## 1. Introduction

The number of publications concerning different aspects of nonlinear optics is enormous and hardly observable. We briefly discuss in this chapter the fundamental nonlinear optical phenomena and methods of their analysis. Nonlinear optics is related to the analysis of the nonlinear interaction between light and matter when the light-induced changes of the medium optical properties occur [1, 2]. The nonlinear optical effects are weak, and their observation became possible only after the invention of lasers which provide a highly coherent and intense radiation [2]. A typical nonlinear optical process consists of two stages. First, the intense coherent light induces a nonlinear response of the medium, and then the modified medium influences the optical radiation in a nonlinear way [1]. The nonlinear medium is described by a system of the dynamic equations including the optical field. The optical field itself is described by Maxwell’s equations including the nonlinear polarization of the medium [1, 2]. All media are essentially nonlinear; however, the nonlinear coupling coefficients are usually very small and can be enhanced by the sufficiently strong optical radiation [1, 2]. For this reason, to a first approximation, light and matter can be considered as a system of uncoupled oscillators, and the nonlinear terms are some orders of magnitude smaller than the linear ones [2]. Nevertheless, the nonlinear effects can be important in the long-time and long-distance limits [2]. Generally, the light can be considered as a superposition of plane waves

where

Here we for the sake of definiteness consider the one-dimensional case. The evolution of the waves (1) is described by the system of the coupled equations in the so-called SVE approximation (SVEA) when the higher-order derivatives of the SVE can be neglected according to conditions (2) [1, 2, 3]. The typical nonlinear optical phenomena are self-focusing, self-trapping, sum- and difference-frequency generation, harmonic generation, parametric amplification and oscillation, stimulated light scattering (SLS), and four-wave mixing (FWM) [1].

During the last decades, optical communications and optical signal processing have been rapidly developing [1, 2, 3, 4]. In particular, the nonlinear optical effects in optical waveguides and fibers became especially important and attracted a wide interest [1, 2, 3, 4]. The nonlinear optical interactions in the waveguide devices have been investigated in detail in Ref. [3]. Nonlinear fiber optics as a separate field of nonlinear optics has been reviewed in Ref. [4]. The self-phase modulation (SPM), cross-phase modulation (XPM), FWM, stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), pulse propagation, and optical solitons in optical fibers have been considered in detail [4]. Silicon photonics, i.e., integrated optics in silicon, also attracted a wide interest due to the highly developed silicon technology which permits the combination of the photonic and electronic devices on the same Si platform [5]. The nonlinear optical phenomena in Si nanostructures such as quantum dots (QD), quantum wells (QW), and superlattices had been discussed [6]. It has been shown that the second harmonic generation (SHG) in silicon nanostructures is possible despite the centrosymmetric structure of Si crystals [6].

Nonlinear dynamics in complex optical systems such as solid-state lasers, CO_{2} lasers, and semiconductor lasers is caused by the light-matter interaction [7]. Under certain conditions, the nonlinear optical processes in such optical complex systems result in instabilities and transition to chaos [7].

In this chapter we briefly describe the basic nonlinear optical phenomena. The detailed analysis of these phenomena may be found in [1, 2, 3, 4, 5, 6, 7] and references therein. The chapter is constructed as follows. Maxwell’s equations for a nonlinear medium and nonlinear optical susceptibilities are considered in Section 2. The mechanisms and peculiarities of the basic nonlinear effects mentioned above are discussed in Section 3. Conclusions are presented in Section 4.

## 2. Maxwell’s equations for a nonlinear medium and nonlinear optical susceptibilities

All electromagnetic phenomena are described by macroscopic Maxwell’s equations for the electric and magnetic fields

Here

Here

Here,

Then, the Fourier transform of the nonlinear polarization (1) yields [1]

where

and

The linear and nonlinear optical properties of a medium are described by the linear and nonlinear susceptibilities (12), and the

In some simple cases, the nonlinear susceptibilities can be evaluated by using the anharmonic oscillator model [1, 8]. It is assumed that a medium consists of

Here

The nonlinear terms become essential when the electromagnetic power is large enough in such a way that a medium response cannot be considered linear anymore [8]. We limit our analysis with quadratic and cubic nonlinearities proportional to

## 3. Nonlinear optical effects

Electromagnetic waves in a medium interact through the nonlinear polarization (8) [1]. Typically, a nonlinear optical effect that occurs due to such an interaction is described by the coupled wave equations of the type (7) with the nonlinear susceptibilities (12) as the coupling coefficients [1]. In general case, the coupled wave method can also include waves other than electromagnetic [1]. For instance, in the case of SBS process, the acoustic waves are taken into account, and in the case of SRS process, the molecular vibrations are typically considered [1, 2, 4]. The coupled wave equations are usually solved by using SVEA (2) [1]. In this section, we discuss some important nonlinear optical phenomena caused by the quadratic and cubic susceptibilities

We start with the sum-frequency, difference-frequency, and second harmonic generation. These phenomena are based on the wave mixing by means of the quadratic susceptibility

Similarly, in the case of the difference-frequency generation, we obtain [1]

where the asterisk means the complex conjugation. Consider the particular case of equal frequencies

Sum-frequency generation, difference-frequency generation, and SHG can be also carried out in the waveguide nonlinear optical devices [3]. Typically, a thin film of a nonlinear material such as ZnO and ZnS, ferroelectric materials LiNbO_{3} and LiTaO_{3}, and III-V semiconductor materials GaAs and AlAs can be used as a waveguiding layer [3]. The output power

where

Consider the nonlinear optical effects related to the cubic susceptibility

Self-focusing is an induced lens effects caused by the self-induced wavefront distortion of the optical beam propagating in the nonlinear medium [1]. In such a medium, a refractive index

Here

SPM is also caused by the positive refractive index change (18). It is the temporal analog of self-focusing which leads to the spectral broadening of optical pulses [4]. In optical fibers, for short pulses and sufficiently large fiber length

In the normal-dispersion regime when

Here

Consider now THG. Unlike SHG, it is always allowed [1]. The third harmonic

The cubic susceptibility

SBS is a nonlinear optical effect related to parametric coupling between light and acoustic waves [1]. It is described by the coupled wave equation (7) for the coupled counterpropagating light waves

Consider now the SRS process. SRS can be described in the framework of the quantum mechanics as a two-photon process where one photon with energy

In the framework of the coupled wave description, SRS is a third-order parametric generation process where the optical pump wave

where

FWM is the nonlinear process with four interacting electromagnetic waves [1]. FWM is a third-order process caused by the third-order nonlinear susceptibility

## 4. Conclusions

We briefly discussed the fundamentals of nonlinear optics. The nonlinear optical phenomena are caused by the interaction between light and matter. Generally, all media are nonlinear. However, optical nonlinearity is extremely weak, and the observation of the nonlinear optical effects became possible only after invention of lasers as the sources of the strong enough coherent optical radiation. The nonlinear optical processes are described by Maxwell’s equations with the nonlinear polarization of the medium. The coupled equations for the interacting electromagnetic and material waves are usually solved by using SVEA. Typically, the second- and third-order polarizations are considered. The nonlinear polarization and the optical field in the medium are related by the nonlinear susceptibilities which in general case can be evaluated by the quantum mechanical methods. In some simple cases, the classical model of anharmonic oscillator also can be used. We briefly discussed the fundamental nonlinear phenomena related to the second- and third-order susceptibilities. The former exists only in the media without the inversion symmetry, while the latter exists in any medium.

The typical nonlinear optical phenomena related to the second-order susceptibility are the sum-frequency generation, difference-frequency generation, and SHG. The typical nonlinear optical phenomena related to the third-order susceptibility are self-focusing, SPM, optical soliton formation and propagation, different types of SLS such as SBS and SRS, and FWM. SBS involves the acoustic waves. SRS involves the material excitations such as molecular vibrations. We also discussed some peculiarities of nonlinear optical processes in optical fibers. The nonlinear optical effects are widely used in optical communications and optical signal processing.