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 E ! r !; t and <sup>H</sup> ! r !; t [1–8]. They have the form [4]

$$\nabla \cdot \overrightarrow{B} = \mathbf{0} \tag{3}$$

$$
\nabla \cdot \vec{D} = \rho\_{\text{free}} \tag{4}
$$

$$
\nabla \times \overrightarrow{E} = -\frac{\partial \overrightarrow{B}}{\partial t} \tag{5}
$$

$$
\nabla \times \overrightarrow{H} = \overrightarrow{J} + \frac{\partial \overrightarrow{D}}{\partial t} \tag{6}
$$

Here ρfree is the free charge density consisting of all charges except the bound charges inside atoms and molecules; J ! is the current density; the electric induction is given by D ! ¼ ε<sup>0</sup> E ! þ P ! ; the magnetic induction (magnetic flux density) has the

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

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

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

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

2. Maxwell's equations for a nonlinear medium and nonlinear optical

All electromagnetic phenomena are described by macroscopic Maxwell's

∇� B !

∇� D !

∇� E !

∇� H ! ¼ J ! þ ∂D !

!

! r !; t 

¼ � <sup>∂</sup><sup>B</sup> !

Here ρfree is the free charge density consisting of all charges except the bound

and H ! r !; t 

¼ 0 (3)

¼ ρfree (4)

<sup>∂</sup><sup>t</sup> (5)

<sup>∂</sup><sup>t</sup> (6)

is the current density; the electric induction

; the magnetic induction (magnetic flux density) has the

[1–8]. They

light scattering (SLS), and four-wave mixing (FWM) [1].

Nonlinear Optics ‐ Novel Results in Theory and Applications

result in instabilities and transition to chaos [7].

Section 3. Conclusions are presented in Section 4.

equations for the electric and magnetic fields E

charges inside atoms and molecules; J

¼ ε<sup>0</sup> E ! þ P !

susceptibilities

have the form [4]

is given by D

4

!

form B ! ¼ μ<sup>0</sup> H ! þ M ! ; ε0, μ<sup>0</sup> are the free space permittivity and permeability, respectively; and P ! , M! are the induced electric and magnetic polarizations, respectively. For nonmagnetic media M ! ¼ 0. Equations (3)–(6) describe the vectors 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 has the form [1–8]

$$
\nabla \times \nabla \times \overrightarrow{E} + \frac{1}{c^2} \frac{\partial^2 \overrightarrow{E}}{\partial t^2} = -\mu\_0 \frac{\partial^2 \overrightarrow{P}}{\partial t^2} \tag{7}
$$

Here c is the free space light velocity. The polarization P ! is a complicated nonlinear function of E ! [1]. In the general nonlinear case, the polarization P ! as a function of the electric field E ! can be expanded into a power series of E ! as follows [1, 2]:

$$\begin{aligned} \frac{1}{\varepsilon\_{0}}P\_{\parallel}\left(\vec{r},t\right) &= \int\_{-\infty}^{\infty} \chi\_{\parallel}^{(1)}\left(\vec{r}\cdot\vec{r'},t-t'\right) E\_{k}\left(\vec{r'},t'\right) d\vec{r'}dt' \\ &+ \int\_{-\infty}^{\infty} \chi\_{\parallel}^{(2)}\left(\vec{r}\cdot\vec{r\_{1}},t-t\_{1};\stackrel{\cdot}{r}\,-\vec{r\_{2}},t-t\_{2}\right) E\_{k}\left(\vec{r\_{1}},t\_{1}\right) E\_{l}\left(\vec{r\_{2}},t\_{2}\right) d\vec{r\_{1}} \,dt\_{1} d\vec{r\_{2}} \,dt\_{2} \\ &+ \int\_{-\infty}^{\infty} \chi\_{\slash}^{(3)}\left(\vec{r}\cdot\vec{r\_{1}},t-t\_{1};\stackrel{\cdot}{r}\,-\vec{r\_{2}},t-t\_{2};\stackrel{\cdot}{r}\,-\vec{r\_{3}},t-t\_{3}\right) E\_{k}\left(\vec{r\_{1}},t\_{1}\right) E\_{l}\left(\vec{r\_{2}},t\_{2}\right) \\ &\times E\_{m}d\vec{r\_{1}} \,dt\_{1} d\vec{r\_{2}} \,dt\_{2} d\vec{r\_{3}} \,dt\_{3} + \dots \end{aligned} \tag{8}$$

Here, χð Þ<sup>1</sup> r !; t � � is the linear susceptibility; <sup>χ</sup>ð Þ <sup>n</sup> <sup>r</sup> !; t � �, n . 1 is <sup>n</sup>th-order nonlinear susceptibility [1]. Suppose that the electric field is a group of monochromatic plane waves given by [1]

$$\overrightarrow{E}\left(\overrightarrow{r},t\right) = \sum\_{n} \overrightarrow{E\_{0n}}\left(\overrightarrow{k\_{n}},\alpha\_{n}\right) \exp\left[i\left(\overrightarrow{k\_{n}}\cdot\overrightarrow{r}\right) - i\alpha\_{n}t\right] \tag{9}$$

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

$$
\overrightarrow{P}\left(\overrightarrow{k},\,\boldsymbol{\omega}\right) = \overrightarrow{P}^{(1)}\left(\overrightarrow{k},\,\boldsymbol{\omega}\right) + \overrightarrow{P}^{(2)}\left(\overrightarrow{k},\,\boldsymbol{\omega}\right) + \overrightarrow{P}^{(3)}\left(\overrightarrow{k},\,\boldsymbol{\omega}\right) + \dots \tag{10}
$$

where

$$\begin{aligned} P\_j^{(1)}\left(\overrightarrow{k},\,\boldsymbol{\alpha}\right) &= \chi\_{jk}^{(1)}\left(\overrightarrow{k},\,\boldsymbol{\alpha}\right) E\_k\left(\overrightarrow{k},\,\boldsymbol{\alpha}\right); \\\\ P\_j^{(2)}\left(\overrightarrow{k},\,\boldsymbol{\alpha}\right) &= \chi\_{jk}^{(2)}\left(\overrightarrow{k} = k\_n + k\_m, \boldsymbol{\alpha} = \boldsymbol{\alpha}\_n + \boldsymbol{\alpha}\_m\right) E\_k\left(\overrightarrow{k}\_n,\,\boldsymbol{\alpha}\_n\right) E\_l\left(\overrightarrow{k}\_m,\,\boldsymbol{\alpha}\_m\right); \\\\ P\_j^{(3)}\left(\overrightarrow{k},\,\boldsymbol{\alpha}\right) &= \chi\_{jk}^{(3)}\left(\overrightarrow{k} = k\_n + k\_m + k\_p, \boldsymbol{\alpha} = \boldsymbol{\alpha}\_n + \boldsymbol{\alpha}\_m + \boldsymbol{\alpha}\_p\right) \\\\ &\quad \times E\_k\left(\overrightarrow{k}\_n,\,\boldsymbol{\alpha}\_n\right) E\_l\left(\overrightarrow{k}\_m,\,\boldsymbol{\alpha}\_m\right) E\_s\left(\overrightarrow{k}\_p,\,\boldsymbol{\alpha}\_p\right) \end{aligned} \tag{11}$$

and

$$\begin{aligned} \chi^{(n)}\left(\overrightarrow{k} = \overrightarrow{k\_1} + \overrightarrow{k\_2} + \dots + \overrightarrow{k\_n}; \boldsymbol{\alpha} = \boldsymbol{\alpha}\_1 + \boldsymbol{\alpha}\_2 + \dots + \boldsymbol{\alpha}\_n\right) \\ \overset{\text{or}}{=}& \left\{ \chi^{(n)}\left(\overrightarrow{r} - \overrightarrow{r\_1}, t - t\_1; \dots; \overrightarrow{r} - \overrightarrow{r\_n}, t - t\_n\right) \right. \\ \left. \times \exp\left\{ -i\left[\left(\overrightarrow{k\_1} \cdot \left(\overrightarrow{r} - \overrightarrow{r\_1}\right)\right) - \boldsymbol{\alpha}\_1(t - t\_1) + \dots + \left(\overrightarrow{k\_n} \cdot \left(\overrightarrow{r} - \overrightarrow{r\_n}\right)\right) - \boldsymbol{\alpha}\_n(t - t\_n)\right] \right\} \right. \\ \left. \times \overrightarrow{d}\overrightarrow{r\_1} dt\_1 \dots d\overrightarrow{r\_n} dt\_n \end{aligned} \tag{12}$$

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 <sup>χ</sup>ð Þ<sup>2</sup> and <sup>χ</sup>ð Þ<sup>3</sup> , respectively. It should be noted that <sup>χ</sup>ð Þ<sup>2</sup> <sup>¼</sup> 0 in the electric dipole approximation for a medium with inversion symmetry [1].

We start with the sum-frequency, difference-frequency, and second harmonic

ð Þ ω<sup>3</sup> where ω<sup>3</sup> ¼ ω<sup>1</sup> þ ω<sup>2</sup> in the cases of sum-frequency [1]. The second-order nonlinear polarization with a sum-frequency ω<sup>3</sup> in such a case has the form [1]

ð Þ2

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

ð Þ2

The efficient nonlinear wave mixing can occur only under the phase-matching conditions. The phase mismatch Δk between the coupled waves is caused by the refractive index dispersion nð Þ ω<sup>i</sup> . The collinear phase matching Δk ¼ 0 can be realized in the medium with an anomalous dispersion or in the birefringent crystals [1]. The detailed analysis of the sum-frequency generation, difference-frequency generation, and SHG in different configurations may be found in [1, 3, 6]. It can be shown that the efficient sum-frequency generation can be realized under the following conditions [1]. The nonlinear optical crystal without the inversion symmetry or with the broken inversion symmetry should have low absorption at the interaction frequencies ω1, <sup>2</sup>,<sup>3</sup> and a sufficiently large quadratic susceptibility χð Þ<sup>2</sup> and should allow the collinear phase matching. The particular phase-matching direction and the coupled wave polarizations should be chosen in order to optimize the

ð Þ2

mode under the no-pump depletion approximation is given by [3]

<sup>P</sup>ð Þ <sup>2</sup><sup>ω</sup> ð Þ¼ <sup>L</sup> <sup>P</sup>ð Þ <sup>ω</sup>

where 2<sup>Δ</sup> <sup>¼</sup> <sup>β</sup>ð Þ <sup>2</sup><sup>ω</sup> � <sup>2</sup>βð Þ <sup>ω</sup> <sup>þ</sup> <sup>K</sup> ; <sup>K</sup> <sup>¼</sup> <sup>2</sup>π=λ; <sup>P</sup>ð Þ <sup>ω</sup>

0 <sup>2</sup>

coupling constant; L is the device length; Δ is the phase mismatch; λ is the pump wavelength; βð Þ <sup>ω</sup> , βð Þ <sup>2</sup><sup>ω</sup> are the propagation constants of the pump and SH waves, respectively; and Λ is the period of the quasi-phase matching (QPM) grating. Waveguide SHG devices can be used in optical signal processing such as laser

k2

<sup>L</sup><sup>2</sup> sin <sup>Δ</sup><sup>L</sup> ΔL <sup>2</sup>

the single-mode laser beams focused into the nonlinear optical crystal [1].

provide the required conversion efficiency. The efficient SHG can be realized with

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 LiNbO3 and LiTaO3, and III-V semiconductor materials GaAs and AlAs can be used as a waveguiding layer [3]. The output power <sup>P</sup>ð Þ <sup>2</sup><sup>ω</sup> ð Þ <sup>L</sup> of the second harmonic (SH)

where the asterisk means the complex conjugation. Consider the particular case of equal frequencies ω<sup>1</sup> ¼ ω<sup>2</sup> ¼ ω. In such a case, the nonlinear polarization (15) has

<sup>j</sup> ð Þ ω<sup>3</sup> ¼ 2ω , and the second harmonic generation (SHG) takes place [1].

!

jklð Þ ω<sup>3</sup> ¼ ω<sup>1</sup> þ ω<sup>2</sup> Ekð Þ ω<sup>1</sup> Elð Þ ω<sup>2</sup> (15)

jklð Þ ω<sup>2</sup> ¼ ω<sup>3</sup> � ω<sup>1</sup> Ekð Þ ω<sup>3</sup> El

eff . The length of the nonlinear crystal must

ð Þ ω<sup>1</sup> , E !

∗

ð Þ ω<sup>2</sup> , and

ð Þ ω<sup>2</sup> (16)

(17)

<sup>0</sup> is the input pump power; k is the

generation. These phenomena are based on the wave mixing by means of the

quadratic susceptibility χð Þ<sup>2</sup> . The three coupled waves are E

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

ð Þ<sup>2</sup> <sup>ð</sup>ω<sup>3</sup> <sup>¼</sup> <sup>ω</sup><sup>1</sup> <sup>þ</sup> <sup>ω</sup>2Þ ¼ <sup>ε</sup>0<sup>χ</sup>

ð Þ<sup>2</sup> <sup>ð</sup>ω<sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>3</sup> � <sup>ω</sup>1Þ ¼ <sup>ε</sup>0<sup>χ</sup>

E !

Pj

Pj

effective nonlinear susceptibility χ

the form Pð Þ<sup>2</sup>

7

The linear and nonlinear optical properties of a medium are described by the linear and nonlinear susceptibilities (12), and the nth-order nonlinear optical effects in such a medium can be obtained theoretically from Maxwell's Eqs. (3)–(6) with the polarization determined by Eq. (8) [1]. We do not present here the analytical properties of the nonlinear susceptibilities which are discussed in detail in Ref. [1].

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 N classical anharmonic oscillators per unit volume [1]. Such an oscillator may describe an electron bound to a core or an infrared-active molecular vibration [1]. The equation of motion of the oscillator in the presence of an applied electric field with the Fourier components at frequencies �ω1, � ω<sup>2</sup> is given by [1]

$$\frac{d^2\mathbf{x}}{dt^2} + \Gamma \frac{d\mathbf{x}}{dt} + \alpha\_0^2 \mathbf{x} + a\mathbf{x}^2 = \frac{q}{m} \left[ E\_1 (e^{-i\alpha \mathbf{y}t} + e^{i\alpha \mathbf{y}t}) + E\_2 (e^{-i\alpha \mathbf{y}t} + e^{i\alpha \mathbf{y}t}) \right] \tag{13}$$

Here x is the oscillator displacement; Γ is the decay factor; ω<sup>0</sup> is the oscillator frequency; q, m are the oscillator charge and mass, respectively; and the anharmonic term ax<sup>2</sup> is small and can be considered as a perturbation in the successive approximation series given by [1, 8]

$$\mathfrak{x} = \mathfrak{x}^{(1)} + \mathfrak{x}^{(2)} + \mathfrak{x}^{(3)} + \dots \tag{14}$$

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 x<sup>2</sup> and x3, respectively [1–8]. The induced electric polarization P can be expressed by using the solutions (13) and (14) as follows: P ¼ Nqx [1]. In general case, the microscopic expressions for nonlinear susceptibilities of a medium are calculated by using the quantum mechanical approach. In particular, the density matrix formalism is a powerful and convenient tool for such calculations [1, 2, 7, 8].
