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 longdistance limits [2]. Generally, the light can be considered as a superposition of plane waves A exp i k! � r ! � � � <sup>ω</sup><sup>t</sup> h i where <sup>k</sup> ! , ω, r !, t are the wave vector, angular frequency, radius vector in the space, and time, respectively [1, 2]. The medium oscillators can be electronic transitions, molecular vibrations and rotations, and acoustic waves [2]. Typically, only a small number of linear and nonlinear oscillator modes are important that satisfy the resonance conditions [1–3]. In such a case, the optical fields can be represented by a finite sum of discrete wave packets E ! ð Þ z; t given by [1–3]

$$\overrightarrow{E}\ (z,t) = \frac{1}{2} \left[ A(z,t) \exp\left(kz - \alpha t\right) + c.c.\right] \tag{1}$$

where c:c: stands for the complex conjugate and A zð Þ ; t is the slowly varying envelope (SVE) such that [1–3]

$$\left|\frac{\partial^2 A}{\partial x^2}\right| \ll \left|k \frac{\partial A}{\partial x}\right|; \left|\frac{\partial^2 A}{\partial t^2}\right| \ll \left|\alpha \frac{\partial A}{\partial t}\right|\tag{2}$$

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) [1–3]. 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].

form B !

[1, 2]:

1 ε0 Pj r !; t � � <sup>¼</sup>

¼ μ<sup>0</sup> H ! þ M !

! , M!

respectively. For nonmagnetic media M

Introductory Chapter: Nonlinear Optical Phenomena DOI: http://dx.doi.org/10.5772/intechopen.83718

!

respectively; and P

has the form [1–8]

nonlinear function of E

function of the electric field E

∞ð

�∞ χ ð Þ1 jk r ! �<sup>r</sup> 0 ! ; t � t 0 � �Ek <sup>r</sup>

�∞ χ ð Þ2 jkl r ! �r<sup>1</sup>

�∞ χ ð Þ3 jklm r ! �r<sup>1</sup>

� Emdr<sup>1</sup>

!; t

matic plane waves given by [1]

P ! k ! ;ω � � <sup>¼</sup> <sup>P</sup>

E ! r !; t � � <sup>¼</sup> <sup>∑</sup>

> ð Þ1 jk k ! ;ω � �Ek <sup>k</sup>

ð Þ2 jkl k !

ð Þ3 jkls k !

� Ek kn ! ;ω<sup>n</sup> � �El km

! dt1dr<sup>2</sup>

þ ∞ð

þ ∞ð

Here, χð Þ<sup>1</sup> r

where

Pð Þ<sup>1</sup> <sup>j</sup> k ! ;ω � � <sup>¼</sup> <sup>χ</sup>

Pð Þ<sup>2</sup> <sup>j</sup> k ! ;ω � � <sup>¼</sup> <sup>χ</sup>

Pð Þ<sup>3</sup> <sup>j</sup> k ! ;ω � � <sup>¼</sup> <sup>χ</sup>

5

; ε0, μ<sup>0</sup> are the free space permittivity and permeability,

!

∇ � ∇� E ! þ 1 c2

Here c is the free space light velocity. The polarization P

0 ! ; t 0 � �d r<sup>0</sup> ! dt<sup>0</sup>

! �r<sup>2</sup> !; t � t<sup>2</sup>

! �r<sup>2</sup>

! dt<sup>3</sup> <sup>þ</sup> …

� � is the linear susceptibility; <sup>χ</sup>ð Þ <sup>n</sup> <sup>r</sup>

n E0<sup>n</sup> !

!ð Þ<sup>1</sup> k ! ;ω � � <sup>þ</sup> <sup>P</sup>

> ! ;ω � �;

¼ kn þ km;ω ¼ ω<sup>n</sup> þ ω<sup>m</sup> � �Ek kn

> ! ;ω<sup>m</sup> � �Es kp

!; t � t2; r

nonlinear susceptibility [1]. Suppose that the electric field is a group of monochro-

kn ! ;ω<sup>n</sup> � � exp i kn

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

!ð Þ<sup>2</sup> k ! ;ω � � <sup>þ</sup> <sup>P</sup>

¼ kn þ km þ kp;ω ¼ ω<sup>n</sup> þ ω<sup>m</sup> þ ω<sup>p</sup> � �

> ! ;ω<sup>p</sup> � �

� �Ek <sup>r</sup><sup>1</sup>

� �Ek <sup>r</sup><sup>1</sup>

!

!; t � t1; r

!; t � t1; r

! dt2dr<sup>3</sup>

tors averaged over the volumes which contain many atoms but have linear dimensions smaller than substantial variations of the applied electric field [8]. Combining Eqs. (3)–(6) we obtain the wave equation for the light propagation in a medium. It

> ∂<sup>2</sup> E ! <sup>∂</sup>t<sup>2</sup> ¼ �μ<sup>0</sup>

are the induced electric and magnetic polarizations,

[1]. In the general nonlinear case, the polarization P

!; t<sup>1</sup> � �El <sup>r</sup><sup>2</sup>

! �r<sup>3</sup> !; t � t<sup>3</sup>

can be expanded into a power series of E

∂<sup>2</sup> P !

!; t<sup>2</sup> � �dr<sup>1</sup>

!; t

! � r ! � � � <sup>i</sup>ωnt

> !ð Þ<sup>3</sup> k ! ;ω

! ;ω<sup>n</sup> � �El km

!; t<sup>1</sup> � �El <sup>r</sup><sup>2</sup>

! dt1dr<sup>2</sup> ! dt<sup>2</sup>

� �, n . 1 is <sup>n</sup>th-order

h i (9)

! ;ω<sup>m</sup> � �;

� � <sup>þ</sup> … (10)

!; t<sup>2</sup> � �

!

¼ 0. Equations (3)–(6) describe the vec-

<sup>∂</sup>t<sup>2</sup> (7)

is a complicated

!

! as a

as follows

(8)

(11)

During the last decades, optical communications and optical signal processing have been rapidly developing [1–4]. In particular, the nonlinear optical effects in optical waveguides and fibers became especially important and attracted a wide interest [1–4]. 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, CO2 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 [1–7] 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.
