensities in the pulse center create a temporal waveguide by increasing the refractive index only in the central part of the pulse. Such a waveguide supports temporal modes just as the core-cladding index difference leads to spatial modes of fibers. Temporal solitons are attractive for optical communications because they are able to maintain their width even in the presence of fiber dispersion. Moreover, the presence of pulses in the neighboring bits perturbs a temporal soliton simply because the combined optical field is not a solution of the nonlinear SchröSdinger (NLS) equation. Neighboring solitons either come closer or move apart because of a nonlinear interaction among them.

This chapter discusses several related concepts of optical solitons. The nonlinear effects in liquid crystals arise mainly from thermal and reorientation processes. While the thermal effects are similar to those observed in other materials,the reorientation effect is a unique characteristic of the liquid-crystalline phase. The cubic, Kerr-like nonlinearity induced by the reorientation effect in the nematic phase of liquid crystals is responsible for numerous nonlinear effects that are not observed in other materials. Because of a spatially nonlocal response of a liquid-crystalline medium, the spatial solitons generated inside nematic liquid crystals represent a novel class of self-trapped optical beams that exist only in nonlocal nonlinear media. Optical waveguides that form when an optical beam is self-trapped inside a nonlinearmedium are of transient nature, in the sense that they disappear when the input beam propagating as a spatial soliton is turned off. Many applications use self-inducedwaveguides for guiding other beams of different polarizations or wavelengths. Theseapplications would benefit if soliton-induced changes in the refractive index were topersist even after the beam producing them was turned off, resulting in the “freezing” of a soliton-induced waveguide.

Vector soliton is a broad term that covers different types of multicomponent solitons. Two-component vector solitons can be spatial or temporal and form using two orthogonally polarized components of a single optical field or two fields of different frequencies but the same polarization. Vector solitons can be incoherently or coherent coupled. They can be incoherently coupled, in the sense that the coupling is phase insensitive. Another important class of vector solitons is associated with the coherent coupling among the optical fields; i.e., the coupling depends on relative phases of the interacting fields. Coherent interaction occurs when the nonlinear medium is weakly anisotropic or weakly birefringent. Coherently coupled vector solitons possess many properties that are different from the case of incoherent coupling. The concept of two-component vector solitons can be easily generalized to multicomponent vector solitons. Such temporal solitons can form, for example, when multi frequency composite optical pulses are transmitted over the same fiber in wavelength-division-multiplexed (WDM) channels of optical communication systems or abit-parallel-wavelength scheme is used for high-speed interconnect applications.

The fundamental physical mechanism behind the transverse instability induced by the self-focusing of soliton stripes is similar to the mechanism governing the self-focusing and modulation instabilities of small-amplitude, quasi-harmonic, wave packets.The instability occurs when transverse modulations on the wave front of a planar solitonstripe decrease the local value of the soliton energy. Two analytical methods can be used for studying the transverse instability of solitons, namely, ray-optics approach and linear stability analysis. The ray -optics approach is based on the assumption that a transversely modulatedplane wave remains locally close to its steady-state profile, so each individual segment of the wave evolves along an individual ray used for analyzing the self-focusinginstabilities of a soliton stripe. It makes use of a linear eigenvalue problem that is obtained by linearizing the (2 + 1)-dimensional NLS equation near the exactone-dimensional soliton solution. Because two-dimensional spatial solitons are unstable in a Kerr medium, the soliton interaction and spiraling can be observed only in non-Kerr media. The interaction can bemodeled theoretically either by using an NLS equation with saturable nonlinearity or by employing the cubic-quintic NLS equation. If the two-dimensional spatial solitons are treated as “effective particles” their interaction potential can be calculated by using the collective-coordinate approach.

A new kind of soliton, known as the Bragg soliton or the gap soliton, can form in a nonlinear media whose refractive index varies weakly in a periodic fashion along its length. Physically, Bragg solitons represent specific combinations of counterpropagating waves that pair in such a way that they move together at the reduced speed. One can understand the reduced speed of a Bragg soliton by noting that the counterpropagating waves form a single entity that moves at a common speed. The relative amplitudes of the two waves participating in soliton formation determine the soliton speed. The interaction of gap solitons with the grating defects displays features, such as, the capture, reflection, and transmission of gap solitons. A gap soliton, incident on a defect, can transfer its energy to a nonlinear defect mode formed at the defect location, provided such a mode exists with the same frequency (resonance) and has an intensity less than that of the gap soliton (energetic accessibility).

This chapter focuses on solitons forming inside photonic crystals. Photonic crystals can be viewed as an optical analog of semiconductors, in the sense that they modify the propagation characteristics of light just as an atomic lattice modifies the properties of electrons through a bandgap structure. Photonic crystals with embedded nonlinear impurities or made of a nonlinear material are called nonlinear photonic crystals, and they create an ideal environment for the generation and observation of localized modes in the form of solitons. These advantages could be employed to design very small all-opticallogical gates using readily available materials and perhaps combining several thousands of such devices on a chip of a few square centimeters. Such solitons can be viewed as an extension of the concepts of the discrete and gap solitons to two (and even three) spatial dimensions. Moreover, in the case of short pulses propagating inside photonic-crystal waveguides and circuits, the concept of such solitons can be further extended to the temporal domain.