**2. Hydrodynamics of SmALC in the external electric field**

In this section we briefly discuss the SmALC hydrodynamics and derive the equation of motion for the smectic layer normal displacement *u x*ð Þ , *y*, *z*, *t* in the external electric field *E* ! ð Þ *x*, *y*, *z*, *t* . SmALC can be described by the one-dimensional periodic density wave due to its layered structure.

Smectic layer oscillations *u x*ð Þ , *y*, *z*, *t* in the external electric field *E* ! ð Þ *x*, *y*, *z*, *t* are shown in **Figure 2**. Hydrodynamics of SmALC in general case is very complicated because SmALC is a strongly anisotropic viscous liquid including the layer oscillations, the mass density, and the elongated molecule orientation variations [29–31]. However, the elastic constant related to the SmALC bulk compression is much larger than the elastic constant *<sup>B</sup>*<sup>≈</sup> <sup>10</sup><sup>6</sup> � <sup>10</sup><sup>7</sup> J m�<sup>3</sup> related to the smectic layer compression [29–31]. The layers can oscillate without the change of the mass density [29–31]. For this reason two uncoupled acoustic modes can propagate in SmALC: the ordinary longitudinal sound wave caused by the mass density variation and the second-sound (SS) wave caused by the layer oscillations [29–31]. SS wave is characterized by strongly anisotropic dispersion relation being neither purely transverse nor longitudinal. It propagates in the direction oblique to the layer plane

#### **Figure 2.**

*The SmALC layer oscillations u x*ð Þ , *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup> in the external electric field E*! ð Þ *x*, *y*, *z*, *t . k* !*<sup>S</sup> is the second-sound (SS) wave vector, vz is the hydrodynamic velocity perpendicular to the layer plane; ε*<sup>∥</sup> *and ε*<sup>⊥</sup> *are the diagonal components of the permittivity tensor parallel and perpendicular to the optical axis, respectively.*

*Stimulated Scattering of Surface Plasmon Polaritons in a Plasmonic Waveguide with a Smectic… DOI: http://dx.doi.org/10.5772/intechopen.89483*

and vanishes for the wave vector *k* !*<sup>S</sup>* perpendicular or parallel to the layer plane [29]. 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 phase transition has the form [29–31]

$$\begin{array}{c} \text{div } \overrightarrow{v} = \mathbf{0} \end{array} \tag{1}$$

$$
\rho \frac{\partial v\_i}{\partial t} = -\frac{\partial \Pi}{\partial \mathbf{x}\_i} + \Lambda\_i + \frac{\partial \sigma'\_{ik}}{\partial \mathbf{x}\_k} \tag{2}
$$

$$
\Lambda\_i = -\frac{\delta F}{\delta u\_i} \tag{3}
$$

$$
\sigma'\_{ik} = a\_0 \delta\_{ik} A\_{ll} + a\_1 \delta\_{ix} A\_{xx} + a\_4 A\_{ik} + a\_5 (\delta\_{ix} A\_{xk} + \delta\_{kx} A\_{xi}) + a\_7 \delta\_{ix} \delta\_{kx} A\_{ll} \tag{4}
$$

$$A\_{ik} = \frac{1}{2} \left( \frac{\partial v\_i}{\partial \mathbf{x}\_k} + \frac{\partial v\_k}{\partial \mathbf{x}\_i} \right) \tag{5}$$

$$v\_x = \frac{\partial u}{\partial t} \tag{6}$$

Here, *v* ! is the hydrodynamic velocity, *ρ*≈103kgm�<sup>3</sup> is the SmALC mass density, Π is the pressure, Λ ! is the generalized force density, *σ*<sup>0</sup> *ik* is the viscous stress tensor, *α<sup>i</sup>* ≈10�<sup>1</sup> 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>*, 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 has only the Z component according to Eq. (3): Λ ! ¼ ð Þ 0, 0,Λ*<sup>z</sup>* . Eq. (6) is specific for 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 *E* ! ð Þ *x*, *y*, *z*, *t* has the form [29–31]

$$F = \frac{1}{2}B\left(\frac{\partial u}{\partial \mathbf{z}}\right)^2 + \frac{1}{2}K\left(\frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial \mathbf{y}^2}\right)^2 - \frac{1}{2}\varepsilon\_0 \varepsilon\_{ik} E\_i E\_k \tag{7}$$

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* ⊥ <sup>2</sup> ≪ *B* where *k<sup>S</sup>* ⊥, the SS wave vector component is parallel to the layer plane. The permittivity tensor *εik* is given by [30]

$$\begin{aligned} \varepsilon\_{\text{xx}} &= \varepsilon\_{\text{\mathcal{V}}} = \varepsilon\_{\perp} + a\_{\perp} \frac{\partial u}{\partial \mathbf{z}}; \varepsilon\_{\text{xx}} = \varepsilon\_{\parallel} + a\_{\parallel} \frac{\partial u}{\partial \mathbf{z}}; \\ \varepsilon\_{\text{xx}} = \varepsilon\_{\text{xx}} &= -\varepsilon\_{\text{d}} \frac{\partial u}{\partial \mathbf{x}}; \varepsilon\_{\text{yx}} = \varepsilon\_{\text{xy}} = -\varepsilon\_{\text{d}} \frac{\partial u}{\partial \mathbf{y}}; \varepsilon\_{\text{d}} = \varepsilon\_{\parallel} - \varepsilon\_{\perp} \end{aligned} \tag{8}$$

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

In this section we briefly discuss the SmALC hydrodynamics and derive the equation of motion for the smectic layer normal displacement *u x*ð Þ , *y*, *z*, *t* in the

shown in **Figure 2**. Hydrodynamics of SmALC in general case is very complicated because SmALC is a strongly anisotropic viscous liquid including the layer oscillations, the mass density, and the elongated molecule orientation variations [29–31]. However, the elastic constant related to the SmALC bulk compression is much larger than the elastic constant *<sup>B</sup>*<sup>≈</sup> <sup>10</sup><sup>6</sup> � <sup>10</sup>7J m�<sup>3</sup> related to the smectic layer compression [29–31]. The layers can oscillate without the change of the mass density [29–31]. For this reason two uncoupled acoustic modes can propagate in SmALC: the ordinary longitudinal sound wave caused by the mass density variation and the second-sound (SS) wave caused by the layer oscillations [29–31]. SS wave is characterized by strongly anisotropic dispersion relation being neither purely transverse nor longitudinal. It propagates in the direction oblique to the layer plane

ð Þ *x*, *y*, *z*, *t* . SmALC can be described by the one-dimensional

ð Þ *x*, *y*, *z*, *t . k*

!*<sup>S</sup>*

*is the second-sound (SS)*

!

ð Þ *x*, *y*, *z*, *t* are

**2. Hydrodynamics of SmALC in the external electric field**

Smectic layer oscillations *u x*ð Þ , *y*, *z*, *t* in the external electric field *E*

conclusions are presented in Section 5.

!

periodic density wave due to its layered structure.

*The SmALC layer oscillations u x*ð Þ , *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup> in the external electric field E*!

*wave vector, vz is the hydrodynamic velocity perpendicular to the layer plane; ε*<sup>∥</sup> *and ε*<sup>⊥</sup> *are the diagonal components of the permittivity tensor parallel and perpendicular to the optical axis, respectively.*

external electric field *E*

*Nanoplasmonics*

**Figure 2.**

**138**

where *ε*∥, *ε*<sup>⊥</sup> are the diagonal components of the permittivity tensor *εik* along and perpendicular to the optical axis and *a*<sup>⊥</sup> � 1, *a*<sup>∥</sup> � 1 are the phenomenological dimensionless coefficients [29, 30]. SmALC is an optically uniaxial medium with the optical Z axis perpendicular to the smectic layer plane [29–31]. Combining Eqs. (1)–(8), we obtain the equation of motion for the smectic layer normal displacement *u x*ð Þ , *y*, *z*, *t* in the electric field *E* ! ð Þ *x*, *y*, *z*, *t* [16, 17]:

$$\begin{split} & -\rho \nabla^2 \frac{\partial^2 u}{\partial t^2} + \left[ a\_1 \nabla\_\perp^2 \frac{\partial^2}{\partial x^2} + \frac{1}{2} (a\_4 + a\_{56}) \nabla^2 \nabla^2 \right] \frac{\partial u}{\partial t} + B \nabla\_\perp^2 \frac{\partial^2 u}{\partial x^2} \\ & = \frac{\varepsilon\_0}{2} \nabla\_\perp^2 \left[ \frac{\partial}{\partial \mathbf{z}} \left( a\_\perp \left( E\_\mathbf{x}^2 + E\_\mathbf{y}^2 \right) + a\_{\parallel} E\_\mathbf{z}^2 \right) - 2 \varepsilon\_d \left( \frac{\partial}{\partial \mathbf{x}} (E\_\mathbf{x} E\_\mathbf{z}) + \frac{\partial}{\partial \mathbf{y}} \left( E\_\mathbf{y} E\_\mathbf{z} \right) \right) \right] \end{split} \tag{9}$$

Here ∇<sup>2</sup> <sup>⊥</sup>*<sup>u</sup>* <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>u</sup>=∂x*<sup>2</sup> <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup> *u=∂y*2. In the absence of the external electric field, the homogeneous solution of the equation of motion (9) represents the SS wave with the dispersion relation [29]:

$$\mathfrak{Q}\_{\mathbb{S}} = \mathfrak{s}\_0 \frac{k\_{\perp}^{\mathbb{S}} k\_x^{\mathbb{S}}}{k^{\mathbb{S}}}; \mathfrak{s}\_0 = \sqrt{\frac{\mathsf{B}}{\rho}}\tag{10}$$

the manipulation and routing of SPPs can demonstrate a subwavelength beyond the diffraction limit together with large bandwidth and high operation rate typical for photonics [36]. The plasmonic devices can be integrated into nanophotonic chips due to their small scale and the compatibility with the VLSI electronic technology [36]. Plasmonic devices are the promising candidates for future integrated photonic circuits for broadband light routing, switching, and interconnecting [36]. It has been shown that different plasmonic structures can provide SPP light waveguiding determining the SPP mode properties [36]. MIM waveguide representing a dielectric sandwiched between two metal slabs attracted a research interest as a basic component of nanoscale plasmonic integrated circuits [37]. LC-tunable waveguides have been proposed as a core element of low-power variable attenuators, phaseshifters, switches, filters, tunable lenses, beam steers, and modulators [37, 38]. Typically NLCs have been used due to their strong optical anisotropy, responsivity to external electric and magnetic fields, and low power [37, 38]. Different types of NLC plasmonic waveguides have been proposed and investigated theoretically [36–38]. Recently, SmALCs attracted attention due to their layered structure and reconfigurable layer curvature [39]. The possibility of the dynamic variation of smectic layer configuration by external fields is intensively studied [39]. We investigated theoretically SLS in the optical slab waveguide with the SmALC core where the third-order optical nonlinearity mechanism was related to the smectic layer dynamic grating created by the interfering waveguide modes [27]. We also consid-

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

ered theoretically the MIM waveguide with the SmALC core [24, 26].

<sup>∇</sup><sup>2</sup>*<sup>u</sup>* <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup>

**Figure 3.**

**141**

*<sup>u</sup>=∂x*<sup>2</sup> <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup>

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

tivity tensor (8) takes the form

*∂u ∂z*

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

*u=∂z*2, ∇<sup>2</sup>

<sup>⊥</sup>*<sup>u</sup>* <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup>

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

The structure of such a symmetric waveguide of the thickness 2d is shown in **Figure 3** [24, 26]. The plane of the waveguide is perpendicular to the SmALC optical axis *Z*. The SmALC in the waveguide core is homeotropically oriented, i.e., the smectic layers are parallel to the waveguide claddings *z* ¼ �*d*, while the SmALC elongated molecules are mainly parallel to the *Z* axis [29]. Typically the waveguide dimension in the *Y* axis direction is much larger than *d*, and the dependence on the coordinate *y* in Eqs. (8) and (9) can be omitted. Than we obtain *u* ¼ *u x*ð Þ , *z*, *t* ,

*u=∂x*2, *kS*

*∂u ∂z*

*The MIM waveguide with the homeotropically oriented SmALC core and counter-propagating SPPs.*

⊥ <sup>2</sup>

; *εxz* ¼ *εzx* ¼ �*ε<sup>a</sup>*

<sup>¼</sup> *kS x* <sup>2</sup>

> *∂u ∂x*

, and the SmALC permit-

;*ε<sup>a</sup>* ¼ *ε*<sup>∥</sup> � *ε*<sup>⊥</sup> (12)

Here, *k<sup>S</sup>* ⊥ � �<sup>2</sup> <sup>¼</sup> *<sup>k</sup><sup>S</sup> x* � �<sup>2</sup> <sup>þ</sup> *<sup>k</sup><sup>S</sup> y* � �<sup>2</sup> and Ω*<sup>S</sup>* and *s*<sup>0</sup> are the SS frequency and velocity, respectively [29]. It is seen from Eq. (10) that the SS frequency Ω*<sup>S</sup>* ¼ 0 for the propagation direction along the smectic layer plane and perpendicular to it. The decay constant Γ is given by

$$\Gamma = \frac{1}{2\rho} \left[ a\_1 \frac{\left(k\_\perp^S\right)^2 \left(k\_x^S\right)^2}{\left(k^S\right)^2} + \frac{1}{2} (a\_4 + a\_{56}) \left(k^S\right)^2 \right] \tag{11}$$

If the viscosity terms responsible for the SS wave decay can be neglected, then the homogeneous part of Eq. (9) reduces to the SS wave equation with the dispersion relation (10) [29–31]:

$$
\rho \nabla^2 \frac{\partial^2 u}{\partial t^2} = B \nabla\_\perp^2 \frac{\partial^2 u}{\partial x^2}
$$

We use equation of motion (9) for the evaluation of the light-enhanced dynamic grating *u x*ð Þ , *y*, *z*, *t* .
