**1. Introduction**

The observation of nonlinear optical effects started with the discovery of the laser in 1960 as a source of the high-intensity coherent radiation [1, 2]. Nonlinear optics is the study of the nonlinear light and matter interaction effects such as the light self-focusing and self-trapping, second harmonic generation (SHG), thirdharmonic generation (THG), four-wave mixing (FWM), parametric processes, different types of stimulated light scattering (SLS), soliton generation and propagation [1, 2]. The nonlinear optical phenomena are widely used in modern communication systems for such applications as the generation of ultra-short pulses, all-optical signal processing and ultrafast switching [2, 3]. Recently, new fields of nonlinear optics emerged such as strong-field nano-optics, nonlinear plasmonics, and nonlinear metamaterials [3–8].

In the case of the strong-field nano-optics the optical field interacts with matter at the wavelength or even at the subwavelength scale [5]. Strong electromagnetic fields of a light wave with the wavelength *λ* 1*μm* interacting with electrons are confined to a few nanometers which result in a substantial enhancement of the local electric field [4]. Typically, an increase of the nonlinear optical response at a nanoscale is caused by the plasmonic effects, i.e., the coherent oscillations of the conduction electrons near the surface of noble-metal structures [3–6]. At the extended metal surfaces, the surface plasmon polaritons (SPPs) can occur [3]. SPP is a surface electromagnetic wave propagating at the metal-dielectric interface [3–5]. In the case of the metal nanoparticles there exist localized surface plasmons (LSPs) with resonances depending on the nanoparticle size and shape [3]. Nonlinear optical effects can be significantly enhanced by plasmonic excitations in two ways: (i) the plasmonic structures provide the optical field enhancement near the metal-dielectric interface due to SPPs or LSPs; (ii) the SPP and LSP parameters are very sensitive to refractive indices of the metal and the surrounding dielectric medium [3–6]. Plasmonic excitations are characterized by the timescale of several femtoseconds which makes it possible the ultrafast optical signal processing [3].

Metamaterials are artificial materials with desirable properties [7]. For this reason, the magnitude of the metamaterial optical nonlinearity can be substantially increased [7]. Generally, all fundamental nonlinear optical phenomena such as selfaction effects, wavelength conversion, nonlinear surface waves, nonlinear guided waves and solitons are possible in nonlinear metamaterials [7]. The nonlinear metamaterials are closely related to plasmonics, active media, and nonlinearity based on liquid crystals (LCs) [7, 8]. LCs possess simultaneously the properties of a liquid and a crystal [8–10]. Thermotropic LCs self-assemble in a different ordered

arrangements of their crystalline axis depending on the temperature [8–10]. The orientational order of LC may be changed by a moderate external electric field [8–10]. For this reason, liquid crystalline media are characterized by strong optical nonlinearity [8–10]. Nematic LCs (NLCs) characterized by the ordering of molecular elongated axes are mainly used in applications [8].

In this chapter, we discuss the fundamentals of plasmonics and the specific features of the nonlinear optical effects in plasmonic nanostructures. The chapter is organized as follows. In Section 2 the interaction of the electromagnetic field with the free electrons in metals is described based on Maxwell's equations and equation of motion for a free electron in the external electric field. The dielectric function of the free electron gas is obtained based on the Drude model. In Section 3 SPPs at the interface of a metal and a dielectric are investigated. In Section 4 the nonlinear phenomena in the plasmonic structures are briefly discussed. The details of this topic may be found in the corresponding references. Conclusions are presented in Section 5.
