Nonlinear Optical Phenomena in Smectic A Liquid Crystals Nonlinear Optical Phenomena in Smectic A Liquid Crystals

Liquid crystals (LC) are the materials characterized by extremely high optical nonlinearity. Their physical properties such as temperature, molecular orientation, density, and electronic structure can be easily perturbed byan applied optical field. In particular, in smectic A LC (SALC), there is a specific mechanism of the cubic optical nonlinearity determined by the smectic layer normal displacement. The physical processes related to this mecha- nism are characterized by a comparatively large cubic susceptibility, short time response, strong dependence on the optical wave polarization and propagation direction, resonant spectral form, low scattering losses as compared to other LC phases, and weak temperature dependence in the region far from the phase transition. We investigated theoretically the nonlinear optical phenomena caused by this type of the cubic nonlinearity in SALC. It has been shown that the light self-focusing, self-trapping, Brillouin-like stimulated light scattering (SLS), and four-wave mixing (FWM) related to the smectic layer normal displacement are strongly manifested in SALC. We obtained the exact analytical solutions in some cases and made the numerical evaluations of the basic parameters such as the optical beam width and SLS gain. constants, elastic constants, viscosities, LC applications in nanophotonics, plasmonics, and metamaterials attracted a wide interest due to the combination of LC large nonlinearity and strong localized electric fields of SPP. Until now, NLC applications in nanophotonics and plasmonics have been investigated. We studied theoretically the nonlinear optical processes at the interface of a metal and a homeotropically oriented SALC. In such a case, SPP penetrating into SALC create the spatially localized surface dynamic grating of smectic layer normal displacement. We have shown that for optical frequencies of about 10 15 s (cid:2) 1 and coupled SPP frequency difference of about 10 8 s (cid:2) 1 , the SALC-metal system cubic susceptibility may be one to two orders of magnitude larger than the cubic susceptibility of isotropic organic liquids. We solved the wave Eq. (13) for the counter-propagating SPP in SALC with the spatially localized nonlinear polarization and obtained the explicit expressions (55) – (57) for the magnitudes and phases of the coupled SPP SVA. It has been shown that the Rayleigh stimulated scattering of SPP on the surface smectic layer oscillations occurs. The rise time of the amplified SPP of about 10 ns can be achieved, which is much faster than the Brillouin relaxation constant in NLC. plasmonic waveguides with NLC for nanophotonic and plasmonic investigated. We proposed an MIM waveguide with an SALC core. We evaluated the electric field of the strongly localized SPP even mode, the smectic layer normal strain and the nonlinear polarization in the MIM core. The evaluation of the SPP SVA shows that the strong SPM process takes place. The SPM is enhanced by the geometric factor caused by the strong SPP localization in the MIM core.


Introduction
Liquid crystals (LC) are characterized by the physical properties intermediate between ordinary isotropic fluids and solids [1]. LC flow like liquids but also exhibit some properties of crystals [1,2]. The various phases in which such materials can exist are called mesophases [1,2]. The LC molecules are large, anisotropic, and complex [2]. Dielectric constants, elastic constants, viscosities, absorption spectra, transition temperatures, anisotropies, and optical nonlinearities of LC are determined by the structure of these molecules [1,2]. There exist three different types of LC: lyotropic, polymeric, and thermotropic [1,2]. Lyotropic LC are obtained when an appropriate concentration of a material is dissolved in a solvent [2]. They can demonstrate a one-, two-, or three-dimensional positional order [2]. Liquid crystalline polymers are built up by the joining together the rigid mesogenic monomers [2]. Thermotropic LC exhibit different mesophases depending on temperature [1,2]. Typically, they consist of organic molecules elongated in one direction and represented as rigid rods [2]. There are two types of LC sample orientation with respect to the boundary: (i) a homeotropic orientation when the long molecular axes are perpendicular to the boundary and (ii) a planar orientation when the long molecular axes are parallel to the boundary [1].
In this work, we consider only thermotropic LC, which are divided into three groups according to their symmetry: nematic LC (NLC), cholesteric LC (CLC), and smectic LC (SLC) [1,2]. NLC are characterized by some long-range order in the direction of the molecular long axes, while the centers of gravity of the molecules do not have any long range order [1,2]. The general direction of the molecules is defined by a unit vector function n ! x; y; z; t ð Þ ; n ! x; y; z; t ð Þ ¼ 1 called director [1,2]. NLC molecules are centrosymmetric such that the n ! and À n ! directions are equivalent; NLC are optically uniaxial media with a comparatively large birefringence of about 0.2 [1,2]. LC consisting of chiral molecules yield CLC phase with the helical structure [1,2]. The molecule centers of gravity in CLC do not have a long range order like in NLC, while the direction of the molecular orientation rotates in space about the helical axis Z with a period of about 300 nm [1,2]. The smectic LC (SLC) are characterized by the positional long range order in the direction of the elongated molecular axis and exhibit a layer structure [1,2]. The layer thickness d ≈ 2nm is approximately equal to the length of the constituent molecule [1,2]. SLC can be considered as natural nanostructures. Inside a layer the molecules form a two-dimensional liquid [1,2]. The layers can easily move one along another because the elastic constant B ≈ 10 6 À 10 7 Jm À3 related to the layer compression is two orders of magnitude less than the elastic constant related to the bulk compression [1]. There exist different phases of SLC: (i) smectic A LC (SALC) where the molecule long axes are perpendicular to the layer plane; (ii) smectic B LC with the in-layer hexagonal ordering of the molecules; (iii) smectic C LC where the molecules are tilted with respect to the layers; (iv) smectic C * LC consisting of the chiral molecules and possessing the spontaneous polarization; (v) different exotic smectic mesophases [1]. In this work, we consider only SALC. The SALC layered structure can be described by the one-dimensional mass density wave characterized by the complex order parameter. The modulus of this order parameter describes the mass density and its phase is related to smectic layer displacement u x; y; z; t ð Þalong the direction perpendicular to the layer plane [1]. SALC is an optically uniaxial medium [2].
LC are highly nonlinear optical materials due to their complex physical structures, and their temperature, molecular orientation, mass density, electronic structure can be easily perturbed by an external optical field [2][3][4]. Almost all known nonlinear optical phenomena have been observed in LC in time scale range from picoseconds to hours, involving laser powers from 10 6 Watt to 10 À9 Watt, in different configurations such as bulk media, optical waveguides, optical resonators and cavities, and spatial light modulators [3]. For instance, a typical LC slab optical waveguide is a thin film of LC with a thickness of about 1μm sandwiched between two glass slides of lower refracted index than LC [2]. Stimulated light scattering (SLS), self-phase modulation (SPM), self-focusing, spatial soliton formation, optical wave mixing, harmonic generation, optical phase conjugation, and other nonlinear optical effects in LC have been investigated [3]. NLC is the most useful and widely studied type of LC [2][3][4]. However, the practical integrated electro-optical applications of NLC are limited by their large losses of about 20 dB/cm and relatively slow responses [2]. The scattering losses in SALC are much lower, and they can be useful in nonlinear optical applications [2]. Recently, the LC applications in plasmonics attracted a wide interest due to the combination of the surface plasmon polaritons (SPP) strong electric fields and the unique electro-optical properties of LC [4].
We investigated theoretically the nonlinear optical phenomena in SALC related to the specific mechanism of the cubic nonlinearity, which is determined by the smectic layer normal displacement u x; y; z; t ð Þin the electric field of optical waves and SPP [5][6][7][8][9][10][11][12][13]. This mechanism combining the properties of the orientational and electrostrictive nonlinearities [2] occurs without the mass density change, strongly depends on the optical wave polarization and propagation direction, and has a resonant form of the frequency dependence. It is characterized by a comparatively short response time similar to acousto-optic processes [2,14].
The theoretical analysis of the nonlinear optical phenomena in SALC related to the layer displacement was based on the simultaneous solution of the Maxwell equations for the optical waves propagating in SALC and the equation of motion for the SALC layers in the electric field of these waves. We used the slowly varying amplitude approximation (SVAA) [14]. We investigated the following nonlinear optical effects in SALC based on the layer displacement nonlinearity: self-focusing and self-trapping, SLS, and four-wave mixing (FWM) [5][6][7][8][9][10]. We applied the developed theory of the nonlinear optical phenomena in SALC to the SPP interactions in SALC [11][12][13]. The SPP stimulated scattering in SALC and the metal/insulator/metal (MIM) plasmonic waveguide with the SALC core are theoretically studied [11][12][13]. The detailed calculations and complicated explicit analytical expressions can be found in Refs [5][6][7][8][9][10][11][12][13]. In this chapter, we describe the general approach to the theoretical analysis of the nonlinear optical phenomena in SALC and present the main results.
The chapter is constructed as follows. The equation of motion for the smectic layer normal displacement u x; y; z; t ð Þin the electric field is derived in Section 2. The self-focusing and selftrapping of the optical wave in SALC are considered in Section 3. The SLS in SALC is investigated in Section 4. The FWM in SALC is analyzed in Section 5. The SPP interaction in SALC is discussed in Section 6. The conclusions are presented in Section 7.

The smectic layer equation of motion
The structure of the homeotropically oriented SALC in an external electric field E ! x; y; z; t ð Þis presented in Figure 1.
The hydrodynamics of SALC is described by the following system of Eq. [1] div v Here, v ! is the hydrodynamic velocity, r is the mass density, Π is the pressure, Λ ! is the generalized force density, σ 0 ik is the viscous stress tensor, α i are the viscosity Leslie coefficients, and F is the free energy density of SALC. Typically, SALC is supposed to be incompressible liquid according to Eq. (1) [1]. For this reason, we assume that the pressure Π ¼ 0 and the SALC-free energy density F do not depend on the bulk compression [1]  where K is the Frank elastic constant associated with the SALC purely orientational energy, ε 0 is the free space permittivity, and ε ik is the SALC permittivity tensor including the nonlinear terms related to the smectic layer strains. It is given by [1] Here, ε ⊥ , ε ∥ are the diagonal components of the uniaxial SALC permittivity tensor perpendicular and parallel to the optical axis Z, respectively, and a ⊥ $ 1; a ∥ $ 1 are the phenomenological dimensionless coefficients. For the smectic layer displacement u x; y; z; t ð Þdepending on z , the purely orientational second term in the free energy density F (7) can be neglected. Indeed, for the typical values of B and K $ 10 À11 N [1], the following inequality is valid: Kk 2 S⊥ ≪ B where k S⊥ is the in-plane component of the smectic layer displacement wave vector. The contribution of the first term containing the normal layer strain is dominant. We consider the smectic layer normal displacement with k Sz 6 ¼ 0. Taking into account the assumptions mentioned above and combining Eqs. (1)-(9), we obtain the equation of motion for the smectic layer normal displacement u x; y; z; t ð Þ[5, 6, 10] .Here, ∇ 2 ⊥ ¼ ∂ 2 u=∂x 2 þ ∂ 2 u=∂y 2 . If the external electric field is absent and the viscosity terms responsible for the decay of the smectic layer displacement are neglected, Eq. (10) coincides with the equation of the so-called second sound (SS) [1] Generally, for an arbitrary direction of the wave vector k ! S in SALC, there exist two practically uncoupled acoustic modes: (i) the ordinary longitudinal sound wave caused by the mass density oscillations; (ii) SS wave caused by the smectic layer oscillations [1]. The SS propagation may be considered separately from ordinary sound since B is much less than the elastic constant of the mass density oscillations [1]. The SS dispersion relation corresponding to Eq. (11) has the form [1] Here, Ω S , k ! S , s 0 are SS frequency, wave vector and phase velocity, respectively. It is seen from Eq. (11) that SS is neither longitudinal, nor purely transverse, and it vanishes for the wave vectors k ! S perpendicular or parallel to the layer plane. SS represents the oscillations of the SALC complex order parameter phase [1]. If we take into account the viscosity terms in Eq. (10), then we can obtain the SS relaxation time τ S given by SS has been observed experimentally [15][16][17].

Self-focusing and self-trapping of optical beams in SALC
We first consider the self-action effects of the optical waves propagating in an anisotropic inhomogeneous nonlinear medium. The light beam propagation through a nonlinear medium is accompanied by the intensity-dependent phase shift on the wavefront of the beam [2]. Selffocusing of light results from the wavefront distortion inflicted on the beam by itself while propagating in a nonlinear medium [14]. In such a case, the field-induced refractive change Δn has the form Δn ¼ n 2 E j j 2 where n 2 ¼ const [14]. A light beam with a finite cross section also diffracts [14]. At a certain optical power level, the beam self-focusing and diffraction can be balanced in such a way that the beam propagates in the nonlinear medium with a plane wavefront and a constant transverse intensity profile [1]. This phenomenon is called selftrapping of an optical beam [1]. The optical wave propagation in a nonlinear medium is described by the following wave equation for the electric field E ! x; y; z; t ð Þ [14] curlcurl E Here, μ 0 is the free space permeability, D  [2,18]. Taking into account the SALC symmetry, we can choose the xz plane as a propagation plane. Then, the ordinary wave is polarized along the Y axis, and its electric field is given by The extraordinary wave is polarized in the XZ plane having a component along the optical axis Z. The electric field of the extraordinary wave has the form Here, e In the linear approximation, substituting Eqs. (14)-(16) into the wave Eq. (13) we obtain the dispersion relations for the ordinary and extraordinary waves, respectively [7,18] It should be noted that the ordinary and extraordinary beams in the uniaxial medium propagate in different directions and the vectors E ! e and D ! L e are not parallel [18]. The extraordinary wave propagates in the direction of the beam vector s ! ⊥E ! e, which is determined by the angle θ e ¼ arctan ε ⊥ =ε ∥ À Á tanθ 1 È É with respect to the Z axis [18]. Here, θ 1 is the angle between k ! e and the Z axis.
We consider separately the self-focusing and self-trapping of the ordinary and extraordinary beams [7]. We start with the analysis of the slab-shaped ordinary beam with the dimension in the Y direction much greater than in the incidence XZ plane. In such a case, the dependence on the coordinate y may be neglected [7,9]. Substituting expression (14) into the equation of motion (9), we obtain [7,9] Expression (18) shows that the nonlinearity related to the smectic layer normal strain is the Kerr-type nonlinearity [14]. We introduce now the coordinates x 0 ; z 0 ð Þ parallel and normal to the ordinary beam propagation direction, respectively [7,9] Here, θ o is the angle between k ! o and the Z axis. We use the SVAA for the ordinary beam amplitude A o [14] We are interested in the spatially localized solutions with the following boundary conditions [7,9] lim Then, substituting expressions (14), (18), (19) and the first ones of Eq. (16), (17) into Eq. (13) and taking into account the SVAA conditions (20), we obtain the truncated equation for the SVA Eq. (22) is the nonlinear Schrodinger equation (NSE) [19]. The coefficient of the last term in the left-hand side (LHS) of Eq. (22) is positive definite The self-trapped beam (23) is the so-called spatial soliton with the width The self-trapped ordinary beam normalized intensity spatial distribution is shown in Figure 3.
The self-trapping of the extraordinary beam (15) can be realized only when the anisotropy angle θ 1 À θ e ð Þis small enough: tan θ 1 À θ e ð Þ≪ k e w o ð Þ À1 [7]. For the typical values of ε ⊥ , ε ∥ [2], the following condition is valid: 0 ≤ tan θ 1 À θ e ð Þ≤ 0:12, and the self-trapping condition for the extraordinary beam can be satisfied [7]. Then, using the procedure described above for the ordinary beam, we obtain the spatial soliton of the extraordinary beam. It has the form [7, 10] Here, x 00 ¼ xsinθ e þ zcosθ e ; z 00 ¼ Àxcosθ e þ zsinθ e are the coordinates parallel and perpendicular to the beam vector, respectively, , and k e∥ is the wave vector component parallel to the beam vector. The width w e of the extraordinary beam spatial soliton is given by [7,10] For the typical values of the SALC parameters [1,2], the optical beam electric field A o, e j j max $ 10 5 V=m, ω $ 10 15 s À1 and small angle θ e the spatial soliton width is w o, e $ 10 À4 m [7]. SALC samples with a thickness of 10 À4 m have been demonstrated experimentally [2,17].
The optical wave self-trapping can occur also at the interface between the linear medium in the region z < 0 with the permittivity ε s and the SALC cladding z > 0 ð Þ. For the light wave Þpropagating along the interface parallel to the X axis, the self-trapped solution represents a bright surface wave with the amplitude A z ð Þ given by [7] A z The cubic susceptibility of SALC χ 3 ð Þ SALC related to the smectic layer compression is larger than χ 3 ð Þ related to the Kerr nonlinearity in organic liquids [14], but it is much less than the giant orientational nonlinearity (GON) in NLC [2]. However, the optical beam intensity in SALC may be much greater than in NLC, which are extremely sensitive to the strong optical fields [2]. In such cases, the approach based on the purely orientational mechanism of the optical nonlinearity is invalid.

Stimulated light scattering (SLS) in SALC
SLS is a process of parametric coupling between light waves and the material excitations of the medium [14]. We consider the SLS in SALC related to the smectic layer normal displacement and SS excited by the interfering optical waves [5,6,[8][9][10]. We have taken into account the combined effect of SALC layered structure and anisotropy. It should be noted that SS propagates in SALC without the change of the mass density in such a way that the SS wave and the ordinary sound wave are decoupled [1].
In general case when the coupled optical waves have arbitrary polarizations and propagation directions, each optical wave in SALC z > 0 ð Þsplits into the extraordinary and ordinary ones with the same frequency and different wave vectors due to the strong anisotropy of SALC [6,10,18]. The polarizations of these waves are shown in Figure 4. The XZ plane is chosen to be the propagation plane of the waves E Here, ω 1 > ω 2 and Δω ¼ ω 1 À ω 2 ≪ ω 1 . Each pair of the waves (27) Here, Δ k The parametric amplification of the fundamental optical waves E ! o, e 2 with the lower frequency ω 2 by the other pair of optical waves E ! o, e 1 with the higher frequency ω 1 occurs in SALC due to the SLS on the light-induced smectic layer dynamic grating (28) [6,10]. It is actually the Stokes SLS [14]. The fundamental optical waves also create Stokes and anti-Stokes small harmonics with the combination frequencies and wave vectors. The analysis of SLS in SALC is based on the simultaneous solution of the smectic layer equation of motion (9), the wave Eq. (13) for ordinary waves (14) and extraordinary waves (15) with the permittivity tensor (8). The nonlinear part of the permittivity tensor (8) ε N ik in the three-dimensional case can be written as follows [6] ε

Combining Eqs. (27)-(30)
, we obtain the nonlinear part of the electric induction, or the nonlinear polarization D NL i ¼ ε 0 ε N ik E k [6]. This nonlinear polarization consists of two types of terms: (i) four harmonics, which are phase-matched with fundamental waves (27); (ii) all other terms with the combination frequencies and wave vectors, which give rise to the small scattered Stokes and anti-Stokes harmonics similar to the Brillouin scattering [6,10,14]. The combination of the anisotropy and nonlinearity also results in the creation of the small additional components of the waves E ! o, e 1 and E ! o 2 polarized in the XZ plane and XY plane, respectively [6].
We start with the analysis of the parametric coupling among the waves (27). Substituting expressions (27)   Eqs. (31) and (32) describe the parametric energy exchange between the fundamental waves, Eqs. (33) and (34) describe the cross-phase modulation (XPM) of these waves [6]. Combining Eqs. (31) and (32), we obtain the Manley-Rowe relation, which expresses the conservation of the total photon number [14]. In our case, it has the form [6] The solution of the system of Eqs. (31)-(34) can be written in the integral form [6,10]  (37) Here, the dimensionless variables are given by [6]: Comparison of Eq. (36), (37), and (40) shows that for z ! ∞ w o, e 1 ! 0 and w o 2 þ w e 2 ! 1. Hence, the pumping waves with the larger frequency ω 1 are depleted, the signal waves with smaller frequency ω 2 < ω 1 are amplified with the saturation at the sufficiently large distances, and the system is stable. The gain terms β j reach their maximal values close to the SS resonance condition Δω ≈ Ω j, which can be satisfied for Δω $ ω 1 s 0 =c ≪ ω 1 [6]. In such a case, β j ≫ δ j , the parametric amplification process is dominant while XPM can be neglected.
In general case, the exact analytical solution of Eqs. (31)-(34) is hardly possible. However, the explicit expressions for the coupled wave SVA have been obtained when both waves are propagating in the same XZ plane [5,6]. For instance, assume that the pumping extraordinary wave with the frequency ω 1 > ω 2 is mainly polarized in the XZ plane, the signal ordinary wave with the frequency ω 2 is mainly polarized along the Y axis, and the intensities of the components with other polarizations are small in such a way that w e 1 ≫ w o 1 ; w o 2 ≫ w e 2 . Then, in the first approximation, the normalized intensities w e 1 , w o 2 of the main components have the form [6,10] w e 1 ¼ It is seen from Eq. (42) that for η ! ∞ w e 1 ! 0; w o 2 ! J 1 , and the crossing point The coordinate dependence of the normalized intensities w e 1 , w o 2 is presented in Figure 5. The numerical estimations show that for the typical values of SALC parameters [1][2][3] in the resonant case the coupling constant per unit optical intensity β 1 =P opt $ 0:01 ð À10Þcm=MW [6]. For the optical intensity P opt $ 10 6 À 10 7 Wcm À2 , the SLS gain β 1max $ 10 2 cm À1 , which is at least an order of magnitude larger than the gain at Brillouin SLS in isotropic organic liquids [14]. Such optical intensities are feasible [20,21].
The explicit expressions of the small component intensities w o 1 and w e 2 can be obtained in the second approximation. They have the form [6] w o It is easy to see from Eq. (43) that for η ! ∞ w o The evaluation of the phases γ o, e 1, 2 shows that the pumping wave phases γ o, e 1 rapidly increase that results in the fast oscillations of the depleted amplitudes A o, e 1 z ð Þ [6]. The phases γ o, e 2 of the signal waves tend to the constant values at sufficiently large η [6]. The Brillouin-like SLS also results in the generation of six Stokes small harmonics with the frequency ω 2 À Δω ð Þand combination wave vectors, six anti-Stokes small harmonics with the frequency ω 1 þ Δω ð Þand combination wave vectors, and eight small harmonics with the fundamental frequencies ω 1, 2 and combination wave vectors [6].

The nondegenerate FWM in SALC
Consider now the nondegenerate FWM in SALC [8][9][10]. Assume that four coupled fundamental optical waves have different close frequencies ω n such that Δω mn ¼ ω m À ω n $ s 0 ω n =c ≪ ω n . For the sake of definiteness, we suppose that ω 1 < ω 2 < ω 3 < ω 4 . These fundamental waves have the form [8][9][10] The interfering waves (44) create a dynamic grating of the smectic layer normal displacement of the type (28), but this time each harmonic has a different frequency Δω mn ¼ ω m À ω n , m, n ¼ 1, …, 4. We discuss two cases: (i) all waves (44) are polarized in the directions perpendicular to the propagation plane and propagate as ordinary waves; (ii) all waves (44) are polarized in the propagation plane and behave as extraordinary waves [8][9][10]. Using the SVAA and the theory developed in the previous section, we obtain the truncated equations for the slowly varying magnitudes A m z ð Þ j jand phases γ m z ð Þ similar to Eqs. (31)-(34). The analysis of these equations shows that the wave with the lowest frequency ω 1 is amplified up to the saturation level determined by the integral of motion I 0 similar to the one from Eq. (35) [8][9][10] Here, the factors l m are defined above for the ordinary or extraordinary wave, respectively. Three other waves with the higher frequencies ω 2, 3, 4 undergo the depletion like the pumping waves [8][9][10]. The depletion of the waves E ! 2, 3, 4 is accompanied by the XPM with the rapidly increasing phases while the phase of the signal wave E In the special case when some ordinary optical waves (44) have perpendicular polarizations vectors e ! m⊥ ⊥ e ! n⊥ the polarization-decoupled FWM is possible [8][9][10] [8].
In the important case when the pumping wave E ! 4 and the signal wave E ! 1 are much stronger than the idler waves E ! 2, 3, the approximate solution can be obtained similarly to the solution (42) and (43) in the case of SLS [8][9][10]. It has been shown that this solution is stable in the case of FWM [8].
In the particular case when the fundamental waves (44) are counter-propagating, the phase conjugation is possible as a result of the nondegenerate FWM in SALC [8][9][10]. Optical phase conjugation (OPC) is the wavefront reversion property of a backward propagating optical wave with respect to a forward propagating wave [22]. The optical waves are phase conjugated to each other if their complex amplitudes are conjugated with respect to their phase factors [22]. Typically, OPC results from nonlinear optical processes such as FWM and SLS [20]. LC are commonly used for FWM and OPC [22].
Suppose that the waves E Here, Δ k ! ¼ Δ k ! 32 is the wave vector mismatch of the FWM process. In the case of OPC caused by SLS the frequency balance condition between the waves with the same vectors is necessary.
We assume that ω 3 À ω 1 ¼ ω 4 À ω 2 . Suppose that the pumping waves E ! 2, 3 are much stronger than the probe wave E ! 4 and the phase-conjugate wave E ! 1 propagating in the negative direction as it is seen from Eq. (46). In such a case, using the constant pumping approximation (CPA) [14] where A 2, 3 ¼ const , we obtain the following solution for the probe wave and the phase-conjugate wave SVA A 1, 4 [8][9][10] Analysis of the truncated equations for A 1,4 shows that there exists the solution with the gain Reg < 0 corresponding to the amplification of the phase-conjugate wave E ! 1 [8][9][10]. Such a case can be characterized as a kind of the Brillouin-enhanced FWM (BEFWM) based on the optical nonlinearity related to the smectic layer normal displacement [8][9][10]. Numerical estimations show that the amplification of the phase-conjugate wave E ! 1 is possible for the typical values of SALC parameters and for the pumping wave intensity of about 100 MWcm À2 [9], which is feasible [20,21]. OPC in the homeotropically oriented SALC film with the thickness of 250 μm had been demonstrated experimentally [20].
The components of the nonlinear electric induction D ! NL , which are not phase matched to the fundamental waves (46) give rise to 12 doubly degenerate combination harmonics of the type.

Nonlinear interaction of surface plasmon polaritons (SPP) in SALC
Integration of strongly nonlinear LC with plasmonic structures and metamaterials would enable active switching and tuning operations with low threshold [4]. LC may be also used in reconfigurable metamaterials for tuning the resonant frequency, the transmission/ reflection coefficient, and the refractive index [23]. Combination of metamaterials and active plasmonic structures with NLC has been investigated [4,23]. In this section, we discuss the nonlinear optical effects caused by the SPP mixing in SALC, which is characterized by low losses and a strong nonlinearity related to the smectic layer normal displacement without the change of the mass density [11][12][13]. Consider the interface z ¼ 0 between a homeotropically oriented SALC z > 0 ð Þand a metal z < 0 ð Þshown in Figure 6 [11,12]. The SALC optical Z axis and the X axis are chosen to be perpendicular and parallel to the interface z ¼ 0, respectively. SPP from the metal penetrate into SALC. The permittivity of the metal ε m ω ð Þ determined by the Drude model is given by is the plasma frequency, n 0 is the free electron density in the metal, e, m are the electron charge and mass, respectively, ω, τ are the SPP angular frequency and lifetime, respectively [24,25]. The efficient SLS in SALC takes place for the counter-propagating SPP with close frequencies Figure 6. The counter-propagating SPP at the interface between a metal z < 0 ð Þand a homeotropically oriented SALC z > 0 ð Þ . ω 1 > ω 2 such that Δω ¼ ω 1 À ω 2 ≪ ω 1 [12]. The spatially localized electric fields of these SPP in SALC have the form [24,25] γ 1 t and large nonlinearity [28]. Photonic components based on plasmonic waveguides with NLC core have been theoretically investigated in a number of articles (see, for example, [28][29][30][31] and references therein).
We consider the nonlinear optical processes in an MIM waveguide with the SALC core [13].
The structure of such a waveguide is shown in Figure 8. SPP propagating in the metal claddings and in SALC core are TM waves [24,25]. The SPP electric and magnetic fields in

Conclusions
In SALC, there exists a specific mechanism of the optical nonlinearity related to the normal displacement u x; y; z; t ð Þ of smectic layers. This mechanism combines the properties of the orientational mechanism typical for LC and of the electrostrictive mechanism. In particular, the smectic layer oscillations occur without the mass density change. Under the resonant condition (11), the SS acoustic wave propagates in SALC in the direction oblique to the layer plane. The cubic nonlinearity related to this mechanism is characterized by a strong anisotropy, a short time response, a weak temperature dependence, a resonant frequency dependence, and a strong dependence on the optical wave polarization and propagation direction. The cubic susceptibility related to the smectic layer displacement is larger than the Kerr type susceptibility in ordinary organic liquids. It should be noted that the nonlinear optics of NLC has been mainly studied. However, SALC are promising candidates for nonlinear optical applications due to their low losses and higher degree of the long range order.
We derived the equation of motion (9) of the smectic layer displacement u x; y; z; t ð Þin the electric field of optical waves. We investigated theoretically the nonlinear optical phenomena in SALC based on this specific mechanism. We solved simultaneously the equation of motion (9) and the wave Eq. (13) for the optical waves including the nonlinear polarization. The solution was based on the SVAA.
In an optically uniaxial SALC, an ordinary wave and an extraordinary one can propagate. Both the ordinary and extraordinary optical beams propagating in SALC undergo self-focusing and self-trapping and form spatial solitons. The optical wave self-trapping can occur at the interface between the linear medium and SALC. We obtained the analytical solutions for the SVA of the self-trapped beams.
SLS of two arbitrary polarized optical waves in SALC transforms into the partially frequency degenerate FWM because each optical wave splits into the ordinary and extraordinary waves. The coupled optical waves create a dynamic grating of the smectic layer normal displacement u x; y; z; t ð Þand undergo the parametric energy exchange and XPM. The signal optical waves with the lower frequency are amplified up to a saturation level determined by the Manley-Rowe relation, while the pumping optical waves with higher frequency are depleted. It has been shown that the system of the coupled optical waves and the dynamic grating is stable. The analytical expressions for the magnitudes and phases of SVA have been obtained in the limiting case when the waves are mainly polarized either perpendicular to the propagation plane, or in it. The SLS gain coefficient is significantly larger than the one in the case of the Brillouin SLS in isotropic organic liquids. The SLS in SALC also results in the generation of the Stokes and anti-Stokes harmonics with the combination wave vectors.
The nondegenerate FWM in SALC results in the amplification of the signal optical wave with the lowest frequency and depletion of three other waves with higher frequencies. The polarizationdecoupled FWM may take place when the polarizations of some optical waves are perpendicular to one another. If the coupled optical waves are counter propagating and their frequencies satisfy the balance conditions typical for OPC process then BEFWM takes place accompanied by the amplification of the phase-conjugate wave. The spectrum of the scattered harmonics consists of 24 Stokes and anti-Stokes terms with combination frequencies and wave vectors.
LC applications in nanophotonics, plasmonics, and metamaterials attracted a wide interest due to the combination of LC large nonlinearity and strong localized electric fields of SPP. Until now, NLC applications in nanophotonics and plasmonics have been investigated. We studied theoretically the nonlinear optical processes at the interface of a metal and a homeotropically oriented SALC. In such a case, SPP penetrating into SALC create the spatially localized surface dynamic grating of smectic layer normal displacement. We have shown that for optical frequencies of about 10 15 s À1 and coupled SPP frequency difference of about 10 8 s À1 , the SALC-metal system cubic susceptibility may be one to two orders of magnitude larger than the cubic susceptibility of isotropic organic liquids. We solved the wave Eq. (13) for the counter-propagating SPP in SALC with the spatially localized nonlinear polarization and obtained the explicit expressions (55)-(57) for the magnitudes and phases of the coupled SPP SVA. It has been shown that the Rayleigh stimulated scattering of SPP on the surface smectic layer oscillations occurs. The rise time of the amplified SPP of about 10 ns can be achieved, which is much faster than the Brillouin relaxation constant in NLC.