**3. SPP modes in a MIM waveguide with SmALC core**

LC slab optical waveguide represents a LC layer of a thickness about 1 μm confined between two glass slides of lower refractive index than LC [14]. LC as a waveguide core provides the photonic signal modulation and switching by using the electro-optic or nonlinear optical effects of LC mesophases [35]. For instance, the large optical nonlinearities were implemented in order to create optical paths by photonic control of solitons in NLC [35]. Various electrode geometries may create due to the electro-optic effect periodically modulated LC core waveguides which can serve as efficient guided distributed Bragg reflectors with the tuning ranges of about 100–1550 nm optical wavelength range [35]. Plasmonic waveguides based on *Stimulated Scattering of Surface Plasmon Polaritons in a Plasmonic Waveguide with a Smectic… DOI: http://dx.doi.org/10.5772/intechopen.89483*

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

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* ,

∇2 *<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=∂z*2, ∇<sup>2</sup> <sup>⊥</sup>*<sup>u</sup>* <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup> *u=∂x*2, *kS* ⊥ <sup>2</sup> <sup>¼</sup> *kS x* <sup>2</sup> , and the SmALC permittivity tensor (8) takes the form

$$\varepsilon\_{\rm xx} = \varepsilon\_{\perp} + a\_{\perp} \frac{\partial u}{\partial \mathbf{z}}; \ \varepsilon\_{\rm xx} = \varepsilon\_{\parallel} + a\_{\parallel} \frac{\partial u}{\partial \mathbf{z}}; \ \varepsilon\_{\rm xx} = \varepsilon\_{\rm xx} = -\varepsilon\_{a} \frac{\partial u}{\partial \mathbf{x}}; \varepsilon\_{a} = \varepsilon\_{\parallel} - \varepsilon\_{\perp} \tag{12}$$

**Figure 3.** *The MIM waveguide with the homeotropically oriented SmALC core and counter-propagating SPPs.*

where *ε*∥, *ε*<sup>⊥</sup> are the diagonal components of the permittivity tensor *εik* along and

!

∇2

� 2*ε<sup>a</sup>*

homogeneous solution of the equation of motion (9) represents the SS wave with

*kS* ⊥*kS z kS* ; *<sup>s</sup>*<sup>0</sup> <sup>¼</sup>

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

> *kS z* � �<sup>2</sup>

1 2

> ⊥ *∂*2 *u ∂z*<sup>2</sup>

We use equation of motion (9) for the evaluation of the light-enhanced dynamic

LC slab optical waveguide represents a LC layer of a thickness about 1 μm confined between two glass slides of lower refractive index than LC [14]. LC as a waveguide core provides the photonic signal modulation and switching by using the electro-optic or nonlinear optical effects of LC mesophases [35]. For instance, the large optical nonlinearities were implemented in order to create optical paths by photonic control of solitons in NLC [35]. Various electrode geometries may create due to the electro-optic effect periodically modulated LC core waveguides which can serve as efficient guided distributed Bragg reflectors with the tuning ranges of about 100–1550 nm optical wavelength range [35]. Plasmonic waveguides based on

*kS* � �<sup>2</sup> <sup>þ</sup>

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 disper-

ð Þ *x*, *y*, *z*, *t* [16, 17]:

<sup>þ</sup> *<sup>B</sup>*∇<sup>2</sup> ⊥ *∂*2 *u ∂z*<sup>2</sup>

> *∂ ∂y*

*EyEz*

(10)

*<sup>∂</sup><sup>x</sup>* ð Þþ *ExEz*

ffiffiffi *B ρ*

s

*u=∂y*2. In the absence of the external electric field, the

and Ω*<sup>S</sup>* and *s*<sup>0</sup> are the SS frequency and velocity,

3 7

<sup>5</sup> (11)

ð Þ *<sup>α</sup>*<sup>4</sup> <sup>þ</sup> *<sup>α</sup>*<sup>56</sup> *<sup>k</sup><sup>S</sup>* � �<sup>2</sup>

*∂t*

*∂*

� � � � � � (9)

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 dis-

ð Þ *<sup>α</sup>*<sup>4</sup> <sup>þ</sup> *<sup>α</sup>*<sup>56</sup> <sup>∇</sup><sup>2</sup>

� � *∂u*

<sup>þ</sup> *<sup>a</sup>*∥*E*<sup>2</sup> *z*

Ω*<sup>S</sup>* ¼ *s*<sup>0</sup>

*kS* ⊥ � �<sup>2</sup>

> *<sup>ρ</sup>*∇<sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *u <sup>∂</sup>t*<sup>2</sup> <sup>¼</sup> *<sup>B</sup>*∇<sup>2</sup>

**3. SPP modes in a MIM waveguide with SmALC core**

placement *u x*ð Þ , *y*, *z*, *t* in the electric field *E*

⊥ *∂*2 *∂z*<sup>2</sup> þ 1 2

*<sup>x</sup>* <sup>þ</sup> *<sup>E</sup>*<sup>2</sup> *y* � �

> <sup>þ</sup> *<sup>k</sup><sup>S</sup> y* � �<sup>2</sup>

> > 2 6 4

� �

*a*<sup>⊥</sup> *E*<sup>2</sup>

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

<sup>¼</sup> *<sup>k</sup><sup>S</sup> x* � �<sup>2</sup>

> <sup>Γ</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup>*<sup>ρ</sup> <sup>α</sup>*<sup>1</sup>

�*ρ*∇<sup>2</sup> *<sup>∂</sup>*<sup>2</sup>

¼ *ε*0 <sup>2</sup> <sup>∇</sup><sup>2</sup> ⊥ *∂ ∂z*

*Nanoplasmonics*

Here ∇<sup>2</sup>

Here, *k<sup>S</sup>* ⊥ � �<sup>2</sup>

*u <sup>∂</sup>t*<sup>2</sup> <sup>þ</sup> *<sup>α</sup>*1∇<sup>2</sup>

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

the dispersion relation [29]:

decay constant Γ is given by

sion relation (10) [29–31]:

grating *u x*ð Þ , *y*, *z*, *t* .

**140**

The permittivity *εm*ð Þ *ω* of the metal claddings is described by the Drude model [7, 8]:

$$\varepsilon\_m(o) = 1 - \frac{o\_p^2}{[o^2 + (io/\tau)]} \tag{13}$$

(20) into the homogeneous part of the wave equation (14) for the claddings and SmALC core, respectively, we obtain the following expressions for the complex

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

*k*2

q

*k*2 *<sup>x</sup> ε*⊥*=ε*<sup>∥</sup>

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where *c* is the free space light velocity. The boundary conditions for the fields

Substituting expressions (17)–(20) into Eqs. (23) and (24), we obtain the dispersion relation for the SPP TM modes in the MIM waveguide given by [24, 26]

!<sup>2</sup>

*kS z ε*⊥

*km z <sup>ε</sup>m*ð Þ *<sup>ω</sup>* � *<sup>k</sup><sup>S</sup>*

!�<sup>2</sup>

*z ε*⊥

*<sup>z</sup>* � <sup>10</sup><sup>4</sup>*m*�<sup>1</sup> and Re *kx* � <sup>10</sup><sup>7</sup>

*<sup>z</sup>* <sup>≈</sup> Re *<sup>k</sup><sup>S</sup>*

� <sup>10</sup>�<sup>4</sup>*<sup>m</sup>* <sup>≫</sup> *<sup>d</sup>* � <sup>10</sup>�<sup>6</sup>*m*, and

*<sup>z</sup>* [24, 26]. The SPP

*m* ≫ *λSPP* ¼

*z <sup>ε</sup>m*ð Þ *<sup>ω</sup>* <sup>þ</sup>

Dispersion relation obtained for the general case of different claddings [7] coincides with expression (25) for the symmetric structure with the same claddings. The results of the numerical solution of Eq. (25) for the typical values of the MIM waveguide parameters and the SPP frequencies *ω* corresponding to the optical wavelength range *λopt* � 1 � 1*:*6 μm and 2*d* � 1 μm are presented in **Figures 4** and **5**.

*<sup>z</sup>* � 106*m*�<sup>1</sup> <sup>≫</sup> Im*k<sup>S</sup>*

*m*�<sup>1</sup> [24, 26]. In such a case, the SPP oscillation length in the Z

*z* � ��<sup>1</sup>

� � � *<sup>ω</sup>*2*ε*⊥*=c*<sup>2</sup>

*H*1*y*ð Þ¼ *z* ¼ *d HSAy*ð Þ *z* ¼ *d* ; *H*2*y*ð Þ¼ *z* ¼ �*d HSAy*ð Þ *z* ¼ �*d* (23) *E*1*x*ð Þ¼ *z* ¼ *d ESAx*ð Þ *z* ¼ *d* ; *E*2*x*ð Þ¼ *z* ¼ �*d ESAx*ð Þ *z* ¼ �*d* (24)

*<sup>x</sup>* � *εm*ð Þ *ω ω*2*=c*<sup>2</sup>

(21)

(22)

(25)

*m*�<sup>1</sup>

wave numbers *km*

*<sup>z</sup>* and *<sup>k</sup><sup>S</sup>*

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

*<sup>z</sup>* [24, 26]:

*kS z* ¼

(17)–(20) at the interfaces *z* ¼ �*d* have the form [7, 8]

exp �4*k<sup>S</sup>*

direction is defined by the relationship 2*π* Im*k<sup>S</sup>*

*<sup>z</sup>* can be neglected inside the MIM waveguide, and *<sup>k</sup><sup>S</sup>*

*z(a) and Imk<sup>S</sup>*

*<sup>z</sup> (b).*

propagation length in the *<sup>X</sup>* direction *LSPP* <sup>¼</sup> ð Þ Im*kx* �<sup>1</sup> � <sup>10</sup>�<sup>4</sup> � <sup>10</sup>�<sup>3</sup>

<sup>2</sup>*π*ð Þ Re *kx* �<sup>1</sup> <sup>&</sup>lt; <sup>10</sup>�<sup>6</sup>*<sup>m</sup>* where *<sup>λ</sup>SPP* is the SPP wavelength. Hence, at the optical wavelength-scale distances, Im*kx* can be neglected, and *kx* ≈ Re *kx* [24, 26].

These results show that Re *k<sup>S</sup>*

<sup>≫</sup> Im*kx* � <sup>10</sup><sup>3</sup>

Im*k<sup>S</sup>*

**Figure 4.**

**143**

*The spectral dependence of Re k<sup>S</sup>*

*zd* � � <sup>¼</sup> *<sup>k</sup><sup>m</sup>*

*km z* ¼

where *<sup>ω</sup><sup>p</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>n</sup>*0*e*2*=*ð Þ *<sup>ε</sup>*0*<sup>m</sup>* <sup>p</sup> is the plasma frequency of the free electron gas; *<sup>n</sup>*<sup>0</sup> is the free electron density in the metal;*e*, *m* are the electron charge and mass, respectively; and *ω*, *τ* are the SPP angular frequency and lifetime, respectively [7, 8]. The electric field *E* ! ð Þ *x*, *z*, *t* of the optical wave propagating in a nonlinear medium is described by the following wave equation including the nonlinear part of the electric induction *D* !*NL* [6]:

$$
overline{\vec{E}} + \mu\_0 \frac{\partial^2 \vec{D}^L}{\partial t^2} = -\mu\_0 \frac{\partial^2 \vec{D}^{NL}}{\partial t^2} \tag{14}$$

Here *μ*<sup>0</sup> is the free space permeability and *D* !*<sup>L</sup>* is the nonlinear part of the electric induction. The SPP can propagate in the plasmonic waveguide only as a transverse magnetic (TM) mode with the electric and magnetic fields given by *E* ! *TM* ¼ ð Þ *Ex*, 0, *Ez* ; *H* ! *TM* <sup>¼</sup> 0, *Hy*, 0 � � [7]. In such a case, we obtain for *<sup>D</sup>* !*<sup>L</sup>* and *D* !*NL* in SmALC using Eq. (12)

$$D\_\mathbf{x}^L = \varepsilon\_0 \varepsilon\_\perp E\_\mathbf{x}; D\_\mathbf{z}^L = \varepsilon\_0 \varepsilon\_\parallel E\_\mathbf{z} \tag{15}$$

$$^jD\_\mathbf{x}^{\rm NL} = \varepsilon\_0 \left( a\_\perp \frac{\partial u}{\partial \mathbf{z}} E\_\mathbf{x} - \varepsilon\_\mathbf{z} \frac{\partial u}{\partial \mathbf{x}} E\_\mathbf{z} \right); \\ D\_\mathbf{z}^{\rm NL} = \varepsilon\_0 \left( a\_\parallel \frac{\partial u}{\partial \mathbf{z}} E\_\mathbf{z} - \varepsilon\_\mathbf{z} \frac{\partial u}{\partial \mathbf{x}} E\_\mathbf{x} \right) \tag{16}$$

The linear part *D* !*<sup>L</sup> <sup>m</sup>* of the electric induction in the metal claddings has the form: *D* !*<sup>L</sup> <sup>m</sup>* ¼ *ε*0*εm*ð Þ *ω E* ! [7]. The SPP TM mode electric and magnetic fields for j j *z* >*d* in the metal claddings *H* ! 1,2ð Þ *x*, *z*, *t* , *E* ! 1,2ð Þ *x*, *z*, *t* and for j j *z* ≤*d* in SmALC *H* ! *SA*ð Þ *x*, *z*, *t* , *E* ! *SA*ð Þ *x*, *z*, *t* have the form [24, 26]

$$\overrightarrow{H}\_{1,2}(\mathbf{x}, \mathbf{z}, t) = \frac{1}{2}\overrightarrow{a}\_{\rangle}H\_{1,20} \exp\left(\mp k\_x^m z + ik\_x \mathbf{x} - i\alpha t\right) + c.c., |\mathbf{z}| > d\tag{17}$$

$$\overrightarrow{E}\_{1,2}(\mathbf{x}, z, t) = \frac{1}{2} \left[ \overrightarrow{a}\_x E\_{1,2x0} + \overrightarrow{a}\_z E\_{1,2x0} \right] \exp\left( \mp k\_x^m z + i k\_x \mathbf{x} - i \alpha t \right) + c.c., |z| > d \quad \text{(18)}$$

$$\overrightarrow{H}\_{\text{SA}}(\mathbf{x}, \mathbf{z}, t) = \frac{1}{2}\overrightarrow{a}\_{\text{J}} \left[ A \exp\left(k\_{\mathbf{z}}^{\text{S}} \mathbf{z}\right) + B \exp\left(-k\_{\mathbf{z}}^{\text{S}} \mathbf{z}\right) \right] \exp\left(ik\_{\mathbf{x}}\mathbf{x} - i\alpha t\right) + c.c., |\mathbf{z}| \le d \quad \text{(19)}$$

$$\begin{split} \overrightarrow{E}\_{SA}(\mathbf{x}, \mathbf{z}, t) &= \frac{1}{2} \left\{ \overrightarrow{a}\_{\mathbf{x}} \frac{k\_x^S}{i\alpha \varepsilon\_0 \varepsilon\_\perp} \left[ A \exp\left(k\_x^S \mathbf{z}\right) - B \exp\left(-k\_x^S \mathbf{z}\right) \right] \right. \\ &\left. - \overrightarrow{a}\_{\mathbf{z}} \frac{k\_x}{i\alpha \varepsilon\_0 \varepsilon\_\parallel} \left[ A \exp\left(k\_x^S \mathbf{z}\right) + B \exp\left(-k\_x^S \mathbf{z}\right) \right] \right\} \exp\left. i\left(k\_x \mathbf{x} - \alpha t\right) + c.c. \text{ } \left| \mathbf{z} \right| \leq d. \end{split} \tag{20}$$

Here c.c. stands for complex conjugate. The SPP fields (17)–(20) are confined in the *Z* direction. In the linear approximation substituting expressions (15), (18), and *Stimulated Scattering of Surface Plasmon Polaritons in a Plasmonic Waveguide with a Smectic… DOI: http://dx.doi.org/10.5772/intechopen.89483*

(20) into the homogeneous part of the wave equation (14) for the claddings and SmALC core, respectively, we obtain the following expressions for the complex wave numbers *km <sup>z</sup>* and *<sup>k</sup><sup>S</sup> <sup>z</sup>* [24, 26]:

$$k\_x^m = \sqrt{k\_x^2 - \varepsilon\_m(\alpha)\alpha^2/c^2} \tag{21}$$

$$k\_{\varepsilon}^{S} = \sqrt{k\_{\text{x}}^{2} \left(\varepsilon\_{\perp}/\varepsilon\_{\parallel}\right) - \alpha^{2} \varepsilon\_{\perp}/c^{2}}\tag{22}$$

where *c* is the free space light velocity. The boundary conditions for the fields (17)–(20) at the interfaces *z* ¼ �*d* have the form [7, 8]

$$H\_{\rm 1y}(z=d) = H\_{\rm SAy}(z=d);\ H\_{\rm 2y}(z=-d) = H\_{\rm SAy}(z=-d) \tag{23}$$

$$E\_{\rm Lx}(\mathbf{z} = d) = E\_{\rm SAT}(\mathbf{z} = d);\ E\_{\rm Lx}(\mathbf{z} = -d) = E\_{\rm SAT}(\mathbf{z} = -d) \tag{24}$$

Substituting expressions (17)–(20) into Eqs. (23) and (24), we obtain the dispersion relation for the SPP TM modes in the MIM waveguide given by [24, 26]

$$\exp\left(-4k\_x^S d\right) = \left(\frac{k\_x^m}{\varepsilon\_m(a)} + \frac{k\_x^S}{\varepsilon\_\perp}\right)^2 \left(\frac{k\_x^m}{\varepsilon\_m(a)} - \frac{k\_x^S}{\varepsilon\_\perp}\right)^{-2} \tag{25}$$

Dispersion relation obtained for the general case of different claddings [7] coincides with expression (25) for the symmetric structure with the same claddings. The results of the numerical solution of Eq. (25) for the typical values of the MIM waveguide parameters and the SPP frequencies *ω* corresponding to the optical wavelength range *λopt* � 1 � 1*:*6 μm and 2*d* � 1 μm are presented in **Figures 4** and **5**.

These results show that Re *k<sup>S</sup> <sup>z</sup>* � 106*m*�<sup>1</sup> <sup>≫</sup> Im*k<sup>S</sup> <sup>z</sup>* � <sup>10</sup><sup>4</sup>*m*�<sup>1</sup> and Re *kx* � <sup>10</sup><sup>7</sup> *m*�<sup>1</sup> <sup>≫</sup> Im*kx* � <sup>10</sup><sup>3</sup> *m*�<sup>1</sup> [24, 26]. In such a case, the SPP oscillation length in the Z direction is defined by the relationship 2*π* Im*k<sup>S</sup> z* � ��<sup>1</sup> � <sup>10</sup>�<sup>4</sup>*<sup>m</sup>* <sup>≫</sup> *<sup>d</sup>* � <sup>10</sup>�<sup>6</sup>*m*, and Im*k<sup>S</sup> <sup>z</sup>* can be neglected inside the MIM waveguide, and *<sup>k</sup><sup>S</sup> <sup>z</sup>* <sup>≈</sup> Re *<sup>k</sup><sup>S</sup> <sup>z</sup>* [24, 26]. The SPP propagation length in the *<sup>X</sup>* direction *LSPP* <sup>¼</sup> ð Þ Im*kx* �<sup>1</sup> � <sup>10</sup>�<sup>4</sup> � <sup>10</sup>�<sup>3</sup> *m* ≫ *λSPP* ¼ <sup>2</sup>*π*ð Þ Re *kx* �<sup>1</sup> <sup>&</sup>lt; <sup>10</sup>�<sup>6</sup>*<sup>m</sup>* where *<sup>λ</sup>SPP* is the SPP wavelength. Hence, at the optical wavelength-scale distances, Im*kx* can be neglected, and *kx* ≈ Re *kx* [24, 26].

**Figure 4.** *The spectral dependence of Re k<sup>S</sup> z(a) and Imk<sup>S</sup> <sup>z</sup> (b).*

The permittivity *εm*ð Þ *ω* of the metal claddings is described by the Drude model

*p*

*<sup>n</sup>*0*e*2*=*ð Þ *<sup>ε</sup>*0*<sup>m</sup>* <sup>p</sup> is the plasma frequency of the free electron gas; *<sup>n</sup>*<sup>0</sup> is

ð Þ *x*, *z*, *t* of the optical wave propagating in a nonlinear medium is

*<sup>∂</sup>t*<sup>2</sup> ¼ �*μ*<sup>0</sup>

!*<sup>L</sup>*

*<sup>z</sup>* ¼ *ε*<sup>0</sup> *a*<sup>∥</sup>

*∂*2 *D* !*NL*

*∂u ∂z*

*<sup>m</sup>* of the electric induction in the metal claddings has the form:

[7]. The SPP TM mode electric and magnetic fields for j j *z* >*d* in the

1,2ð Þ *x*, *z*, *t* and for j j *z* ≤*d* in SmALC

exp ∓ *km*

h i � �

h i � �

*z z*

� *<sup>B</sup>* exp �*kS*

<sup>þ</sup> *<sup>B</sup>* exp �*kS*

Here c.c. stands for complex conjugate. The SPP fields (17)–(20) are confined in the *Z* direction. In the linear approximation substituting expressions (15), (18), and

*z z*

*z z*

<sup>þ</sup> *<sup>B</sup>* exp �*kS*

*z z* � �

*z z* � � ½ � *<sup>ω</sup>*<sup>2</sup> <sup>þ</sup> ð Þ *<sup>i</sup>ω=<sup>τ</sup>* (13)

*<sup>∂</sup>t*<sup>2</sup> (14)

is the nonlinear part of the electric

!*<sup>L</sup>*

*<sup>z</sup>* ¼ *ε*0*ε*∥*Ez* (15)

*∂u ∂x Ex*

*Ez* � *ε<sup>a</sup>*

*<sup>z</sup> <sup>z</sup>* <sup>þ</sup> *ikxx* � *<sup>i</sup>ω<sup>t</sup>* � � <sup>þ</sup> *<sup>c</sup>:c:*, j j *<sup>z</sup>* <sup>&</sup>gt;*<sup>d</sup>* (17)

*<sup>z</sup> <sup>z</sup>* <sup>þ</sup> *ikxx* � *<sup>i</sup>ω<sup>t</sup>* � � <sup>þ</sup> *<sup>c</sup>:c:*, j j *<sup>z</sup>* <sup>&</sup>gt;*<sup>d</sup>* (18)

exp ð Þþ *ikxx* � *iωt c:c:*, j j *z* ≤ *d* (19)

g exp *i k*ð Þþ *xx* � *ωt c:c:*; j j *z* ≤ *d*

(20)

� �

! *TM* ¼

and *D* !*NL* in

(16)

*<sup>ε</sup>m*ð Þ¼ *<sup>ω</sup>* <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

described by the following wave equation including the nonlinear part of the

þ *μ*<sup>0</sup> *∂*2 *D* !*<sup>L</sup>*

magnetic (TM) mode with the electric and magnetic fields given by *E*

*TM* <sup>¼</sup> 0, *Hy*, 0 � � [7]. In such a case, we obtain for *<sup>D</sup>*

*<sup>x</sup>* <sup>¼</sup> *<sup>ε</sup>*0*ε*⊥*Ex*; *<sup>D</sup><sup>L</sup>*

; *DNL*

induction. The SPP can propagate in the plasmonic waveguide only as a transverse

the free electron density in the metal;*e*, *m* are the electron charge and mass, respectively; and *ω*, *τ* are the SPP angular frequency and lifetime, respectively [7, 8]. The

[7, 8]:

*Nanoplasmonics*

electric field *E*

ð Þ *Ex*, 0, *Ez* ; *H*

!

SmALC using Eq. (12)

*DNL*

*<sup>m</sup>* ¼ *ε*0*εm*ð Þ *ω E*

metal claddings *H*

*H* !

1,2ð Þ¼ *x*, *z*, *t*

*SA*ð Þ¼ *x*, *z*, *t*

*SA*ð Þ¼ *x*, *z*, *t*

*SA*ð Þ *x*, *z*, *t* , *E*

*D* !*<sup>L</sup>*

*H* !

> *E* !

*H* !

*E* !

**142**

*<sup>x</sup>* ¼ *ε*<sup>0</sup> *a*<sup>⊥</sup>

!

!

1,2ð Þ¼ *x*, *z*, *t*

1 <sup>2</sup> *<sup>a</sup>* !

1 2 *a* !

1 2 f*a* ! *x kS z iωε*0*ε*<sup>⊥</sup>

�*a* ! *z kx ωε*0*ε*<sup>∥</sup>

The linear part *D*

!

*∂u ∂z*

1,2ð Þ *x*, *z*, *t* , *E*

1 2 *a* !

*xE*1,2*x*<sup>0</sup> þ *a*

*<sup>y</sup> A* exp *kS*

!*<sup>L</sup>*

electric induction *D*

where *<sup>ω</sup><sup>p</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

!*NL*

[6]:

Here *μ*<sup>0</sup> is the free space permeability and *D*

*curlcurlE*!

*DL*

*∂u ∂x Ez*

*Ex* � *ε<sup>a</sup>*

!

*SA*ð Þ *x*, *z*, *t* have the form [24, 26]

! *zE*1,2*z*<sup>0</sup>

*z z* � �

h i

*yH*1,20 exp ∓ *km*

h i � �

*A* exp *kS*

*A* exp *kS*

� �

!

**Figure 5.** *The spectral dependence of Re kx (a) and Imkx (b).*

Consequently, for a given optical frequency *ω*, a single localized TM mode can exist in the SmALC core of the MIM waveguide with the electric field *E* ! *SA*ð Þ *x*, *z*, *t* given by [24, 26]

$$\begin{split} \overrightarrow{E}\_{SA} &= E\_0 \left[ \overrightarrow{a}\_x \cosh\left(\left(\operatorname{Re} k\_x^S\right) z\right) - \overrightarrow{a}\_x i \frac{k\_x \varepsilon\_\perp}{k\_x^S \varepsilon\_\parallel} \sinh\left(\left(\operatorname{Re} k\_x^S\right) z\right) \right] \\ &\times \exp\left[i\left(\left(\operatorname{Re} k\_x\right) x - \alpha t\right)\right] + c.c. \end{split} \tag{26}$$

*G kx*, *k<sup>S</sup>*

þ 1 2

**Figure 6.**

*D* !*NL*

part *D* !*NL*

**145**

*<sup>z</sup>* , Δ*ω*

�*i*Δ*ω*f�*α*1ð Þ 2 Re *kx* <sup>2</sup> 2 Re *<sup>k</sup><sup>S</sup>*

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

<sup>¼</sup> <sup>4</sup>*ρ*ð Þ <sup>Δ</sup>*<sup>ω</sup>* <sup>2</sup> �ð Þ Re *kx* <sup>2</sup> <sup>þ</sup> Re *kS*

ð Þ� *<sup>α</sup>*<sup>4</sup> <sup>þ</sup> *<sup>α</sup>*<sup>56</sup> ð Þ 2 Re *kx* <sup>2</sup> <sup>þ</sup> 2 Re *kS*

*z* <sup>2</sup>

<sup>2</sup> <sup>2</sup>

in the *Z* direction and oscillates in the propagation direction *X*.

**4. Nonlinear interaction of SPPs in the MIM waveguide**

*∂*2 *ESA*1,20 *∂t*2

  *z*

g � *<sup>B</sup>*ð Þ 2 Re *kx* <sup>2</sup> 2 Re *kS*

*z* <sup>2</sup> (30)

(31)

<sup>2</sup>

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

*z*

<sup>Δ</sup>*<sup>ω</sup>* � <sup>10</sup>8s�1, and it can be neglected. The normalized smectic layer displacement *u x*ð Þ , *z*, *t* ¼ *t*<sup>0</sup> *=U*<sup>0</sup> for the optical wavelength *λopt* ¼ 1*:*6μm is shown in **Figure 6**. It is seen from **Figure 6** that the dynamic grating is localized inside the MIM waveguide

The light-enhanced dynamic grating (28) results in the nonlinear polarization

propagating SPPs (27), we should solve wave Eq. (14) including the nonlinear term

defined by Eq. (16). In order to investigate the interaction of the counter-

. We use the SVA approximation for the SPP electric field amplitudes *ESA*1,20ðÞ¼ *t* j j *ESA*1,20ð Þ*t* exp *iθSA*1,2ð Þ*t* where j j *ESA*1,20ð Þ*t* and *θSA*1,2ð Þ*t* are the SVA magnitudes and phases, respectively [6]. For the distances of the order of magnitude of the SPP wavelength *λSPP* <1 μm, the dependence of SAVs on the *x* coordinate can be neglected. We assume according to the SVA approximation that

> ≪ *ω*<sup>1</sup>

*∂ESA*1,20 *∂t*

 

 

of the electric induction in SmALC. Then, substituting relationships (15),

Substituting expressions (27) and (28) into Eqs. (16), we evaluate the nonlinear

of Eq. (9) is overdamped for the typical values of SmALC parameters and

*The normalized smectic layer displacement u x*ð Þ , *z*, *t* ¼ *t*<sup>0</sup> *for the optical wavelength λopt* ¼ 1*:*6 *μm.*

Expression (28) is the enhanced solution of Eq. (9). The homogeneous solution

The numerical estimations show that for the SPP modes with the close optical frequencies *<sup>ω</sup>*1,2 � <sup>10</sup>15s�<sup>1</sup> and the frequency difference <sup>Δ</sup>*<sup>ω</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> � <sup>10</sup><sup>8</sup>*<sup>s</sup>* �1 ≪ *ω*1, the wave numbers of the both SPPs *kS <sup>z</sup>*1,2 and *kx*1,2 are practically equal. As a result, only counter-propagating SPP modes can strongly interact in the MIM core creating the dynamic grating of smectic layers as it is seen from Eq. (9). The electric field of the counter-propagating SPP modes of the type (26) in the MIM waveguide SmALC core has the form

$$\begin{split} \overrightarrow{E}\_{SA1,2} &= E\_{SA1,20} \left[ \overrightarrow{a}\_x \cosh\left(\left(\text{Re}\,k\_x^S\right)\mathbf{z}\right) \mp \overrightarrow{a}\_x i \frac{k\_x e\_\perp}{k\_x^S e\_\parallel} \sinh\left(\left(\text{Re}\,k\_x^S\right)\mathbf{z}\right) \right] \\ &\times \exp\left[\pm i(\text{Re}\,k\_x)\mathbf{x} - i o\_{1,2} t\right] + c.c. \end{split} \tag{27}$$

Substituting expression (27) into equation of motion (9), we obtain the expression of the smectic layer displacement localized dynamic grating *u x*ð Þ , *z*, *t* :

$$u(\mathbf{x}, \mathbf{z}, t) = U\_0 \sinh\left(2\left(\operatorname{Re} k\_x^S\right)\mathbf{z}\right) \exp\left[i\left(2\left(\operatorname{Re} k\_x\right)\mathbf{x} - \Delta w t\right)\right] + \mathbf{c}.\tag{28}$$

Here

$$\begin{aligned} U\_0 &= -\frac{4\epsilon\_0 E\_{SA10} E\_{SA20}^\*}{G\left(k\_x, k\_x^S, \Delta a\right)} \left(\operatorname{Re} k\_x\right)^2 \left(\operatorname{Re} k\_x^S\right) h; \\\ h &= \left\{ a\_\perp - a\_\parallel \frac{|k\_x|^2 \epsilon\_\perp^2}{\left|k\_x^S\right|^2 \epsilon\_\parallel^2} - 2\epsilon\_a \frac{(\operatorname{Re} k\_x)^2 \epsilon\_\perp}{\left(\operatorname{Re} k\_x^S\right)^2 \epsilon\_\parallel} \right\} \end{aligned} \tag{29}$$

**144**

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

#### **Figure 6.**

Consequently, for a given optical frequency *ω*, a single localized TM mode can exist

� *a* ! *zi kxε*<sup>⊥</sup> *kS z ε*∥

The numerical estimations show that for the SPP modes with the close optical

result, only counter-propagating SPP modes can strongly interact in the MIM core creating the dynamic grating of smectic layers as it is seen from Eq. (9). The electric field of the counter-propagating SPP modes of the type (26) in the MIM waveguide

frequencies *<sup>ω</sup>*1,2 � <sup>10</sup>15s�<sup>1</sup> and the frequency difference <sup>Δ</sup>*<sup>ω</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> � <sup>10</sup><sup>8</sup>*<sup>s</sup>*

*z* � �

sion of the smectic layer displacement localized dynamic grating *u x*ð Þ , *z*, *t* :

*z*

*z* � �

� �

*<sup>U</sup>*<sup>0</sup> ¼ � <sup>4</sup>*ε*0*ESA*10*E*<sup>∗</sup>

*h* ¼ *a*<sup>⊥</sup> � *a*<sup>∥</sup>

8 ><

>:

*G kx*, *k<sup>S</sup>*

*z* � � ∓ *a* ! *zi kxε*<sup>⊥</sup> *kS z ε*∥

Substituting expression (27) into equation of motion (9), we obtain the expres-

*SA*20

� � ð Þ Re *kx* <sup>2</sup> Re *<sup>k</sup><sup>S</sup>*

� 2*ε<sup>a</sup>*

*<sup>z</sup>* , Δ*ω*

j j *kx* <sup>2</sup> *ε*2 ⊥

*kS z* � � � � 2 *ε*2 ∥

� � " #

� � " #

!

*z* � �

*<sup>z</sup>*1,2 and *kx*1,2 are practically equal. As a

sinh Re *k<sup>S</sup>*

exp ½*i*ð Þ 2 Re ð Þ *kx x* � Δ*ωt* � þ *c:c:* (28)

*z* � �

*ε*⊥

*ε*∥

ð Þ Re *kx* <sup>2</sup>

Re *k<sup>S</sup> z* � �<sup>2</sup> *h*;

9 >=

>;

*z* � �

*z*

*z*

sinh Re *k<sup>S</sup>*

*SA*ð Þ *x*, *z*, *t* given

(26)

�1

(27)

(29)

in the SmALC core of the MIM waveguide with the electric field *E*

*z* � �

*z* � �

*<sup>x</sup>* cosh Re *k<sup>S</sup>*

� exp ½*i*ð Þ ð Þ Re *kx x* � *ωt* � þ *c:c:*

*<sup>x</sup>* cosh Re *k<sup>S</sup>*

� exp ½�*i*ð Þ Re *kx x* � *iω*1,2*t*� þ *c:c:*

by [24, 26]

**Figure 5.**

*Nanoplasmonics*

*E* !

*SA* ¼ *E*<sup>0</sup> *a*

SmALC core has the form

*SA*1,2 ¼ *ESA*1,20 *a*

*E* !

Here

**144**

!

*The spectral dependence of Re kx (a) and Imkx (b).*

≪ *ω*1, the wave numbers of the both SPPs *kS*

!

*u x*ð Þ¼ , *<sup>z</sup>*, *<sup>t</sup> <sup>U</sup>*<sup>0</sup> sinh 2 Re *<sup>k</sup><sup>S</sup>*

*The normalized smectic layer displacement u x*ð Þ , *z*, *t* ¼ *t*<sup>0</sup> *for the optical wavelength λopt* ¼ 1*:*6 *μm.*

$$\begin{split} G\left(k\_{\rm x},k\_{\rm x}^{\rm S},\Delta\boldsymbol{\alpha}\right) &= 4\rho(\Delta\boldsymbol{\alpha})^{2}\left[-\left(\operatorname{Re}\,k\_{\rm x}\right)^{2}+\left(\operatorname{Re}\,k\_{\rm x}^{\rm S}\right)^{2}\right] \\ &- i\Delta\boldsymbol{\alpha}\left\{-\alpha\_{1}(2\operatorname{Re}\,k\_{\rm x})^{2}\left(2\operatorname{Re}\,k\_{\rm x}^{\rm S}\right)^{2} \\ &+\frac{1}{2}\left(\boldsymbol{\alpha}\_{4}+\boldsymbol{\alpha}\_{56}\right)\left[-\left(2\operatorname{Re}\,k\_{\rm x}\right)^{2}+\left(2\operatorname{Re}\,k\_{\rm x}^{\rm S}\right)^{2}\right]^{2}\right\}-B(2\operatorname{Re}\,k\_{\rm x})^{2}\left(2\operatorname{Re}\,k\_{\rm x}^{\rm S}\right)^{2} \end{split} \tag{30}$$

Expression (28) is the enhanced solution of Eq. (9). The homogeneous solution of Eq. (9) is overdamped for the typical values of SmALC parameters and <sup>Δ</sup>*<sup>ω</sup>* � <sup>10</sup>8s�1, and it can be neglected. The normalized smectic layer displacement *u x*ð Þ , *z*, *t* ¼ *t*<sup>0</sup> *=U*<sup>0</sup> for the optical wavelength *λopt* ¼ 1*:*6μm is shown in **Figure 6**. It is seen from **Figure 6** that the dynamic grating is localized inside the MIM waveguide in the *Z* direction and oscillates in the propagation direction *X*.

### **4. Nonlinear interaction of SPPs in the MIM waveguide**

The light-enhanced dynamic grating (28) results in the nonlinear polarization defined by Eq. (16). In order to investigate the interaction of the counterpropagating SPPs (27), we should solve wave Eq. (14) including the nonlinear term *D* !*NL* . We use the SVA approximation for the SPP electric field amplitudes *ESA*1,20ðÞ¼ *t* j j *ESA*1,20ð Þ*t* exp *iθSA*1,2ð Þ*t* where j j *ESA*1,20ð Þ*t* and *θSA*1,2ð Þ*t* are the SVA magnitudes and phases, respectively [6]. For the distances of the order of magnitude of the SPP wavelength *λSPP* <1 μm, the dependence of SAVs on the *x* coordinate can be neglected. We assume according to the SVA approximation that

$$\left|\frac{\partial^2 E\_{\rm SA1,20}}{\partial t^2}\right| \ll \alpha\_1 \left|\frac{\partial E\_{\rm SA1,20}}{\partial t}\right|\tag{31}$$

Substituting expressions (27) and (28) into Eqs. (16), we evaluate the nonlinear part *D* !*NL* of the electric induction in SmALC. Then, substituting relationships (15),

### *Nanoplasmonics*

(16), and (27) into wave equation (14), taking into account the dispersion relation (22), neglecting the terms *∂*<sup>2</sup> *ESA*1,20*=∂t* <sup>2</sup> according to condition (31), combining the phase-matched terms with the frequencies *ω*1,2, and dividing the real and imaginary parts, we derive the equations for the SVA magnitudes j j *ESA*1,20ð Þ*t* and phases *θSA*1,2ð Þ*t* . They have the form

$$\begin{split} & \frac{1}{\alpha\_{1,2}} \frac{\partial |E\_{\rm SAT,20}(t)|^{2}}{\partial t} F\_{1}(\mathbf{z}) \\ &= \mp \frac{8\varepsilon\_{0} \text{Im} G\left(k\_{\rm x}, k\_{\rm x}^{\rm S}, \Delta\alpha\right) \left|\operatorname{Re} k\_{\rm x}\right|^{2} h |E\_{\rm SAT,0}(t)|^{2} |E\_{\rm SAT,20}(t)|^{2}}{\varepsilon\_{\perp} \left|\operatorname{G}\left(k\_{\rm x}, k\_{\rm x}^{\rm S}, \Delta\alpha\right)\right|^{2}} F\_{2}(\mathbf{z}) \\ & \quad \frac{1}{\alpha\_{1,2}} \frac{\partial \theta\_{1,2}}{\partial t} F\_{1}(\mathbf{z}) \\ &= -\frac{4\varepsilon\_{0} \operatorname{Re} G\left(k\_{\rm x}, k\_{\rm x}^{\rm S}, \Delta\alpha\right) |E\_{\rm SAT,10}(t)|^{2} |\operatorname{Re} k\_{\rm x}|^{2} h}{\varepsilon\_{\perp} \left|G\left(k\_{\rm x}, k\_{\rm x}^{\rm S}, \Delta\alpha\right)\right|^{2}} F\_{2}(\mathbf{z}) \end{split} \tag{33}$$

Here we assumed that the factor exp ½ � �ð Þ Im*kx x* ≈ 1 for the distances *<sup>x</sup>* <sup>≪</sup> ð Þ Im*kx* �<sup>1</sup> . The functions *F*1,2ð Þ*z* describing the SPP mode localization inside the MIM waveguide are given by

$$F\_1(\mathbf{z}) = \cosh^2\left(k\_x^S \mathbf{z}\right) + \frac{\left|k\_x\right|^2}{\left|k\_x^S\right|^2} \sinh^2\left(k\_x^S \mathbf{z}\right) \tag{34}$$

$$\cosh^2\left(k\_x^S \mathbf{z}\right) \left[\cosh\left(2k\_x^S \mathbf{z}\right)\left(a\_\perp \left(k\_x^S\right)^2 + e\_a \frac{e\_\perp}{e\_a} k\_x^2\right) - e\_a \frac{e\_\perp}{e\_a} k\_x^2\right]$$

where the localization factor *FN kx*, *kS*

*The spectral dependence of the localization factor FN kx*, *k<sup>S</sup>*

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

*FN kx*, *k<sup>S</sup> z* � �

sinh 4*k<sup>S</sup>*

<sup>þ</sup> sinh 2*k<sup>S</sup>*

� sinh 2*kS*

2 6 4

**Figure 7**.

j j *ESA*1,20ð Þ*<sup>t</sup>* <sup>2</sup> [6]:

**147**

*zd* � � <sup>þ</sup> <sup>4</sup>*kS*

4

0

B@

0

B@

*zd* � � *<sup>a</sup>*<sup>⊥</sup> <sup>þ</sup> *<sup>a</sup>*<sup>∥</sup>

*zd* � � <sup>1</sup> <sup>þ</sup>

It is seen from **Figure 7** that *FN kx*, *kS*

*∂ ∂t*

h i

*zd*

6 4

*k*2 *x kS z* � �<sup>2</sup>

The spectral dependence of the localization factor *FN kx*, *k<sup>S</sup>*

j j *ESA*10ð Þ*<sup>t</sup>* <sup>2</sup> *ω*1

¼ f

**Figure 7.**

*z* � � is given by

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

1

CA <sup>þ</sup> <sup>2</sup>*k<sup>S</sup>*

*z*

!

We obtain from Eq. (39) the Manley-Rowe relation for the SVA magnitudes

the range of the optical wavelengths essential for optical communications. The addition of Eq. (36) results in the following conservation condition [6]:

*ε*⊥*k*<sup>2</sup> *x ε*<sup>∥</sup> *k<sup>S</sup> z* � �<sup>2</sup> <sup>þ</sup> *<sup>ε</sup><sup>a</sup>*

*z* � �*.*

CA � <sup>2</sup>*k<sup>S</sup>*

*zd* � � <sup>1</sup> � *<sup>k</sup>*<sup>2</sup>

<sup>þ</sup> j j *ESA*20ð Þ*<sup>t</sup>* <sup>2</sup> *ω*2

� � <sup>2</sup>

*zd* � �*ε<sup>a</sup>*

0

B@

*k*2 *x kS z*

*k*2 *x kS z*

*x kS z* � �<sup>2</sup>

� �<sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>ε</sup>*<sup>⊥</sup>

� �<sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>ε</sup>*<sup>⊥</sup>

1

3 7 5

�1

*z*

CA

� � is varying by an order of magnitude in

*ε*∥

*ε*∥ � �

� � is presented in

¼ 0 (39)

3 7 5

g

(38)

*a*<sup>⊥</sup> � *a*<sup>∥</sup>

*ε*⊥*k*<sup>2</sup> *x ε*<sup>∥</sup> *kS z* � �<sup>2</sup>

1

$$\begin{split} F\_2(\boldsymbol{z}) &= \cosh^2 \left( k\_x^S \boldsymbol{z} \right) \left[ \cosh \left( 2k\_x^S \boldsymbol{z} \right) \left( \boldsymbol{a}\_\perp \left( k\_x^S \right)^\* + \varepsilon\_a \frac{\boldsymbol{e}\_\perp}{\varepsilon\_\parallel} k\_x^2 \right) - \varepsilon\_a \frac{\boldsymbol{e}\_\perp}{\varepsilon\_\parallel} k\_x^2 \right] \\ &- k\_x^2 \sinh^2 \left( k\_x^S \boldsymbol{z} \right) \left[ \cosh \left( 2k\_x^S \boldsymbol{z} \right) \left( \boldsymbol{a}\_\parallel \frac{\boldsymbol{e}\_\perp}{\varepsilon\_\parallel} - \boldsymbol{e}\_\perp \right) - \varepsilon\_a \right] \end{split} \tag{35}$$

Here we neglected the small quantities Im*kx* and Im*kS <sup>z</sup>* assuming that for *<sup>x</sup>* <sup>≪</sup> ð Þ Im*kx* �<sup>1</sup> and j j *<sup>z</sup>* <sup>≤</sup>*d*, we may use the relationships *kx* <sup>≈</sup> Re *kx* and *kS <sup>z</sup>* <sup>≈</sup> Re *<sup>k</sup><sup>S</sup> z* . We integrate both parts of Eqs. (32) and (33) over the MIM waveguide thickness �*d*≤*z*≤ *d* [40]. After the integration, Eqs. (32) and (33) take the form

$$\begin{split} \frac{1}{\alpha\_{1,2}} \frac{\partial |E\_{\rm SA1,20}(t)|^{2}}{\partial t} \\ &= \mp \frac{8\epsilon\_{0} \text{Im} G\left(k\_{\rm x}, k\_{\rm x}^{\rm S}, \Delta\alpha\right) k\_{\rm x}^{2} h\left(k\_{\rm x}^{\rm S}\right)^{2}}{\epsilon\_{\perp} \left| G\left(k\_{\rm x}, k\_{\rm x}^{\rm S}, \Delta\alpha\right) \right|^{2}} \\ &\times |E\_{\rm SA10}(t)|^{2} |E\_{\rm SA20}(t)|^{2} F\_{\rm N}\left(k\_{\rm x}, k\_{\rm x}^{\rm S}\right) \end{split} \tag{36}$$
 
$$\begin{split} \frac{1}{\alpha\_{1,2}} \frac{\partial \theta\_{1,2}}{\partial t} \\ &= -\frac{4\epsilon\_{0} \operatorname{Re} G\left(k\_{\rm x}, k\_{\rm x}^{\rm S}, \Delta\alpha\right) k\_{\rm x}^{2} h\left(k\_{\rm x}^{\rm S}\right)^{2}}{\epsilon\_{\perp} \left| G\left(k\_{\rm x}, k\_{\rm x}^{\rm S}, \Delta\alpha\right) \right|^{2}} |E\_{\rm SA2,10}(t)|^{2} F\_{\rm N}\left(k\_{\rm x}, k\_{\rm x}^{\rm S}\right) \end{split} \tag{37}$$

1

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

**Figure 7.** *The spectral dependence of the localization factor FN kx*, *k<sup>S</sup> z* � �*.*

where the localization factor *FN kx*, *kS z* � � is given by

$$\begin{split} &F\_{N}\left(k\_{x},k\_{x}^{\mathcal{S}}\right) \\ &= \left\{\frac{\sinh\left(4k\_{x}^{\mathcal{S}}d\right)+4k\_{x}^{\mathcal{S}}d}{4}\right\}\left[a\_{\perp}-a\_{\parallel}\frac{\varepsilon\_{\perp}k\_{x}^{\mathcal{S}}}{\varepsilon\_{\parallel}\left(k\_{x}^{\mathcal{S}}\right)^{2}}+\varepsilon\_{\text{a}}\frac{k\_{x}^{2}}{\left(k\_{x}^{\mathcal{S}}\right)^{2}}\left(1+\frac{\varepsilon\_{\perp}}{\varepsilon\_{\parallel}}\right)\right] \\ &+\sinh\left(2k\_{x}^{\mathcal{S}}d\right)\left(a\_{\perp}+a\_{\parallel}\frac{\varepsilon\_{\perp}k\_{x}^{\mathcal{S}}}{\varepsilon\_{\parallel}\left(k\_{x}^{\mathcal{S}}\right)^{2}}\right)-\left(2k\_{x}^{\mathcal{S}}d\right)\varepsilon\_{\text{a}}\frac{k\_{x}^{2}}{\left(k\_{x}^{\mathcal{S}}\right)^{2}}\left(1+\frac{\varepsilon\_{\perp}}{\varepsilon\_{\parallel}}\right)\right. \\ &\times\left[\sinh\left(2k\_{x}^{\mathcal{S}}d\right)\left(1+\frac{k\_{x}^{2}}{\left(k\_{x}^{\mathcal{S}}\right)^{2}}\right)+\left(2k\_{x}^{\mathcal{S}}d\right)\left(1-\frac{k\_{x}^{2}}{\left(k\_{x}^{\mathcal{S}}\right)^{2}}\right)\right]^{-1} \end{split} \tag{38}$$

The spectral dependence of the localization factor *FN kx*, *k<sup>S</sup> z* � � is presented in **Figure 7**.

It is seen from **Figure 7** that *FN kx*, *kS z* � � is varying by an order of magnitude in the range of the optical wavelengths essential for optical communications. The addition of Eq. (36) results in the following conservation condition [6]:

$$\frac{\partial}{\partial t} \left( \frac{|E\_{\rm SA10}(t)|^2}{o\_1} + \frac{|E\_{\rm SA20}(t)|^2}{o\_2} \right) = \mathbf{0} \tag{39}$$

We obtain from Eq. (39) the Manley-Rowe relation for the SVA magnitudes j j *ESA*1,20ð Þ*<sup>t</sup>* <sup>2</sup> [6]:

(16), and (27) into wave equation (14), taking into account the dispersion relation

phase-matched terms with the frequencies *ω*1,2, and dividing the real and imaginary parts, we derive the equations for the SVA magnitudes j j *ESA*1,20ð Þ*t* and phases

<sup>2</sup> according to condition (31), combining the

j j *ESA*20ð Þ*<sup>t</sup>* <sup>2</sup>

(32)

(33)

(34)

(35)

(36)

(37)

*<sup>z</sup>* <sup>≈</sup> Re *<sup>k</sup><sup>S</sup> z* .

<sup>2</sup> *F*2ð Þ*z*

½ � Re *kx* <sup>2</sup> *h*

<sup>2</sup> *F*2ð Þ*z*

*z z* 

� *ε<sup>a</sup>*

� *ε<sup>a</sup> ε*⊥ *ε*∥ *k*2 *x*

*<sup>z</sup>* assuming that for

*ESA*1,20*=∂t*

*F*1ð Þ*z*

*F*1ð Þ*z*

4*ε*<sup>0</sup> Re *G kx*, *k<sup>S</sup>*

*<sup>F</sup>*1ð Þ¼ *<sup>z</sup>* cosh <sup>2</sup> *kS*

cosh 2*kS*

Here we neglected the small quantities Im*kx* and Im*kS*

*<sup>z</sup>* , Δ*ω* 

½ � Re *kx* <sup>2</sup>

*<sup>z</sup>* , Δ*ω* 

*ε*<sup>⊥</sup> *G kx*, *k<sup>S</sup>*

Here we assumed that the factor exp ½ � �ð Þ Im*kx x* ≈ 1 for the distances

*z z* 

*z z* 

We integrate both parts of Eqs. (32) and (33) over the MIM waveguide thickness

*ε*<sup>⊥</sup> *G kx*, *k<sup>S</sup>*

*z z* 

cosh 2*k<sup>S</sup>*

*<sup>x</sup>* <sup>≪</sup> ð Þ Im*kx* �<sup>1</sup> and j j *<sup>z</sup>* <sup>≤</sup>*d*, we may use the relationships *kx* <sup>≈</sup> Re *kx* and *kS*

�*d*≤*z*≤ *d* [40]. After the integration, Eqs. (32) and (33) take the form

8*ε*0Im*G kx*, *kS*

*k*2 *xh kS z* <sup>2</sup>

> 

*<sup>∂</sup>*j j *ESA*1,20ð Þ*<sup>t</sup>* <sup>2</sup> *∂t*

�j j *ESA*10ð Þ*<sup>t</sup>* <sup>2</sup>

*<sup>z</sup>* , Δ*ω* 

*<sup>z</sup>* , Δ*ω* 

*<sup>z</sup>* , Δ*ω* 

*ε*<sup>⊥</sup> *G kx*, *k<sup>S</sup>*

*h E*j j *SA*10ð Þ*<sup>t</sup>* <sup>2</sup>

j j *ESA*2,10ð Þ*<sup>t</sup>* <sup>2</sup>

 

. The functions *F*1,2ð Þ*z* describing the SPP mode localization inside the

<sup>2</sup> sinh <sup>2</sup> *<sup>k</sup><sup>S</sup>*

þ *ε<sup>a</sup> ε*⊥ *ε*∥ *k*2 *x*

<sup>þ</sup> j j *kx* <sup>2</sup> *kS z* 

*a*<sup>⊥</sup> *k<sup>S</sup> z* <sup>2</sup>

> *a*∥ *ε*⊥ *ε*∥ � *ε<sup>a</sup>*

*<sup>z</sup>* , Δ*ω* 

j j *ESA*20ð Þ*<sup>t</sup>* <sup>2</sup>

*<sup>z</sup>* , Δ*ω* 

<sup>2</sup> j j *ESA*2,10ð Þ*<sup>t</sup>* <sup>2</sup>

*k*2 *xh kS z* <sup>2</sup>

> 2

*FN kx*, *k<sup>S</sup> z* 

> *FN kx*, *kS z*

*<sup>z</sup>* , Δ*ω* 

 

(22), neglecting the terms *∂*<sup>2</sup>

*θSA*1,2ð Þ*t* . They have the form

*<sup>∂</sup>*j j *ESA*1,20ð Þ*<sup>t</sup>* <sup>2</sup> *∂t*

> 1 *ω*1,2

¼ �

MIM waveguide are given by

*<sup>F</sup>*2ð Þ¼ *<sup>z</sup>* cosh <sup>2</sup> *<sup>k</sup><sup>S</sup>*

�*k*<sup>2</sup>

1 *ω*1,2

¼ �

**146**

*∂θ*1,2 *∂t*

4*ε*<sup>0</sup> Re *G kx*, *kS*

*ε*<sup>⊥</sup> *G kx*, *kS*

8*ε*0Im*G kx*, *kS*

*∂θ*1,2 *∂t*

*z z* 

> *z z*

> > 1 *ω*1,2

¼ ∓

*<sup>x</sup>* sinh <sup>2</sup> *<sup>k</sup><sup>S</sup>*

1 *ω*1,2

*Nanoplasmonics*

¼ ∓

*<sup>x</sup>* <sup>≪</sup> ð Þ Im*kx* �<sup>1</sup>

*Nanoplasmonics*

$$\frac{\left|E\_{\rm SA10}(t)\right|^2}{a\_1} + \frac{\left|E\_{\rm SA20}(t)\right|^2}{a\_2} = const = I\_0 \tag{40}$$

exchange between the SPPs interfering on the smectic layer dynamic grating. Indeed, *I*<sup>1</sup> ! 0 and *I*<sup>2</sup> ! 1 for *g* >0 and *t* ! ∞. Actually, the SLS of the orientational type takes place [6]. The *SPP*<sup>1</sup> with the normalized intensity *I*<sup>1</sup> plays a role of the pumping wave, while the *SPP*<sup>2</sup> is a signal wave. The temporal dependence of *<sup>I</sup>*1,2ð Þ*<sup>t</sup>* for the pumping wave amplitude j j¼ *ESA*10ð Þ*<sup>t</sup>* <sup>10</sup>6*V=<sup>m</sup>* is shown in **Figure 9**. It is seen form **Figure 6** that for *I*1ð Þ 0 > *I*2ð Þ 0 , the characteristic time *t*<sup>0</sup> exists

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

when *I*1ð Þ¼ *t*<sup>0</sup> *I*2ð Þ *t*<sup>0</sup> . Using expressions (44) and (45), we obtain

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

*I*1,2ðÞ¼ *t*

*θSA*1ðÞ�*t θSA*1ð Þ¼� 0

*θSA*2ð Þ�*t θSA*2ð Þ¼ 0

**Figure 9.**

**149**

j j¼ *ESA*10ð Þ*<sup>t</sup>* <sup>10</sup><sup>6</sup>*V=m.*

the phases *θSA*1,2ð Þ*t* . They are given by the following expressions:

*<sup>t</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> *g*

> 1 2

ln *<sup>I</sup>*1ð Þ <sup>0</sup> *I*2ð Þ 0

Substitute expression (46) into Eqs. (44) and (45). Then they take the form

<sup>1</sup><sup>∓</sup> tanh *<sup>g</sup>*

The time duration of the energy exchange between the SPPs is about 1*ns* as it is seen from **Figure 9**. Substituting relationships (43)–(45) into Eq. (37), we evaluate

Re *G kx*, *kS*

2Im*G kx*, *kS*

*The temporal dependence of the SPP normalized intensities I*1,2ð Þ*t for pumping wave amplitude*

2

Re *G kx*, *k<sup>S</sup>*

2Im*G kx*, *k<sup>S</sup>*

ð Þ *t* � *t*<sup>0</sup>

*<sup>z</sup>* , Δ*ω* � �

*<sup>z</sup>* , Δ*ω* � �

(48)

(49)

� ln ½ � *I*2ð Þ 0 exp ð Þþ *gt I*1ð Þ 0

*<sup>z</sup>* , Δ*ω* � �

*<sup>z</sup>* , Δ*ω* � �

� ln 1½ � � *I*1ð Þþ 0 *I*1ð Þ 0 exp ð Þ �*gt*

� � (46)

h i h i (47)

We introduce the dimensionless quantities

$$I\_{1,2}(t) = \frac{\left|E\_{\text{SA1,20}}(t)\right|^2}{o\_{1,2}I\_0};\ I\_1(t) + I\_2(t) = \mathbf{1}\tag{41}$$

Substituting relationship (41) into Eq. (36), we obtain

$$\frac{\partial I\_{1,2}}{\partial t} = \mp \text{g}I\_1I\_2 \tag{42}$$

where the gain *g* has the form

$$\mathbf{g} = \frac{8e\_0 \text{Im} G\left(k\_{\text{x}}, k\_{\text{x}}^S, \Delta\alpha\right) k\_{\text{x}}^2 h\left(k\_{\text{x}}^S\right)^2 \alpha\_1 \alpha\_2 I\_0}{e\_\perp \left| G\left(k\_{\text{x}}, k\_{\text{x}}^S, \Delta\alpha\right) \right|^2} F\_N\left(k\_{\text{x}}, k\_{\text{x}}^S\right) \tag{43}$$

The spectral dependence of the gain *g* is shown in **Figure 8**. The solution of Eq. (41) has the form

$$I\_1(t) = \frac{I\_1(0) \exp\left(-gt\right)}{1 - I\_1(0)[1 - \exp\left(-gt\right)]}\tag{44}$$

$$I\_2(t) = \frac{1 - I\_1(0)}{1 - I\_1(0)[1 - \exp\left(-gt\right)]} \tag{45}$$

It is easy to see from Eqs. (44) and (45) that the solutions *I*1,2ð Þ*t* satisfy the Manley-Rowe relation (40). Expressions (44) and (45) describe the energy

**Figure 8.** *The spectral dependence of the gain* <sup>g</sup> *for the SPP electric field amplitude E*j j¼ *SA*10ð Þ*<sup>t</sup>* <sup>10</sup><sup>6</sup>*V=m.*

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

exchange between the SPPs interfering on the smectic layer dynamic grating. Indeed, *I*<sup>1</sup> ! 0 and *I*<sup>2</sup> ! 1 for *g* >0 and *t* ! ∞. Actually, the SLS of the orientational type takes place [6]. The *SPP*<sup>1</sup> with the normalized intensity *I*<sup>1</sup> plays a role of the pumping wave, while the *SPP*<sup>2</sup> is a signal wave. The temporal dependence of *<sup>I</sup>*1,2ð Þ*<sup>t</sup>* for the pumping wave amplitude j j¼ *ESA*10ð Þ*<sup>t</sup>* <sup>10</sup>6*V=<sup>m</sup>* is shown in **Figure 9**.

It is seen form **Figure 6** that for *I*1ð Þ 0 > *I*2ð Þ 0 , the characteristic time *t*<sup>0</sup> exists when *I*1ð Þ¼ *t*<sup>0</sup> *I*2ð Þ *t*<sup>0</sup> . Using expressions (44) and (45), we obtain

$$t\_0 = \frac{1}{\mathcal{g}} \ln \left[ \frac{I\_1(\mathbf{0})}{I\_2(\mathbf{0})} \right] \tag{46}$$

Substitute expression (46) into Eqs. (44) and (45). Then they take the form

$$I\_{1,2}(t) = \frac{1}{2} \left[ \mathbf{1} \mp \tanh \left[ \frac{\mathbf{g}}{2} (t - t\_0) \right] \right] \tag{47}$$

The time duration of the energy exchange between the SPPs is about 1*ns* as it is seen from **Figure 9**. Substituting relationships (43)–(45) into Eq. (37), we evaluate the phases *θSA*1,2ð Þ*t* . They are given by the following expressions:

$$
\theta\_{\rm SA1}(t) - \theta\_{\rm SA1}(0) = -\frac{\mathrm{Re}\,\mathsf{G}\left(k\_x, k\_x^S, \Delta\omega\right)}{2\mathrm{Im}\,\mathsf{G}\left(k\_x, k\_x^S, \Delta\omega\right)}\tag{48}
$$

$$
\times \ln\left[I\_2(\mathbf{0})\exp\left(\mathfrak{g}t\right) + I\_1(\mathbf{0})\right]
$$

$$
\theta\_{\rm SA2}(t) - \theta\_{\rm SA2}(0) = \frac{\mathrm{Re}\,\mathsf{G}\left(k\_x, k\_x^S, \Delta\omega\right)}{2\mathrm{Im}\,\mathsf{G}\left(k\_x, k\_x^S, \Delta\omega\right)}\tag{49}
$$

$$
\times \ln\left[1 - I\_1(\mathsf{0}) + I\_1(\mathsf{0})\exp\left(-\mathfrak{g}t\right)\right]
$$

#### **Figure 9.**

j j *ESA*10ð Þ*<sup>t</sup>* <sup>2</sup> *ω*1

*<sup>I</sup>*1,2ðÞ¼ *<sup>t</sup>* j j *ESA*1,20ð Þ*<sup>t</sup>* <sup>2</sup>

Substituting relationship (41) into Eq. (36), we obtain

8*ε*0Im*G kx*, *k<sup>S</sup>*

*ω*1,2*I*<sup>0</sup>

*∂I*1,2

*<sup>z</sup>* , Δ*ω* 

*ε*<sup>⊥</sup> *G kx*, *k<sup>S</sup>*

*I*1ðÞ¼ *t*

*I*2ðÞ¼ *t*

*k*2 *xh k<sup>S</sup> z* <sup>2</sup>

*<sup>z</sup>* , Δ*ω* 

The spectral dependence of the gain *g* is shown in **Figure 8**. The solution of

It is easy to see from Eqs. (44) and (45) that the solutions *I*1,2ð Þ*t* satisfy the Manley-Rowe relation (40). Expressions (44) and (45) describe the energy

*The spectral dependence of the gain* <sup>g</sup> *for the SPP electric field amplitude E*j j¼ *SA*10ð Þ*<sup>t</sup>* <sup>10</sup><sup>6</sup>*V=m.*

 

*I*1ð Þ 0 exp ð Þ �*gt*

1 � *I*1ð Þ 0

We introduce the dimensionless quantities

where the gain *g* has the form

*g* ¼

Eq. (41) has the form

*Nanoplasmonics*

**Figure 8.**

**148**

<sup>þ</sup> j j *ESA*20ð Þ*<sup>t</sup>* <sup>2</sup> *ω*2

¼ *const* ¼ *I*<sup>0</sup> (40)

; *I*1ð Þþ*t I*2ðÞ¼ *t* 1 (41)

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> <sup>∓</sup>*gI*1*I*<sup>2</sup> (42)

*ω*1*ω*2*I*<sup>0</sup>

<sup>2</sup> *FN kx*, *<sup>k</sup><sup>S</sup>*

<sup>1</sup> � *<sup>I</sup>*1ð Þ <sup>0</sup> ½ � <sup>1</sup> � exp ð Þ �*gt* (44)

<sup>1</sup> � *<sup>I</sup>*1ð Þ <sup>0</sup> ½ � <sup>1</sup> � exp ð Þ �*gt* (45)

*z* 

(43)

*The temporal dependence of the SPP normalized intensities I*1,2ð Þ*t for pumping wave amplitude* j j¼ *ESA*10ð Þ*<sup>t</sup>* <sup>10</sup><sup>6</sup>*V=m.*

The temporal dependence of the SPP SVA phases *θSA*1,2ð Þ*t* is shown in **Figure 10**.

It is seen from expressions (48) and (49) that SLS of the SPPs in the MIM waveguide is accompanied by XPM. For the large time intervals *t* ! ∞, the phase of the pumping wave increases linearly:

$$
\theta\_{\rm SA1}(t) - \theta\_{\rm SA1}(0) \to -\frac{\text{Re}\, G\left(k\_x, k\_x^{\rm S}, \Delta\alpha\right)}{2\text{Im}\, G\left(k\_x, k\_x^{\rm S}, \Delta\alpha\right)} \text{gt} \tag{50}
$$

Such a behavior corresponds to the rapid oscillations of the depleted pumping wave amplitude. The signal wave phase for *t* ! ∞ tends to a constant value:

$$
\theta\_{\text{SA2}}(t) - \theta\_{\text{SA2}}(\mathbf{0}) \to \frac{\text{Re}\,\mathbf{G}\left(k\_{\text{x}}, k\_{\text{x}}^{\text{S}}, \Delta\alpha\right)}{2\text{Im}\,\mathbf{G}\left(k\_{\text{x}}, k\_{\text{x}}^{\text{S}}, \Delta\alpha\right)} \ln\left[1 - I\_{1}(\mathbf{0})\right] \tag{51}
$$

Substituting expressions (41) and (47) into Eq. (29), we obtain the explicit expression for the dynamic grating amplitude. It takes the form

$$U\_0 = -\frac{2\epsilon\_0 I\_0 k\_x^2 k\_x^S \sqrt{\alpha\_1 \alpha\_2} h}{G\left(k\_x, k\_x^S, \Delta \alpha\right) \cosh\left[\frac{\mathfrak{g}}{2}(t - t\_0)\right]} \exp\left[i(\theta\_{\rm SA1} - \theta\_{\rm SA2})\right] \tag{52}$$

The temporal dependence of the amplitude (52) normalized absolute value j j *U*0*=U*0 max is presented in **Figure 11**. Here

$$|U\_{0\max}| = \frac{2\varepsilon\_0 I\_0 \left| k\_x^2 k\_x^S \right| \sqrt{\alpha\_1 \alpha\_2} |h|}{\left| G \left( k\_x, k\_x^S, \Delta \nu \right) \right|} \tag{53}$$

*vx*ð Þ¼ *x*, *z*, *t* Δ*ωU*<sup>0</sup>

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

**5. Conclusions**

**151**

**Figure 11.**

energy exchange between the interacting SPPs.

*kS z kx*

cosh 2*k<sup>S</sup>*

*z z*

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

*The temporal dependence of the dynamic grating amplitude normalized absolute value U*j j <sup>0</sup>*=U*<sup>0</sup> max *.*

Expressions (28) and (52)–(55) and **Figure 11** show that the orientational and hydrodynamic excitations in SmALC core of the MIM waveguide enhanced by the SPPs are spatially localized and reach their maximum value during the time of the

We investigated theoretically the nonlinear interaction of SPPs in the MIM waveguide with the SmALC core. The third-order nonlinearity mechanism is related to the smectic layer oscillations that take place without the change of the mass density. We solved simultaneously the equation of motion for the smectic layer normal displacement and the Maxwell equations for SPPs including the nonlinear polarization caused by the smectic layer strain. We evaluated the dynamic grating of the smectic layer displacement enhanced by the interfering SPPs. We evaluated the SVAs of the interacting SPPs. It has been shown that the SLS of the orientational type takes place. The pumping wave is depleted, while the signal wave is amplified up to the saturation level defined by the total intensity of the interacting waves. SLS

is accompanied by XPM. The phase of the depleted pumping wave rapidly

the strong energy exchange between the interfering SPPs.

increases, while the phase of the amplified wave tends to a constant value. The SPP characteristic rise time is of the magnitude of 10�<sup>9</sup> s for a feasible SPP electric field of 106 V/m. The smectic layer displacement and hydrodynamic velocity enhanced by SPPs are spatially localized and reach their maximum value during the time of

exp <sup>½</sup>*i*ð Þ <sup>2</sup>*kxx* � <sup>Δ</sup>*ω<sup>t</sup>* � þ *<sup>c</sup>:c:* (55)

We evaluate now the hydrodynamic flow velocity in the MIM wave guide core. Substituting expression (28) into Eqs. (1) and (6), we obtain

$$v\_x(\mathbf{x}, \mathbf{z}, t) = -i\Delta a \, U\_0 \sinh\left(2k\_x^S z\right) \exp\left[i(2k\_x \mathbf{x} - \Delta \alpha t)\right] + c.c.\tag{54}$$

**Figure 10.** *The temporal dependence of the SPP SVA phases θSA*1ð Þ*t (a) and θSA*2ð Þ*t (b).*

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

**Figure 11.** *The temporal dependence of the dynamic grating amplitude normalized absolute value U*j j <sup>0</sup>*=U*<sup>0</sup> max *.*

$$v\_x(\mathbf{x}, z, t) = \Delta \nu U\_0 \frac{k\_x^S}{k\_\mathbf{x}} \cosh\left(2k\_x^S z\right) \exp\left[i(2k\_x \mathbf{x} - \Delta \nu t)\right] + c.c.\tag{55}$$

Expressions (28) and (52)–(55) and **Figure 11** show that the orientational and hydrodynamic excitations in SmALC core of the MIM waveguide enhanced by the SPPs are spatially localized and reach their maximum value during the time of the energy exchange between the interacting SPPs.

### **5. Conclusions**

The temporal dependence of the SPP SVA phases *θSA*1,2ð Þ*t* is shown in **Figure 10**. It is seen from expressions (48) and (49) that SLS of the SPPs in the MIM waveguide is accompanied by XPM. For the large time intervals *t* ! ∞, the phase of

Such a behavior corresponds to the rapid oscillations of the depleted pumping

Re *G kx*, *k<sup>S</sup>*

2Im*G kx*, *k<sup>S</sup>*

Substituting expressions (41) and (47) into Eq. (29), we obtain the explicit

ffiffiffiffiffiffiffiffiffiffi *ω*1*ω*<sup>2</sup> p *h*

<sup>2</sup> ð Þ *t* � *t*<sup>0</sup>

*xkS z* � � �

*G kx*, *kS*

We evaluate now the hydrodynamic flow velocity in the MIM wave guide core.

� ffiffiffiffiffiffiffiffiffiffi *ω*1*ω*<sup>2</sup> <sup>p</sup> j j *<sup>h</sup>*

� � �

*<sup>z</sup>* , Δ*ω* � � � �

cosh *<sup>g</sup>*

The temporal dependence of the amplitude (52) normalized absolute value

�

*z z* � �

wave amplitude. The signal wave phase for *t* ! ∞ tends to a constant value:

Re *G kx*, *k<sup>S</sup>*

2Im*G kx*, *k<sup>S</sup>*

*<sup>z</sup>* , Δ*ω* � �

*<sup>z</sup>* , Δ*ω*

*<sup>z</sup>* , Δ*ω* � �

*<sup>z</sup>* , Δ*ω*

� � *gt* (50)

� � ln 1½ � � *<sup>I</sup>*1ð Þ <sup>0</sup> (51)

� � exp *<sup>i</sup>*ð Þ *<sup>θ</sup>SA*<sup>1</sup> � *<sup>θ</sup>SA*<sup>2</sup> (52)

exp ½*i*ð Þ 2*kxx* � Δ*ωt* � þ *c:c:* (54)

(53)

the pumping wave increases linearly:

*Nanoplasmonics*

*θSA*1ðÞ�*t θSA*1ð Þ!� 0

expression for the dynamic grating amplitude. It takes the form

*<sup>z</sup>* , Δ*ω* � �

Substituting expression (28) into Eqs. (1) and (6), we obtain

*The temporal dependence of the SPP SVA phases θSA*1ð Þ*t (a) and θSA*2ð Þ*t (b).*

*vz*ð Þ¼� *<sup>x</sup>*, *<sup>z</sup>*, *<sup>t</sup> <sup>i</sup>*Δ*ωU*<sup>0</sup> sinh 2*k<sup>S</sup>*

**Figure 10.**

**150**

*xkS z*

j j *<sup>U</sup>*0 max <sup>¼</sup> <sup>2</sup>*ε*0*I*<sup>0</sup> *<sup>k</sup>*<sup>2</sup>

*θSA*2ð Þ�*t θSA*2ð Þ! 0

*<sup>U</sup>*<sup>0</sup> ¼ � <sup>2</sup>*ε*0*I*0*k*<sup>2</sup>

j j *U*0*=U*0 max is presented in **Figure 11**. Here

*G kx*, *k<sup>S</sup>*

We investigated theoretically the nonlinear interaction of SPPs in the MIM waveguide with the SmALC core. The third-order nonlinearity mechanism is related to the smectic layer oscillations that take place without the change of the mass density. We solved simultaneously the equation of motion for the smectic layer normal displacement and the Maxwell equations for SPPs including the nonlinear polarization caused by the smectic layer strain. We evaluated the dynamic grating of the smectic layer displacement enhanced by the interfering SPPs. We evaluated the SVAs of the interacting SPPs. It has been shown that the SLS of the orientational type takes place. The pumping wave is depleted, while the signal wave is amplified up to the saturation level defined by the total intensity of the interacting waves. SLS is accompanied by XPM. The phase of the depleted pumping wave rapidly increases, while the phase of the amplified wave tends to a constant value. The SPP characteristic rise time is of the magnitude of 10�<sup>9</sup> s for a feasible SPP electric field of 106 V/m. The smectic layer displacement and hydrodynamic velocity enhanced by SPPs are spatially localized and reach their maximum value during the time of the strong energy exchange between the interfering SPPs.

*Nanoplasmonics*
