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

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

and

¼ ∞ð

χð Þ <sup>n</sup> k ! ¼ k<sup>1</sup> ! þk<sup>2</sup> !

�∞

� d r !

detail in Ref. [1].

d2 x dt<sup>2</sup> <sup>þ</sup> <sup>Γ</sup>

dx dt <sup>þ</sup> <sup>ω</sup><sup>2</sup>

imation series given by [1, 8]

3. Nonlinear optical effects

6

χð Þ <sup>n</sup> r ! �r<sup>1</sup>

� exp �i k<sup>1</sup>

<sup>1</sup>dt1…d r ! ndtn

! � r ! �r<sup>1</sup> ! � � � �

þ… þ kn !

Nonlinear Optics ‐ Novel Results in Theory and Applications

!; t � t1; …; r

� �

� �

! �rn

;ω ¼ ω<sup>1</sup> þ ω<sup>2</sup> þ … þ ω<sup>n</sup>

� ω1ð Þþ t � t<sup>1</sup> … þ kn

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

In some simple cases, the nonlinear susceptibilities can be evaluated by using the

�iω1<sup>t</sup> <sup>þ</sup> <sup>e</sup> <sup>i</sup>ω1<sup>t</sup> � � <sup>þ</sup> <sup>E</sup><sup>2</sup> <sup>e</sup>

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 approx-

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 formal-

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

�iω2<sup>t</sup> <sup>þ</sup> <sup>e</sup> <sup>i</sup>ω2<sup>t</sup> � � � � (13)

<sup>x</sup> <sup>¼</sup> <sup>x</sup>ð Þ<sup>1</sup> <sup>þ</sup> <sup>x</sup>ð Þ<sup>2</sup> <sup>þ</sup> <sup>x</sup>ð Þ<sup>3</sup> <sup>þ</sup> … (14)

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]

m E<sup>1</sup> e

ism is a powerful and convenient tool for such calculations [1, 2, 7, 8].

<sup>0</sup><sup>x</sup> <sup>þ</sup> ax<sup>2</sup> <sup>¼</sup> <sup>q</sup>

n o h i

! � r ! �rn ! � � � �

� ωnð Þ t � tn

(12)

!; t � tn

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 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 ! ð Þ ω<sup>1</sup> , E ! ð Þ ω<sup>2</sup> , and E ! 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]

$$P\_j^{(2)}(a\nu\_3 = a\_1 + a\_2) = \varepsilon\_0 \chi\_{jkl}^{(2)}(a\nu\_3 = a\_1 + a\_2) E\_k(a\nu\_1) E\_l(a\nu\_2) \tag{15}$$

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

$$P\_j^{(2)}(
\alpha\_2 = 
\alpha\_3 - 
\alpha\_1) = 
\varepsilon\_0 \chi\_{jkl}^{(2)}(
\alpha\_2 = 
\alpha\_3 - 
\alpha\_1) E\_k(
\alpha\_3) E\_l^\*(
\alpha\_2) \tag{16}$$

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 the form Pð Þ<sup>2</sup> <sup>j</sup> ð Þ ω<sup>3</sup> ¼ 2ω , and the second harmonic generation (SHG) takes place [1]. 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 effective nonlinear susceptibility χ ð Þ2 eff . The length of the nonlinear crystal must provide the required conversion efficiency. The efficient SHG can be realized with the single-mode laser beams focused into the nonlinear optical crystal [1].

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

$$P^{(2\alpha)}(L) = \left(P\_0^{(o)}\right)^2 k^2 L^2 \left(\frac{\sin \Delta L}{\Delta L}\right)^2\tag{17}$$

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

printer, laser display, optical memory, short pulse, multicolor, and ultraviolet light generation [3].

Consider the nonlinear optical effects related to the cubic susceptibility χð Þ<sup>3</sup> . These phenomena are much weaker than the second-order ones. However, they can exist in centrosymmetric media where <sup>χ</sup>ð Þ<sup>2</sup> <sup>¼</sup> 0 and may be strongly pronounced under the high enough optical intensity pumping. We briefly discuss self-focusing, SPM, third harmonic generation (THG), SBS, SRS, and FWM.

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 n has the form [1]

$$m = n\_0 + \Delta n \left( \left| E \right|^2 \right) \tag{18}$$

Pj

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

The cubic susceptibility χð Þ<sup>3</sup>

counterpropagating light waves E

ð Þ� ω<sup>1</sup> E !<sup>∗</sup> <sup>2</sup>ð Þ ω<sup>2</sup>

[1]. The pumping wave E<sup>1</sup>

place: ℏð Þ¼ ω<sup>1</sup> � ω<sup>2</sup> Ef � Ei [1].

ð Þ¼ ω<sup>1</sup> ε0χ

ð Þ3 <sup>R</sup><sup>1</sup> j j E<sup>2</sup> 2 E1 !

results in the generation of the anti-Stokes wave Ea

p � ρ<sup>0</sup>

ℏω<sup>1</sup> k<sup>1</sup> ! Þ 

wave E<sup>2</sup> !

> P !ð Þ<sup>3</sup>

> > ð Þ3

where χ

P !ð Þ<sup>3</sup>

9

∂εr <sup>∂</sup><sup>ρ</sup> E<sup>1</sup> !

ð Þ<sup>3</sup> ð Þ¼ <sup>3</sup><sup>ω</sup> <sup>ε</sup>0<sup>χ</sup>

!

light and acoustic waves is caused by the electrostrictive pressure

excitation is overdamped, and the signal Stokes wave E<sup>2</sup>

metric generation process where the optical pump wave E<sup>1</sup>

ð Þ ω1, <sup>2</sup> related to SRS in such a case takes the form [1, 2]

backward direction �z under the conditions that Im χ

!

where the selective amplification is needed [4].

ð Þ3

reason, the laser intensity required for the efficient THG is limited by the optical damage in crystals [1]. The phase matching for the THG is difficult to achieve which results in low efficiency of the THG process [1, 4]. THG can be realized in highly nonlinear optical fibers where the phase matching can be accomplished [4].

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

mass density variation Δρ ωð Þ <sup>a</sup> ¼ ω<sup>1</sup> � ω<sup>2</sup> [1, 2, 4]. The nonlinear coupling between

and permittivity, respectively. The acoustic wave enhanced by the interacting pump and signal (Stokes) wave modulates the mass density of the medium which in turn modulates the refractive index [1, 3, 4]. For the typical values of the attenuation coefficient and the acoustic frequency shift of about 5 GHz, the acoustic wave

ω<sup>1</sup> � ω<sup>2</sup> . 0, and the optical gain is larger than the optical wave damping constant

been successfully demonstrated in optical fibers, and the SBS gain in a fiber can be used for the amplification of the weak signal with the frequency shift equal to the acoustic frequency ω<sup>a</sup> [4]. Brillouin fiber amplifiers may be used for applications

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

is absorbed by the system and another photon with energy ℏω<sup>2</sup> k<sup>2</sup>

In the framework of the coupled wave description, SRS is a third-order para-

ð Þ ω<sup>2</sup> and a material excitation wave [1]. The nonlinear polarization

ð Þ¼ ω<sup>2</sup> ε0χ

<sup>R</sup>1,<sup>2</sup> are the third-order Raman susceptibilities coupling the optical waves

!

ð Þ <sup>ω</sup><sup>1</sup> , P!ð Þ<sup>3</sup>

and providing SRS process [1]. They can be evaluated by using the quantum mechanical methods [1]. Typically, the material excitation wave in the SRS process is considered as molecular vibrations or optical phonons [1, 2, 4]. The specific feature of SRS is the so-called Stokes-anti-Stokes coupling [1, 2]. Indeed, the mixing of the pump wave with the frequency ω<sup>1</sup> and the Stokes wave with the frequency ω<sup>2</sup>

Stokes frequency ω<sup>a</sup> ¼ 2ω<sup>1</sup> � ω<sup>2</sup> . ω<sup>1</sup> [1]. Consequently, the coupled wave analysis of SRS should include the equations for the pump wave, Stokes wave, anti-Stokes

emitted [1]. The system itself makes a transition from the initial state with the energy Ei to the final state with the energy Ef , and the energy conservation takes

where <sup>ρ</sup>0, <sup>ε</sup><sup>r</sup> are the equilibrium medium mass density

jklmð Þ 3ω Ekð Þ ω Elð Þ ω Emð Þ ω (21)

<sup>1</sup>,2ð Þ ω1, <sup>2</sup> and the acoustic wave equation for the

!

ð Þ3

ð Þ ω<sup>1</sup> is decaying in the forward direction z [1]. SBS has

!

ð Þ3 <sup>R</sup><sup>2</sup> j j E<sup>1</sup> 2 E2 !

ð Þ ω<sup>2</sup> would grow in the

! Þ 

ð Þ ω<sup>1</sup> generates a Stokes

ð Þ ω<sup>2</sup> (22)

ð Þ ω<sup>a</sup> ¼ 2ω<sup>1</sup> � ω<sup>2</sup> at the anti-

is

<sup>B</sup> . 0, ω<sup>1</sup> ≫ ω<sup>a</sup> ¼

[1]. For this

is usually small compared to the <sup>χ</sup>ð Þ<sup>2</sup>

Here <sup>n</sup><sup>0</sup> is the refractive index of the unperturbed medium, <sup>Δ</sup>n Ej j<sup>2</sup> � � is the optical field-induced refractive index change, and E is the optical beam electric field. Typically, the field-induced refractive index change can be described as <sup>Δ</sup><sup>n</sup> <sup>¼</sup> <sup>n</sup>2j j <sup>E</sup> <sup>2</sup> like in the case of the so-called Kerr nonlinearity [1, 3]. If <sup>Δ</sup><sup>n</sup> . 0, the central part of the optical beam with a higher intensity has a larger refractive index than the beam edge. Consequently, the central part of the beam travels at a smaller velocity than the beam edge. As a result, the gradual distortion of the original plane wavefront of the beam occurs, and the beam appears to focus by itself [1]. The selffocusing results in the local increase of the optical power in the central part of the beam and possible optical damage of transparent materials limiting the high-power laser performance [1].

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 Lf , the combined effect of the group velocity dispersion (GVD) and SPM should be taken into account [4]. The GVD parameter β<sup>2</sup> is given by [4]

$$\beta\_2 = \frac{1}{c} \left( 2 \frac{dn}{d\alpha} + \alpha \frac{d^2 n}{d\alpha^2} \right) \tag{19}$$

In the normal-dispersion regime when β<sup>2</sup> . 0, the combined effect of the SPM and GVD leads to a pulse compression. In the opposite case of the anomalousdispersion regime β<sup>2</sup> , 0, SPM and GVD under certain conditions can be mutually compensated [4]. In such a case, the pulse propagates in the optical fiber as an optical soliton, i.e., a solitary wave which does not change after mutual collisions [4]. The solitons are described with the nonlinear Schrödinger equation (NLS) which can be solved with the inverse scattering method [4]. The fundamental soliton solution uð Þ ξ; τ has the form [4]

$$\mu(\xi,\tau) = \eta \text{[cosh}\,(\eta\tau)]^{-1} \exp\left(i\eta^2\xi/2\right) \tag{20}$$

Here η is the soliton amplitude; τ ¼ t � β<sup>1</sup> ð Þz =T0; ξ ¼ z=LD; β<sup>1</sup> ¼ 1=vg; vg is the light group velocity in the optical fiber; LD is the dispersion length; and T<sup>0</sup> is the initial width of the incident pulse. The optical solitons can propagate undistorted over long distances, and they can be applied in fiber-optic communications [4].

Consider now THG. Unlike SHG, it is always allowed [1]. The third harmonic E ! ð Þ 3ω is caused by the third-order nonlinear polarization given by [1, 2]

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

printer, laser display, optical memory, short pulse, multicolor, and ultraviolet light

Consider the nonlinear optical effects related to the cubic susceptibility χð Þ<sup>3</sup> . These phenomena are much weaker than the second-order ones. However, they can exist in centrosymmetric media where <sup>χ</sup>ð Þ<sup>2</sup> <sup>¼</sup> 0 and may be strongly pronounced under the high enough optical intensity pumping. We briefly discuss self-focusing,

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

<sup>n</sup> <sup>¼</sup> <sup>n</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>n Ej j<sup>2</sup> � �

Here <sup>n</sup><sup>0</sup> is the refractive index of the unperturbed medium, <sup>Δ</sup>n Ej j<sup>2</sup> � �

optical field-induced refractive index change, and E is the optical beam electric field. Typically, the field-induced refractive index change can be described as <sup>Δ</sup><sup>n</sup> <sup>¼</sup> <sup>n</sup>2j j <sup>E</sup> <sup>2</sup> like in the case of the so-called Kerr nonlinearity [1, 3]. If <sup>Δ</sup><sup>n</sup> . 0, the central part of the optical beam with a higher intensity has a larger refractive index than the beam edge. Consequently, the central part of the beam travels at a smaller velocity than the beam edge. As a result, the gradual distortion of the original plane wavefront of the beam occurs, and the beam appears to focus by itself [1]. The selffocusing results in the local increase of the optical power in the central part of the beam and possible optical damage of transparent materials limiting the high-power

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 Lf , the combined effect of the group velocity dispersion (GVD) and SPM should be taken

> <sup>2</sup> dn <sup>d</sup><sup>ω</sup> <sup>þ</sup> <sup>ω</sup> <sup>d</sup><sup>2</sup>

In the normal-dispersion regime when β<sup>2</sup> . 0, the combined effect of the SPM and GVD leads to a pulse compression. In the opposite case of the anomalousdispersion regime β<sup>2</sup> , 0, SPM and GVD under certain conditions can be mutually compensated [4]. In such a case, the pulse propagates in the optical fiber as an optical soliton, i.e., a solitary wave which does not change after mutual collisions [4]. The solitons are described with the nonlinear Schrödinger equation (NLS) which can be solved with the inverse scattering method [4]. The fundamental

Here η is the soliton amplitude; τ ¼ t � β<sup>1</sup> ð Þz =T0; ξ ¼ z=LD; β<sup>1</sup> ¼ 1=vg; vg is the light group velocity in the optical fiber; LD is the dispersion length; and T<sup>0</sup> is the initial width of the incident pulse. The optical solitons can propagate undistorted over long distances, and they can be applied in fiber-optic communications [4]. Consider now THG. Unlike SHG, it is always allowed [1]. The third harmonic

!

n dω<sup>2</sup>

exp iη<sup>2</sup>

ξ=2 � � (20)

(18)

(19)

is the

SPM, third harmonic generation (THG), SBS, SRS, and FWM.

into account [4]. The GVD parameter β<sup>2</sup> is given by [4]

soliton solution uð Þ ξ; τ has the form [4]

E !

8

<sup>β</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> c

<sup>u</sup>ð Þ¼ <sup>ξ</sup>; <sup>τ</sup> <sup>η</sup>½ � cosh ð Þ ητ �<sup>1</sup>

ð Þ 3ω is caused by the third-order nonlinear polarization given by [1, 2]

medium, a refractive index n has the form [1]

Nonlinear Optics ‐ Novel Results in Theory and Applications

generation [3].

laser performance [1].

$$P\_j^{(3)}(\mathfrak{Z}o) = \varepsilon\_0 \chi\_{jklm}^{(3)}(\mathfrak{Z}o) E\_k(o) E\_l(o) E\_m(o) \tag{21}$$

The cubic susceptibility χð Þ<sup>3</sup> is usually small compared to the <sup>χ</sup>ð Þ<sup>2</sup> [1]. For this reason, the laser intensity required for the efficient THG is limited by the optical damage in crystals [1]. The phase matching for the THG is difficult to achieve which results in low efficiency of the THG process [1, 4]. THG can be realized in highly nonlinear optical fibers where the phase matching can be accomplished [4].

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 E ! <sup>1</sup>,2ð Þ ω1, <sup>2</sup> and the acoustic wave equation for the mass density variation Δρ ωð Þ <sup>a</sup> ¼ ω<sup>1</sup> � ω<sup>2</sup> [1, 2, 4]. The nonlinear coupling between light and acoustic waves is caused by the electrostrictive pressure

p � ρ<sup>0</sup> ∂εr <sup>∂</sup><sup>ρ</sup> E<sup>1</sup> ! ð Þ� ω<sup>1</sup> E !<sup>∗</sup> <sup>2</sup>ð Þ ω<sup>2</sup> where <sup>ρ</sup>0, <sup>ε</sup><sup>r</sup> are the equilibrium medium mass density and permittivity, respectively. The acoustic wave enhanced by the interacting pump and signal (Stokes) wave modulates the mass density of the medium which in turn modulates the refractive index [1, 3, 4]. For the typical values of the attenuation coefficient and the acoustic frequency shift of about 5 GHz, the acoustic wave excitation is overdamped, and the signal Stokes wave E<sup>2</sup> ! ð Þ ω<sup>2</sup> would grow in the backward direction �z under the conditions that Im χ ð Þ3 <sup>B</sup> . 0, ω<sup>1</sup> ≫ ω<sup>a</sup> ¼ ω<sup>1</sup> � ω<sup>2</sup> . 0, and the optical gain is larger than the optical wave damping constant [1]. The pumping wave E<sup>1</sup> ! ð Þ ω<sup>1</sup> is decaying in the forward direction z [1]. SBS has been successfully demonstrated in optical fibers, and the SBS gain in a fiber can be used for the amplification of the weak signal with the frequency shift equal to the acoustic frequency ω<sup>a</sup> [4]. Brillouin fiber amplifiers may be used for applications where the selective amplification is needed [4].

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 ℏω<sup>1</sup> k<sup>1</sup> ! Þ is absorbed by the system and another photon with energy ℏω<sup>2</sup> k<sup>2</sup> ! Þ is emitted [1]. The system itself makes a transition from the initial state with the energy Ei to the final state with the energy Ef , and the energy conservation takes place: ℏð Þ¼ ω<sup>1</sup> � ω<sup>2</sup> Ef � Ei [1].

In the framework of the coupled wave description, SRS is a third-order parametric generation process where the optical pump wave E<sup>1</sup> ! ð Þ ω<sup>1</sup> generates a Stokes wave E<sup>2</sup> ! ð Þ ω<sup>2</sup> and a material excitation wave [1]. The nonlinear polarization P !ð Þ<sup>3</sup> ð Þ ω1, <sup>2</sup> related to SRS in such a case takes the form [1, 2]

$$
\overrightarrow{\boldsymbol{P}}^{(3)}(o\_1) = \varepsilon\_0 \chi\_{R1}^{(3)} |\boldsymbol{E}\_2|^2 \overrightarrow{\boldsymbol{E}}\_1(o\_1), \ \overrightarrow{\boldsymbol{P}}^{(3)}(o\_2) = \varepsilon\_0 \chi\_{R2}^{(3)} |\boldsymbol{E}\_1|^2 \overrightarrow{\boldsymbol{E}}\_2(o\_2) \tag{22}
$$

where χ ð Þ3 <sup>R</sup>1,<sup>2</sup> are the third-order Raman susceptibilities coupling the optical waves and providing SRS process [1]. They can be evaluated by using the quantum mechanical methods [1]. Typically, the material excitation wave in the SRS process is considered as molecular vibrations or optical phonons [1, 2, 4]. The specific feature of SRS is the so-called Stokes-anti-Stokes coupling [1, 2]. Indeed, the mixing of the pump wave with the frequency ω<sup>1</sup> and the Stokes wave with the frequency ω<sup>2</sup> results in the generation of the anti-Stokes wave Ea ! ð Þ ω<sup>a</sup> ¼ 2ω<sup>1</sup> � ω<sup>2</sup> at the anti-Stokes frequency ω<sup>a</sup> ¼ 2ω<sup>1</sup> � ω<sup>2</sup> . ω<sup>1</sup> [1]. Consequently, the coupled wave analysis of SRS should include the equations for the pump wave, Stokes wave, anti-Stokes

wave, and the material excitation wave [1, 2]. The analysis of this problem can be found in Refs. [1, 2]. Usually, the anti-Stokes wave is attenuated [2]. SRS in optical fibers can be used for the development of Raman fiber lasers and Raman fiber amplifiers [4].

FWM is the nonlinear process with four interacting electromagnetic waves [1]. FWM is a third-order process caused by the third-order nonlinear susceptibility χð Þ<sup>3</sup> . It can be easily observed by using the high-intensity lasers, and it has been demonstrated experimentally [1]. FWM is a complicated nonlinear phenomenon because it exhibits different nonlinear effects for different combinations of the coupled wave frequencies, wave vectors, and polarizations. The analysis of FWM is based on the general theory of optical wave mixing [1, 2, 4]. For three input pump waves with frequencies ω1, <sup>2</sup>, 3, the singly resonant, doubly resonant, and triply resonant cases can occur [1]. They correspond to the situations when one, two, or three input frequencies or their algebraic sums approach medium transition frequencies [1]. In such cases the third-order susceptibility χð Þ<sup>3</sup> can be divided into a resonant part χ ð Þ3 R and a nonresonant part χ ð Þ3 NR [1]. The FWM process has some important applications. Due to the wide range of the mixed frequencies, FWM can be used for the generation of the waves from the infrared up to ultraviolet range [1]. For instance, the parametric amplification can be realized when two strong pump waves amplify two counterpropagating weak waves [1]. The frequency degenerate FWM occurs when the frequencies of the four waves are the same. It is used for the creation of a phaseconjugated wave with respect to one of the coupled waves [2]. In such a case, the phase of the output wave is complex conjugate to the phase of the input wave [1, 2]. FWM in optical fibers can be used for signal amplification, phase conjugation, wavelength conversion, pulse generation, and high-speed optical switching [4].
