**1. Introduction**

Nonlinear optical phenomena based on the second- and third-order optical nonlinearity characterized by susceptibilities *χ*ð Þ<sup>2</sup> and *χ*ð Þ<sup>3</sup> , respectively, are widely used in modern communication systems for the optical signal processing due to their ultrafast response time and a large number of different interactions [1–5]. The second-order susceptibility *χ*ð Þ<sup>2</sup> exists in non-centrosymmetric media, while the third-order susceptibility *χ*ð Þ<sup>3</sup> exists in any medium [6]. The second-order susceptibility *χ*ð Þ<sup>2</sup> may be used for the second harmonic generation (SHG), sum, and difference frequency generation; the ultrafast Kerr-type third-order susceptibility *χ*ð Þ<sup>3</sup> results in such effects as four-wave mixing (FWM), self-phase modulation (SPM), cross-phase modulation (XPM), third harmonic generation (THG), bistability, and different types of the stimulated light scattering (SLS) [1–6]. Optical-electricaloptical conversion processes can be replaced with the optical signal processing characterized by the femtosecond response time of nonlinearities in optical materials [2, 3]. All-optical signal processing, ultrafast switching, optical generation of ultrashort pulses, the control over the laser radiation frequency spectrum,

wavelength exchange, coherent detection, multiplexing/demultiplexing, and tunable optical delays can be realized by using the nonlinear optical effects [1–4]. However, optical nonlinearities are weak and usually occur only with high-intensity laser beams [1, 6]. An effective nonlinear optical response can be substantially increased by using the plasmonic effects caused by the coherent oscillations of conduction electrons near the surface of noble metal structures [1]. In the case of the extended metal surfaces, the surface plasmon polaritons (SPPs) may occur [1, 7, 8]. SPPs are electromagnetic excitations propagating at the interface between a dielectric and a conductor, evanescently confined in the perpendicular direction [1, 7]. The SPP electromagnetic field decays exponentially on both sides of the interface which results in the subwavelength confinement near the metal surface [1]. The SPP propagation length is limited by the ohmic losses in metal [1, 7, 8].

experimentally, and it was shown that the scattering losses in SmALC are much lower than in NLC due to a higher degree of the long-range order [15]. SmALC can be useful in nonlinear optical applications and low-loss active waveguide devices for

*Stimulated Scattering of Surface Plasmon Polaritons in a Plasmonic Waveguide with a Smectic…*

the claddings are shown in **Figure 1a** and **b**, respectively.

SmALCs are characterized by a positional long-range order in the direction of the elongated molecular axis and demonstrate a layer structure with a layer thickness *dSmA* ≈2*nm* [14]. Inside a smectic layer, the molecules form a two-dimensional liquid [14]. Actually, SmALC can be considered as a natural nanostructure. The structures of NLC with the elongated molecules directed mainly along the vector

The nonlinear optical phenomena in SmALC such as a light self-focusing, selftrapping, SPM, SLS, and FWM based on the specific mechanism of the third-order optical nonlinearity related to the smectic layer normal displacement had been investigated theoretically [16–28]. In particular it has been shown that at the interface of a metal and SmALC, the counter-propagating SPPs created the dynamic grating of the smectic layer normal displacement *u x*ð Þ , *z*, *t* , and the SLS of the interfering SPPs occurred [22, 23, 26]. We also investigated the behavior of SPP mode in a MIM waveguide with the SmALC core [24, 26]. In such a waveguide, SPP behaves as a strongly localized transverse magnetic (TM) mode which creates the

localized smectic layer normal deformation and undergoes SPM [24, 26]. In this chapter we consider theoretically the interaction of the counterpropagating SPP modes in the MIM waveguide with the SmALC core. The interfering SPP TM modes with the close optical frequencies *ω*1,2 create a localized dynamic grating of the smectic layer normal displacement *u x*ð Þ , *z*, *t* with the frequency Δ*ω* ¼ *ω*<sup>1</sup> � *ω*<sup>2</sup> ≪ *ω*<sup>1</sup> which results in the nonlinear polarization and stimulated scattering of SPPs. We solved simultaneously the equation of motion for smectic layers in the electric field of the interfering SPP modes and the Maxwell equations for the SPPs in the MIM waveguide taking into account the nonlinear polarization. We used the slowly varying amplitude (SVA) approximation for the SPPs [6]. We evaluated the magnitudes and phases of the coupled SPP SVAs. It is shown that the energy exchange between the coupled SPPs and XPM takes place. We also evaluated the SPP-induced smectic layer displacement and SmALC hydrodynamic velocity. We have shown that the high-frequency localized electric field can occur in the MIM

waveguide with the SmALC core due to the flexoelectric effect [28].

*The structure of molecular alignment of a nematic liquid crystal (NLC) (a) and the homeotropically oriented*

*smectic A liquid crystal (SmALC). The molecules are perpendicular to the layer plane (b).*

! and the homeotropically oriented SmALC with the layer plane parallel to

integrated optics [14, 15].

*DOI: http://dx.doi.org/10.5772/intechopen.89483*

director *n*

**Figure 1.**

**137**

Nonlinear optical effects can be enhanced by plasmonic excitations as follows: (i) the coupling of light to surface plasmons results in strong local electromagnetic fields; (ii) typically, plasmonic excitations are highly sensitive to dielectric properties of the metal and surrounding medium [1]. In nonlinear optical phenomena, such a sensitivity can be used for the light-induced nonlinear change in the dielectric properties of one of the materials which result in the varying of the plasmonic resonances and the signal beam propagation conditions [1]. Plasmonic excitations are characterized by timescale of several femtoseconds which permits the ultrafast optical signal processing [1]. The SPP field confinement and enhancement can be changed by modifying the structure of the metal or the dielectric near the interface [1]. For example, plasmonic waveguides can be created [1, 7–9]. Nanoplasmonic waveguides can confine and enhance electric fields near the nanometallic surfaces due to the propagating SPPs [9]. Nanoplasmonic waveguide consists of one or two metal films combined with one or two dielectric slabs [9]. Typically, two types of the plasmonic waveguides exist: (i) an insulator/metal/insulator (IMI) heterostructure where a thin metallic layer is placed between two infinitely thick dielectric claddings and (ii) a metal/insulator/metal (MIM) heterostructure where a thin dielectric layer is sandwiched between two metallic claddings [7]. The MIM waveguides for nonlinear optical applications require highly nonlinear dielectrics [9]. The nonlinear metamaterials can significantly increase the nonlinearity magnitude [10]. Investigation of nonlinear metamaterials is related in particular to nonlinear plasmonics and active media [10]. One of the metamaterial nonlinearity mechanisms is based on liquid crystals (LCs) [10]. Tunability and a strongly nonlinear response of metamaterials can be obtained by their integration with LCs offering a practical solution for controlling metamaterial devices [11].

The integration of LCs with plasmonic and metamaterials may be promising for applications in modern photonics due to the extremely large optical nonlinearity of LCs, strong localized electric fields of surface plasmon polaritons (SPPs), and high operation rates as compared to conventional electro-optic devices [12]. Practically all nonlinear optical processes such as wave mixing, self-focusing, self-guiding, optical bistabilities and instabilities, phase conjugation, SLS, optical limiting, interface switching, beam combining, and self-starting laser oscillations have been observed in LCs [13]. LC can be incorporated into nano- and microstructures such as a MIM plasmonic waveguide. Nematic LCs (NLCs) characterized by the orientation long-range order of the elongated molecules are mainly used in optical applications including plasmonics and nanophotonics [11–14]. For instance, lightinduced control of fishnet metamaterials infiltrated with NLCs was demonstrated experimentally where a metal-dielectric (Au-MgF2) sandwich nanostructure on a glass substrate with the inserted NLC was used [11]. However, the NLC applications are limited by their large losses and relatively slow response [14, 15]. The light scattering in smectic A LC (SmALC) waveguides had been studied theoretically and *Stimulated Scattering of Surface Plasmon Polaritons in a Plasmonic Waveguide with a Smectic… DOI: http://dx.doi.org/10.5772/intechopen.89483*

experimentally, and it was shown that the scattering losses in SmALC are much lower than in NLC due to a higher degree of the long-range order [15]. SmALC can be useful in nonlinear optical applications and low-loss active waveguide devices for integrated optics [14, 15].

SmALCs are characterized by a positional long-range order in the direction of the elongated molecular axis and demonstrate a layer structure with a layer thickness *dSmA* ≈2*nm* [14]. Inside a smectic layer, the molecules form a two-dimensional liquid [14]. Actually, SmALC can be considered as a natural nanostructure. The structures of NLC with the elongated molecules directed mainly along the vector director *n* ! and the homeotropically oriented SmALC with the layer plane parallel to the claddings are shown in **Figure 1a** and **b**, respectively.

The nonlinear optical phenomena in SmALC such as a light self-focusing, selftrapping, SPM, SLS, and FWM based on the specific mechanism of the third-order optical nonlinearity related to the smectic layer normal displacement had been investigated theoretically [16–28]. In particular it has been shown that at the interface of a metal and SmALC, the counter-propagating SPPs created the dynamic grating of the smectic layer normal displacement *u x*ð Þ , *z*, *t* , and the SLS of the interfering SPPs occurred [22, 23, 26]. We also investigated the behavior of SPP mode in a MIM waveguide with the SmALC core [24, 26]. In such a waveguide, SPP behaves as a strongly localized transverse magnetic (TM) mode which creates the localized smectic layer normal deformation and undergoes SPM [24, 26].

In this chapter we consider theoretically the interaction of the counterpropagating SPP modes in the MIM waveguide with the SmALC core. The interfering SPP TM modes with the close optical frequencies *ω*1,2 create a localized dynamic grating of the smectic layer normal displacement *u x*ð Þ , *z*, *t* with the frequency Δ*ω* ¼ *ω*<sup>1</sup> � *ω*<sup>2</sup> ≪ *ω*<sup>1</sup> which results in the nonlinear polarization and stimulated scattering of SPPs. We solved simultaneously the equation of motion for smectic layers in the electric field of the interfering SPP modes and the Maxwell equations for the SPPs in the MIM waveguide taking into account the nonlinear polarization. We used the slowly varying amplitude (SVA) approximation for the SPPs [6]. We evaluated the magnitudes and phases of the coupled SPP SVAs. It is shown that the energy exchange between the coupled SPPs and XPM takes place. We also evaluated the SPP-induced smectic layer displacement and SmALC hydrodynamic velocity. We have shown that the high-frequency localized electric field can occur in the MIM waveguide with the SmALC core due to the flexoelectric effect [28].

**Figure 1.**

*The structure of molecular alignment of a nematic liquid crystal (NLC) (a) and the homeotropically oriented smectic A liquid crystal (SmALC). The molecules are perpendicular to the layer plane (b).*

wavelength exchange, coherent detection, multiplexing/demultiplexing, and tunable optical delays can be realized by using the nonlinear optical effects [1–4]. However, optical nonlinearities are weak and usually occur only with high-intensity laser beams [1, 6]. An effective nonlinear optical response can be substantially increased by using the plasmonic effects caused by the coherent oscillations of conduction electrons near the surface of noble metal structures [1]. In the case of the extended metal surfaces, the surface plasmon polaritons (SPPs) may occur [1, 7, 8]. SPPs are electromagnetic excitations propagating at the interface between a dielectric and a conductor, evanescently confined in the perpendicular direction [1, 7]. The SPP electromagnetic field decays exponentially on both sides of the interface which results in the subwavelength confinement near the metal surface [1]. The SPP prop-

Nonlinear optical effects can be enhanced by plasmonic excitations as follows: (i) the coupling of light to surface plasmons results in strong local electromagnetic fields; (ii) typically, plasmonic excitations are highly sensitive to dielectric properties of the metal and surrounding medium [1]. In nonlinear optical phenomena, such a sensitivity can be used for the light-induced nonlinear change in the dielectric properties of one of the materials which result in the varying of the plasmonic resonances and the signal beam propagation conditions [1]. Plasmonic excitations are characterized by timescale of several femtoseconds which permits the ultrafast optical signal processing [1]. The SPP field confinement and enhancement can be changed by modifying the structure of the metal or the dielectric near the interface [1]. For example, plasmonic waveguides can be created [1, 7–9]. Nanoplasmonic waveguides can confine and enhance electric fields near the nanometallic surfaces due to the propagating SPPs [9]. Nanoplasmonic waveguide consists of one or two metal films combined with one or two dielectric slabs [9]. Typically, two types of

agation length is limited by the ohmic losses in metal [1, 7, 8].

*Nanoplasmonics*

the plasmonic waveguides exist: (i) an insulator/metal/insulator (IMI)

offering a practical solution for controlling metamaterial devices [11].

**136**

heterostructure where a thin metallic layer is placed between two infinitely thick dielectric claddings and (ii) a metal/insulator/metal (MIM) heterostructure where a thin dielectric layer is sandwiched between two metallic claddings [7]. The MIM waveguides for nonlinear optical applications require highly nonlinear dielectrics [9]. The nonlinear metamaterials can significantly increase the nonlinearity magnitude [10]. Investigation of nonlinear metamaterials is related in particular to nonlinear plasmonics and active media [10]. One of the metamaterial nonlinearity mechanisms is based on liquid crystals (LCs) [10]. Tunability and a strongly nonlinear response of metamaterials can be obtained by their integration with LCs

The integration of LCs with plasmonic and metamaterials may be promising for applications in modern photonics due to the extremely large optical nonlinearity of LCs, strong localized electric fields of surface plasmon polaritons (SPPs), and high operation rates as compared to conventional electro-optic devices [12]. Practically all nonlinear optical processes such as wave mixing, self-focusing, self-guiding, optical bistabilities and instabilities, phase conjugation, SLS, optical limiting, interface switching, beam combining, and self-starting laser oscillations have been observed in LCs [13]. LC can be incorporated into nano- and microstructures such as a MIM plasmonic waveguide. Nematic LCs (NLCs) characterized by the orientation long-range order of the elongated molecules are mainly used in optical applications including plasmonics and nanophotonics [11–14]. For instance, lightinduced control of fishnet metamaterials infiltrated with NLCs was demonstrated experimentally where a metal-dielectric (Au-MgF2) sandwich nanostructure on a glass substrate with the inserted NLC was used [11]. However, the NLC applications are limited by their large losses and relatively slow response [14, 15]. The light scattering in smectic A LC (SmALC) waveguides had been studied theoretically and

#### *Nanoplasmonics*

The chapter is constructed as follows. The hydrodynamics of SmALC in the external electric field is considered in Section 2. The SPP modes of the MIM waveguide are derived in Section 3. The SPP SVAs, the smectic layer dynamic grating amplitude, and the SmALC hydrodynamic velocity are evaluated in Section 4. The conclusions are presented in Section 5.

and vanishes for the wave vector *k*

*DOI: http://dx.doi.org/10.5772/intechopen.89483*

phase transition has the form [29–31]

*σ*0

Here, *v*

*α<sup>i</sup>* ≈10�<sup>1</sup>

*E* !

where *k<sup>S</sup>*

**139**

Π is the pressure, Λ

!

ð Þ *x*, *y*, *z*, *t* has the form [29–31]

tivity tensor *εik* is given by [30]

*εxx* ¼ *εyy* ¼ *ε*<sup>⊥</sup> þ *a*<sup>⊥</sup>

*εxz* ¼ *εzx* ¼ �*ε<sup>a</sup>*

*<sup>F</sup>* <sup>¼</sup> <sup>1</sup> 2 *<sup>B</sup> <sup>∂</sup><sup>u</sup> ∂z* <sup>2</sup>

has only the Z component according to Eq. (3): Λ

!*<sup>S</sup>*

*ρ ∂vi*

*<sup>∂</sup><sup>t</sup>* ¼ � <sup>∂</sup><sup>Π</sup> *∂xi*

*Aik* <sup>¼</sup> <sup>1</sup> 2

is the generalized force density, *σ*<sup>0</sup>

þ Λ*<sup>i</sup>* þ

<sup>Λ</sup>*<sup>i</sup>* ¼ � *<sup>δ</sup><sup>F</sup> δui*

*ik* ¼ *α*0*δikAll* þ *α*1*δizAzz* þ *α*4*Aik* þ *α*56ð*δizAzk* þ *δkzAzi*Þ þ *α*7*δizδkzAll* (4)

*∂vi ∂xk* þ *∂vk ∂xi*

*vz* <sup>¼</sup> *<sup>∂</sup><sup>u</sup>*

and *F* is the free energy density of SmALC. Typically, SmALC is supposed to be an incompressible liquid according to Equation (1) [29]. For this reason, we assume that the pressure Π ¼ 0 and the SmALC free energy density *F* do not depend on the bulk compression [29–31]. We are interested in the SS propagation and neglect the ordinary sound mode. The normal layer displacement *u x*ð Þ , *y*, *z*, *t* by definition has only one component along the Z axis. In such a case, the generalized force density

SmALC since it determines the condition of the smectic layer continuity [29–31]. The SmALC free energy density *F* in the presence of the external electric field

Here *<sup>K</sup>* � <sup>10</sup>�<sup>11</sup>*<sup>N</sup>* is the Frank elastic constant associated with the SmALC orientational energy inside layers, *ε*<sup>0</sup> is the free space permittivity, and *εik* is the SmALC permittivity tensor including the terms defined by the smectic layer strains. The purely orientational second term in the free energy density *F* (7) can be neglected since for the typical values of the elastic constants *B* and *KK kS*

⊥, the SS wave vector component is parallel to the layer plane. The permit-

;*εzz* ¼ *ε*<sup>∥</sup> þ *a*<sup>∥</sup>

;*εyz* ¼ *εzy* ¼ �*ε<sup>a</sup>*

*∂u ∂z*

*∂u ∂x*

þ 1 2 *<sup>K</sup> <sup>∂</sup>*<sup>2</sup> *u ∂x*<sup>2</sup> þ

! is the hydrodynamic velocity, *ρ*≈103kgm�<sup>3</sup> is the SmALC mass density,

kg smð Þ�<sup>1</sup> are the viscosity Leslie coefficients, *<sup>δ</sup>ik* <sup>¼</sup> 1, *<sup>i</sup>* <sup>¼</sup> *<sup>k</sup>*;*δik* <sup>¼</sup> 0, *<sup>i</sup>* 6¼ *<sup>k</sup>*,

!

*∂*2 *u ∂y*<sup>2</sup> <sup>2</sup>

� 1 2

*∂u ∂z* ;

;*ε<sup>a</sup>* ¼ *ε*<sup>∥</sup> � *ε*<sup>⊥</sup>

*∂u ∂y*

*∂σ*<sup>0</sup> *ik ∂xk*

SmALC is characterized by the complex order parameter, and SS represents the oscillations of the order parameter phase [29]. SS in SmALC has been observed experimentally by different methods [32–34]. The system of hydrodynamic equations for the incompressible SmALC under the constant temperature far from the

*Stimulated Scattering of Surface Plasmon Polaritons in a Plasmonic Waveguide with a Smectic…*

perpendicular or parallel to the layer plane [29].

*div v*! <sup>¼</sup> <sup>0</sup> (1)

(5)

*<sup>∂</sup><sup>t</sup>* (6)

*ik* is the viscous stress tensor,

¼ ð Þ 0, 0,Λ*<sup>z</sup>* . Eq. (6) is specific for

*ε*0*εikEiEk* (7)

⊥ <sup>2</sup>

≪ *B*

(8)

(2)

(3)
