Open access peer-reviewed chapter

Nonlinear Optical Phenomena in Smectic A Liquid Crystals

By Boris I. Lembrikov, David Ianetz and Yossef Ben Ezra

Submitted: February 28th 2017Reviewed: September 14th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.70997

Downloaded: 143


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 by an 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 mechanism 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.


  • smectic liquid crystals
  • second sound
  • nonlinear optics
  • cubic nonlinearity
  • stimulated scattering of light
  • four-wave mixing
  • surface plasmon polariton

1. 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 nxyzt;nxyzt=1called director [1, 2]. NLC molecules are centrosymmetric such that the nand ndirections 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 Zwith 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 d2nmis 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 B106107Jm3related 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 uxyztalong 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 106 Watt to 10−9 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 1μmsandwiched 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 uxyztin 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 uxyztin the electric field is derived in Section 2. The self-focusing and self-trapping 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.

2. The smectic layer equation of motion

The structure of the homeotropically oriented SALC in an external electric field Exyztis presented in Figure 1.

Figure 1.

Homeotropically oriented SALC in an external electric field E → x y z t .

The hydrodynamics of SALC is described by the following system of Eq. [1]


Here, vis the hydrodynamic velocity, ρis the mass density, Πis the pressure, Λis the generalized force density, σikis the viscous stress tensor, αiare the viscosity Leslie coefficients, δik=1,i=k;δik=0,ik, and Fis 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=0and the SALC-free energy density Fdo not depend on the bulk compression [1]. The normal layer displacement uxyztby definition has only one component along the Z axis. In such a case, the generalized force density Λhas only the Z component according to Eq. (3): Λ=00Λz. Eq. (6) is specific for SALC since it determines the condition of the smectic layer continuity [1]. The SALC free energy density Fin the presence of the external electric field Exyzthas the form [1]


where Kis the Frank elastic constant associated with the SALC purely orientational energy, ε0is the free space permittivity, and εikis the SALC permittivity tensor including the nonlinear terms related to the smectic layer strains. It is given by [1]


Here, ε,εare the diagonal components of the uniaxial SALC permittivity tensor perpendicular and parallel to the optical axis Z, respectively, and a1;a1are the phenomenological dimensionless coefficients. For the smectic layer displacement uxyztdepending on z, the purely orientational second term in the free energy density F(7) can be neglected. Indeed, for the typical values of Band K1011N[1], the following inequality is valid: KkS2Bwhere kSis 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 kSz0. Taking into account the assumptions mentioned above and combining Eqs. (1)(9), we obtain the equation of motion for the smectic layer normal displacement uxyzt[5, 6, 10]


Here, 2=2u/x2+2u/y2. 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 kSin 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 Bis 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, ΩS,kS,s0are 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 kSperpendicular 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 τSgiven by


SS has been observed experimentally [15, 16, 17].

3. 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]. Self-focusing 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 Δnhas the form Δn=n2E2where n2=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 self-trapping of an optical beam [1]. The optical wave propagation in a nonlinear medium is described by the following wave equation for the electric field Exyzt[14]


Here, μ0is the free space permeability, DLand DNLare the linear and nonlinear parts of the electric induction. In SALC as a uniaxial medium two waves with the same frequency ωcan propagate: an ordinary wave with the wave vector koand an extraordinary one with the wave vector ke[2, 18]. Taking into account the SALC symmetry, we can choose the xzplane as a propagation plane. Then, the ordinary wave is polarized along the Yaxis, and its electric field is given by


The extraordinary wave is polarized in the XZplane having a component along the optical axis Z. The electric field of the extraordinary wave has the form


Here, ee=axeex+azeezis the polarization unit vector of the extraordinary wave, ax,zare the unit vectors of the X,Zaxes, and c.c. stands for complex conjugate. The propagation direction and polarization of the ordinary and extraordinary waves in SALC are shown in Figure 2.

Figure 2.

Propagation direction and polarization of the ordinary wave E → o and extraordinary wave E → e in SALC.

The corresponding linear electric induction vectors DoL,DeLare given by [7, 18]


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 Eeand DeLare not parallel [18]. The extraordinary wave propagates in the direction of the beam vector sEe, which is determined by the angle θe=arctanε/εtanθ1with respect to the Zaxis [18]. Here, θ1is the angle between keand the Zaxis.

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 Ydirection much greater than in the incidence XZplane. In such a case, the dependence on the coordinate ymay 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 xzparallel and normal to the ordinary beam propagation direction, respectively [7, 9]


Here, θois the angle between koand the Zaxis. We use the SVAA for the ordinary beam amplitude Ao[14]


We are interested in the spatially localized solutions with the following boundary conditions [7, 9]


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 Aoxz, which has the form [7]


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 ω2ε02a2/4c2Bko>0, which corresponds to the stationary two-dimensional self-focusing of the light beam. The solution of Eq. (22) with the boundary conditions (21) has the form [7]


The self-trapped beam (23) is the so-called spatial soliton with the width wo=2Bεε0aAomaxko1[7].

The self-trapped ordinary beam normalized intensity spatial distribution is shown in Figure 3.

Figure 3.

The self-trapped ordinary beam normalized intensity for the maximum amplitude A o max = 10 5 V / m and θ o = π / 6 .

The self-trapping of the extraordinary beam (15) can be realized only when the anisotropy angle θ1θeis small enough: tanθ1θekewo1[7]. For the typical values of ε,ε[2], the following condition is valid: 0tanθ1θe0.12, 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=xsinθe+zcosθe;z=xcosθe+zsinθeare the coordinates parallel and perpendicular to the beam vector, respectively,

he=aeex2+aeez2sinθe+2εaeexeezcosθe, le=ke1+εa/εsinθe1, and keis the wave vector component parallel to the beam vector. The width weof 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 Ao,emax105V/m, ω1015s1and small angle θethe spatial soliton width is wo,e104m[7]. SALC samples with a thickness of 104mhave been demonstrated experimentally [2, 17].

The optical wave self-trapping can occur also at the interface between the linear medium in the region z<0with the permittivity εsand the SALC cladding z>0. For the light wave Ey=Azexpikoxωtpropagating along the interface parallel to the X axis, the self-trapped solution represents a bright surface wave with the amplitude Azgiven by [7]


The cubic susceptibility of SALC χSALC3related to the smectic layer compression is larger than χ3related 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.

4. 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>0splits 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 E1o,e. In such a case, the extraordinary wave E1eis polarized in the XZ plane, while the ordinary wave E1ois parallel to the Y axis [18]. The ordinary wave E2ois polarized in the XY plane perpendicular to the optical Z axis, and the extraordinary wave E2epossesses a three-dimensional polarization vector e2e[18]. The wave vectors k1,2oand k1eof these waves satisfy the dispersion relations (17) while the three-dimensional wave vector k2esatisfies the dispersion relation k2xe2+k2ye2ε1+k2ze2ε1=ω2/c2[18]. The fundamental ordinary and extraordinary waves have the form, respectively


Figure 4.

The polarizations of the fundamental ordinary waves E → 1 , 2 o and extraordinary waves E → 1 , 2 e in SALC (z > 0).

Here, ω1>ω2and Δω=ω1ω2ω1. Each pair of the waves (27) has the same frequency, and for this reason, we define the nonlinear mixing of these waves as partially frequency degenerate FWM [6]. We assume that the complex amplitudes A1,2o,ez=A1,2o,ezexpiγ1,2o,ezare slowly varying along the optical axis Z: 2A1,2o,e/z2k1,2zo,eA1,2o,e/z. As a result, the nonlinear two-wave mixing analyzed in Ref. [5] transforms into a partially degenerate FWM [6, 10]. We substitute the waves (27) into equation of motion (9). The interfering optical waves (27) with close frequencies ω1,2create a dynamic grating of the smectic layer normal displacement uxyztconsisting of four propagating harmonics with the same frequency and different wave vectors. It has the form [6]


Here, Δk1=k1ek2o;Δk2=k1ek2e;Δk3=k1ok2o;Δk4=k1ok2e; h1=aΔk1ze1xee2xoεaΔk1xe1zee2xo+Δk1ye1zee2yo, h2=aΔk2ze1xee2xe+aΔk2ze1zee2zeεaΔk2xe1xee2ze+e1zee2xe+Δk2ye1zee2ye; h3=aΔk3ze2yo; h4=aΔk4ze2yeεaΔk4ye2ze; M1=A1eA2o;M2=A1eA2eM3=A1oA2o,M4=A1oA2e, and


The parametric amplification of the fundamental optical waves E2o,ewith the lower frequency ω2by the other pair of optical waves E1o,ewith the higher frequency ω1occurs 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) εikNin the three-dimensional case can be written as follows [6]


Combining Eqs. (27)(30), we obtain the nonlinear part of the electric induction, or the nonlinear polarization DiNL=ε0εikNEk[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 E1o,eand E2opolarized 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)(30) and the phase-matched part of DNLinto wave Eq. (13), taking into account SVAA for the complex amplitudes A1,2o,ez, and separating the real and imaginary parts, we obtain the truncated equations for the magnitudes A1,2o,ezand phases γ1,2o,ezof these SVA [6]


Here, l1,2o=k1,2zo;l1,2e=k1,2ze1e1,2zek1,2ee1,2ek1,2ze1.

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]


Here, the dimensionless variables are given by [6]:


Comparison of Eq. (36), (37), and (40) shows that for zw1o,e0and w2o+w2e1. Hence, the pumping waves with the larger frequency ω1are depleted, the signal waves with smaller frequency ω2<ω1are amplified with the saturation at the sufficiently large distances, and the system is stable. The gain terms βjreach their maximal values close to the SS resonance condition ΔωΩj, which can be satisfied for Δωω1s0/cω1[6]. In such a case, βjδj, 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 ω1>ω2is mainly polarized in the XZ plane, the signal ordinary wave with the frequency ω2is mainly polarized along the Y axis, and the intensities of the components with other polarizations are small in such a way that w1ew1o;w2ow2e. Then, in the first approximation, the normalized intensities w1e, w2oof the main components have the form [6, 10]


Here, w1e+w2o=J1=const=w1e0+w2o0, η=β1J1z/2. It is seen from Eq. (42) that for ηw1e0;w2oJ1, and the crossing point z0=β1J11lnw1e0/w2o0exists only for w1e0/w2o0>1. The coordinate dependence of the normalized intensities w1e, w2ois 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 β1/Popt0.0110cm/MW[6]. For the optical intensity Popt106107Wcm2, the SLS gain β1max102cm1, 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].

Figure 5.

The dependence of the normalized pumping and signal intensities w 1 e / J 1 ; w 2 o / J 1 on the dimensionless coordinate η for the pumping-to-signal ratio w 1 e 0 / w 2 o 0 = 1.5 ; 5 (curves 1 and 2, respectively).

The explicit expressions of the small component intensities w1oand w2ecan be obtained in the second approximation. They have the form [6]


It is easy to see from Eq. (43) that for ηw1o0and w2ew2e01+w1e0/w2o0β2/β1=const.

The evaluation of the phases γ1,2o,eshows that the pumping wave phases γ1o,erapidly increase that results in the fast oscillations of the depleted amplitudes A1o,ez[6]. The phases γ2o,eof 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 ω2Δωand combination wave vectors, six anti-Stokes small harmonics with the frequency ω1+Δωand combination wave vectors, and eight small harmonics with the fundamental frequencies ω1,2and combination wave vectors [6].

5. 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 ωnsuch that Δωmn=ωmωns0ωn/cωn. For the sake of definiteness, we suppose that ω1<ω2<ω3<ω4. 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 Δωmn=ωmωn,m,n=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 Amzand phases γmzsimilar to Eqs. (31)(34). The analysis of these equations shows that the wave with the lowest frequency ω1is amplified up to the saturation level determined by the integral of motion I0similar to the one from Eq. (35) [8, 9, 10]


Here, the factors lmare defined above for the ordinary or extraordinary wave, respectively. Three other waves with the higher frequencies ω2,3,4undergo the depletion like the pumping waves [8, 9, 10]. The depletion of the waves E2,3,4is accompanied by the XPM with the rapidly increasing phases while the phase of the signal wave E1saturates at large distances. The analytical solution of the type (42) and (43) has been obtained for the case when the pumping wave E4and the signal wave E1are much stronger than the idler waves E2,3with the intermediate frequencies ω2,3[8, 9, 10].

In the special case when some ordinary optical waves (44) have perpendicular polarizations vectors ementhe polarization-decoupled FWM is possible [8, 9, 10]. Such waves do not excite the dynamic grating since the corresponding coupling constants hmno=aΔkmnzemenvanish [8, 9, 10]. In the case of the extraordinary wave mixing, the polarization-decoupled FWM is impossible because of the SALC anisotropy. If the electric field of the signal ordinary wave E1is perpendicular to the fields of all other waves than this wave propagates though SALC without any change of its SVA: A1o=const. If E1oE2,3oand E1oE4o, then FWM is divided in two separate two-wave mixing processes between the waves E1,4oand the waves E2,3owith the solutions similar to solution (42) [8].

In the important case when the pumping wave E4and the signal wave E1are much stronger than the idler waves E2,3, 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 E1,4are phase-conjugate while the waves E2,3are forward-going and backward-going pumping waves, which have the form [8]


Here, Δk=Δk32is 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 ω3ω1=ω4ω2. Suppose that the pumping waves E2,3are much stronger than the probe wave E4and the phase-conjugate wave E1propagating in the negative direction as it is seen from Eq. (46). In such a case, using the constant pumping approximation (CPA) [14] where A2,3=const, we obtain the following solution for the probe wave and the phase-conjugate wave SVA A1,4[8, 9, 10]


Analysis of the truncated equations for A1,4shows that there exists the solution with the gain Reg<0corresponding to the amplification of the phase-conjugate wave E1[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 E1is possible for the typical values of SALC parameters and for the pumping wave intensity of about 100 MWcm−2 [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 DNL, which are not phase matched to the fundamental waves (46) give rise to 12 doubly degenerate combination harmonics of the type.

AmApAnexpikm+kpknrωm+ωpωntand 12 harmonics of the type Am2Anexpi2kmrωmtknr+ωnt[8, 9, 10].

6. 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=0between a homeotropically oriented SALC z>0and a metal z<0shown 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=0, respectively.

Figure 6.

The counter-propagating SPP at the interface between a metal z < 0 and a homeotropically oriented SALC z > 0 .

SPP from the metal penetrate into SALC. The permittivity of the metal εmωdetermined by the Drude model is given by εmω=1ωp2ω1ω+i/τ1where ωp=n0e2/ε0mis the plasma frequency, n0is the free electron density in the metal, e,mare the electron charge and mass, respectively, ω,τare the SPP angular frequency and lifetime, respectively [24, 25]. The efficient SLS in SALC takes place for the counter-propagating SPP with close frequencies ω1>ω2such that Δω=ω1ω2ω1[12]. The spatially localized electric fields of these SPP in SALC have the form [24, 25]


The SPP are polarized as transverse magnetic (TM) waves with the electric field components Ex,zand the magnetic field component Hy[23, 24]. In an optically uniaxial SALC, SPP propagate as extraordinary waves [18]. The numerical estimations show that for the optical frequency range and the small frequency difference Δω107105ω1the SPP1,2 wave vectors are practically equal [11, 12]. They have the form [11, 12]


Numerical estimations show that for the typical values of ω1,2,ωp,τthe following relations are valid: RekzSImkzS,RekxImkx[11, 12, 24, 25]. For the optical wavelength λopt0.61.33μm, the SPP propagation length Lxand the wavelength λsare given by, respectively: Lx=Imkx184550μm, λs=2π/Rekx0.330.77μmLx[12]. The SPP localization length Lz=RekzS1106mbelongs to the subwavelength scale: ImkzS10m1RekzSand can be neglected [12].

Substituting the SPP fields (48) into the smectic layer equation of motion (9), we obtain the dynamic grating uxztgiven by [11, 12]




Unlike the dynamic grating (28) created by the interfering optical waves, the grating (50) caused by SPP is spatially localized both in the X and in the Z directions [11, 12]. The localized grating (50) can be characterized as an enhanced Rayleigh wave of SS [26]. Analysis of GkxkzSΔω(52) shows that the resonant case ReGkxkzSΔω=0cannot be achieved for the frequency difference Δω107108s1, and the spontaneous SS surface wave can be neglected [12]. The cubic susceptibility of the SALC-metal system χijkl3Δω;i,j,k,l=x,zrelated to the localized grating (50) is essentially complex. For the typical values of the SALC parameters, SPP in silver, ω1=1.4×1015s1and Δω107108s1the numerical estimations yield χxxxx3χzzzz310201019m2/V2[11], which is larger by one-two orders of magnitude than the cubic susceptibilities of some organic liquids and solid materials [27].

We substitute the localized layer displacement uxzt(50) into the expression of the SALC nonlinear permittivity (30), evaluate the nonlinear part of the electric induction DNLfor SPP (48), and by using the standard procedure, we obtain from Eq. (13) the truncated equations for the SPP SVA A1,2t=A1,2texpiγ1,2t. The dependence of SVA A1,2on the xcoordinate can be neglected in the central part of the dynamic grating (50) for the distances of several SPP wavelengths [12]. Integrating the SPP electric field and nonlinear electric induction over z0, we obtain the following truncated equations for the normalized SPP intensities I1,2=A1,22/ω1,2I01.


Here, A12/ω1+A22/ω2=I0=constis the integral of motion obtained from the Manley-Rowe relation [14], and the gain ghas the form [12]


Here, b=2RekzSae1x2+ae1z2+4εaRekxIme1ze1x. Solution of Eq. (53) has the form [12]


It is easy to see from Eq. (55) that I1t+I2t=1.

Expressions (55) show that the energy exchange between SPP takes place. In the limiting case t, we obtain: I1t1;I2t0[12]. The time dependence of the normalized SPP intensities I1t,I2t(55) is presented in Figure 7. The phases γ1,20of the SPP SVA have the form


Figure 7.

The temporal dependence of the SPP normalized intensities I 1 , 2 t for the input electric field of 10 6 V / m and optical wavelengths λ opt 1 = 0.6 μm (curves 1) and λ opt 1 = 1.33 μm (curve 2).

It is easy to see from Eqs. (56) and (57) that for tthe phase γ1tof the amplified SPP I1tends to a constant value γ1tγ10ReGkxkzSΔω2ImGkxkzSΔωlnI10, while the phase of the depleted SPP I2γ2tγ20for large time intervals such that gt1takes the form γ2tγ20ReGkxkzSΔω2ImGkxkzSΔωgtand γ2tfor t. The SVA of the depleted SPP I2undergoes strong XPM and rapidly oscillates in the time domain. The results (55)(57) show that the Rayleigh stimulated scattering [27] of SPP on the smectic layer normal displacement localized grating is accompanied by XPM and the parametric energy exchange between SPP [12]. The rise time of the amplified SPP is about 12μsas it is seen from Figure 7. It is much faster than the thermal response time τR=100μsand the purely orientational response time 25msin NLC [4]. Numerical estimations show that for the SPP electric field of 107V/mthe rise time of about 10 ns can be achieved, which is much less than the Brillouin relaxation time τB200ns[4, 12].

Structures consisting of alternative conducting and dielectric thin films are capable of guiding SPP light waves [24, 25]. Each single interface can sustain bound SPP. When the distance between adjacent interfaces is comparable or smaller than the SPP localization length Lz=RekzS1, the coupled modes occur due to the interaction between SPP [24]. The following specific three-layer guiding systems can be considered: (i) an insulator/metal/insulator (IMI) heterostructure where a thin metallic layer is sandwiched between two infinitely thick dielectric claddings; (ii) a metal/insulator/metal (MIM) heterostructure where a thin dielectric core layer is sandwiched between two infinitely thick metallic claddings [24]. LC can be used as a tunable cladding material or as the guiding core material due to their excellent electro-optic properties 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 the metallic claddings z>d;z<dH1,2xzt,E1,2xzt, and in the SALC core zdHSAxzt,ESAxzthave the form, respectively [24]


Figure 8.

The MIM waveguide with the homeotropically oriented SALC as a core.

The complex wave number kzSof SPP in SALC in the linear approximation is determined by expression (49) and the SPP wave number in the metallic claddings is given by kzm=kx2εmωω2/c2[24]. Using the boundary conditions for the tangential components of the SPP fields (58)(61) at the interface z=dbetween the metallic cladding and the SALC core, we obtain the dispersion relation for the MIM modes [13, 24]


Numerical estimations show that for the typical values of the SPP frequency ω, the plasma frequency ωp, the SPP lifetime τmentioned above, and the MIM thickness 2d=1μmthe following relationships take place:

RekzS106m1ImkzS104m1,Rekx107m1Imkx103m1. The SPP wavelength in the Z direction is given by 2πImkzS1102μmand can be neglected inside the MIM waveguide core with the thickness of 2d1μm. Then, a single localized TM can exist in the MIM waveguide according to the dispersion relation (62). The even TM mode has the form [13]


The distribution of the TM even mode normalized intensity ESA2/E02in the MIM waveguide core is presented in Figure 9. It is seen from Figure 9 that the intensity is filling the MIM waveguide core due to the overlapping of SPP inserted from the metallic claddings z<d;z>d. Substituting the SPP field (63) into equation of motion (9), we obtain the smectic layer normal strain [13].

Figure 9.

Distribution of the SPP normalized intensity E → SA 2 / E 0 2 in the MIM waveguide core (arbitrary units).


The nonlinear polarization in the SALC core caused by the smectic layer strain (64) has the form [13]


Substituting the SPP electric field (63) and nonlinear polarization (65) into Eq. (13) and separating linear and nonlinear parts, we obtain the truncated equation for SVA E0t=E0texpt.


At the distances xLx=Imkx1, the SVA dependence on xcan be neglected since exp2Imkxx1[13]. The dispersion effects can be neglected because the dispersion length LDLx[13]. Integrating both sides of Eq. (66) over zddand separating real and imaginary parts we obtain the following equations for the magnitude Etand phase φtof the SPP SVA.




The solution of Eq. (67) has the form


Eq. (69) shows that the strong SPM of the even SPP mode in the MIM wave guide occurs. It is enhanced by a large geometric factor F2kzSkxd/F1kzSkxd, which can achieve a value of 102104for 2d=1μmand RekzS1063×106m1,Rekx5×106107m1.

7. Conclusions

In SALC, there exists a specific mechanism of the optical nonlinearity related to the normal displacement uxyztof 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 uxyztin 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 uxyztand 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 polarization-decoupled 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 1015 s−1 and coupled SPP frequency difference of about 108 s−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.

The plasmonic waveguides with NLC for nanophotonic and plasmonic have been theoretically 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.

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Boris I. Lembrikov, David Ianetz and Yossef Ben Ezra (December 20th 2017). Nonlinear Optical Phenomena in Smectic A Liquid Crystals, Liquid Crystals, Pankaj Kumar Choudhury, IntechOpen, DOI: 10.5772/intechopen.70997. Available from:

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