Open access peer-reviewed chapter - ONLINE FIRST

Stimulated Scattering of Surface Plasmon Polaritons in a Plasmonic Waveguide with a Smectic A Liquid Crystalline Core

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

Submitted: May 26th 2019Reviewed: September 2nd 2019Published: October 30th 2019

DOI: 10.5772/intechopen.89483

Downloaded: 32

Abstract

We considered theoretically the nonlinear interaction of surface plasmon polaritons (SPPs) in a metal-insulator-metal (MIM) plasmonic waveguide with a smectic liquid crystalline core. The interaction is related to the specific cubic optical nonlinearity mechanism caused by smectic layer oscillations in the SPP electric field. The interfering SPPs create the localized dynamic grating of the smectic layer strain that results in the strong stimulated scattering of SPP modes in the MIM waveguide. We solved simultaneously the smectic layer equation of motion in the SPP electric field and the Maxwell equations for the interacting SPPs. We evaluated the SPP mode slowly varying amplitudes (SVAs), the smectic layer dynamic grating amplitude, and the hydrodynamic velocity of the flow in a smectic A liquid crystal (SmALC).

Keywords

  • surface plasmon polariton (SPP)
  • smectic liquid crystals
  • stimulated light scattering (SLS)
  • plasmonic waveguide

1. Introduction

Nonlinear optical phenomena based on the second- and third-order optical nonlinearity characterized by susceptibilities χ2and χ3, respectively, are widely used in modern communication systems for the optical signal processing due to their ultrafast response time and a large number of different interactions [1, 2, 3, 4, 5]. The second-order susceptibility χ2exists in non-centrosymmetric media, while the third-order susceptibility χ3exists in any medium [6]. The second-order susceptibility χ2may be used for the second harmonic generation (SHG), sum, and difference frequency generation; the ultrafast Kerr-type third-order susceptibility χ3results in such effects as four-wave mixing (FWM), self-phase modulation (SPM), cross-phase modulation (XPM), third harmonic generation (THG), bistability, and different types of the stimulated light scattering (SLS) [1, 2, 3, 4, 5, 6]. Optical-electrical-optical conversion processes can be replaced with the optical signal processing characterized by the femtosecond response time of nonlinearities in optical materials [2, 3]. All-optical signal processing, ultrafast switching, optical generation of ultrashort pulses, the control over the laser radiation frequency spectrum, wavelength exchange, coherent detection, multiplexing/demultiplexing, and tunable optical delays can be realized by using the nonlinear optical effects [1, 2, 3, 4]. However, optical nonlinearities are weak and usually occur only with high-intensity laser beams [1, 6]. An effective nonlinear optical response can be substantially increased by using the plasmonic effects caused by the coherent oscillations of conduction electrons near the surface of noble metal structures [1]. In the case of the extended metal surfaces, the surface plasmon polaritons (SPPs) may occur [1, 7, 8]. SPPs are electromagnetic excitations propagating at the interface between a dielectric and a conductor, evanescently confined in the perpendicular direction [1, 7]. The SPP electromagnetic field decays exponentially on both sides of the interface which results in the subwavelength confinement near the metal surface [1]. The SPP propagation length is limited by the ohmic losses in metal [1, 7, 8].

Nonlinear optical effects can be enhanced by plasmonic excitations as follows: (i) the coupling of light to surface plasmons results in strong local electromagnetic fields; (ii) typically, plasmonic excitations are highly sensitive to dielectric properties of the metal and surrounding medium [1]. In nonlinear optical phenomena, such a sensitivity can be used for the light-induced nonlinear change in the dielectric properties of one of the materials which result in the varying of the plasmonic resonances and the signal beam propagation conditions [1]. Plasmonic excitations are characterized by timescale of several femtoseconds which permits the ultrafast optical signal processing [1]. The SPP field confinement and enhancement can be changed by modifying the structure of the metal or the dielectric near the interface [1]. For example, plasmonic waveguides can be created [1, 7, 8, 9]. Nanoplasmonic waveguides can confine and enhance electric fields near the nanometallic surfaces due to the propagating SPPs [9]. Nanoplasmonic waveguide consists of one or two metal films combined with one or two dielectric slabs [9]. Typically, two types of the plasmonic waveguides exist: (i) an insulator/metal/insulator (IMI) heterostructure where a thin metallic layer is placed between two infinitely thick dielectric claddings and (ii) a metal/insulator/metal (MIM) heterostructure where a thin dielectric layer is sandwiched between two metallic claddings [7]. The MIM waveguides for nonlinear optical applications require highly nonlinear dielectrics [9]. The nonlinear metamaterials can significantly increase the nonlinearity magnitude [10]. Investigation of nonlinear metamaterials is related in particular to nonlinear plasmonics and active media [10]. One of the metamaterial nonlinearity mechanisms is based on liquid crystals (LCs) [10]. Tunability and a strongly nonlinear response of metamaterials can be obtained by their integration with LCs offering a practical solution for controlling metamaterial devices [11].

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

SmALCs are characterized by a positional long-range order in the direction of the elongated molecular axis and demonstrate a layer structure with a layer thickness dSmA2nm[14]. Inside a smectic layer, the molecules form a two-dimensional liquid [14]. Actually, SmALC can be considered as a natural nanostructure. The structures of NLC with the elongated molecules directed mainly along the vector director nand the homeotropically oriented SmALC with the layer plane parallel to the claddings are shown in Figure 1a and b, respectively.

Figure 1.

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

The nonlinear optical phenomena in SmALC such as a light self-focusing, self-trapping, SPM, SLS, and FWM based on the specific mechanism of the third-order optical nonlinearity related to the smectic layer normal displacement had been investigated theoretically [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. In particular it has been shown that at the interface of a metal and SmALC, the counter-propagating SPPs created the dynamic grating of the smectic layer normal displacement uxzt, and the SLS of the interfering SPPs occurred [22, 23, 26]. We also investigated the behavior of SPP mode in a MIM waveguide with the SmALC core [24, 26]. In such a waveguide, SPP behaves as a strongly localized transverse magnetic (TM) mode which creates the localized smectic layer normal deformation and undergoes SPM [24, 26].

In this chapter we consider theoretically the interaction of the counter-propagating SPP modes in the MIM waveguide with the SmALC core. The interfering SPP TM modes with the close optical frequencies ω1,2create a localized dynamic grating of the smectic layer normal displacement uxztwith the frequency Δω=ω1ω2ω1which results in the nonlinear polarization and stimulated scattering of SPPs. We solved simultaneously the equation of motion for smectic layers in the electric field of the interfering SPP modes and the Maxwell equations for the SPPs in the MIM waveguide taking into account the nonlinear polarization. We used the slowly varying amplitude (SVA) approximation for the SPPs [6]. We evaluated the magnitudes and phases of the coupled SPP SVAs. It is shown that the energy exchange between the coupled SPPs and XPM takes place. We also evaluated the SPP-induced smectic layer displacement and SmALC hydrodynamic velocity. We have shown that the high-frequency localized electric field can occur in the MIM waveguide with the SmALC core due to the flexoelectric effect [28].

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

2. Hydrodynamics of SmALC in the external electric field

In this section we briefly discuss the SmALC hydrodynamics and derive the equation of motion for the smectic layer normal displacement uxyztin the external electric field Exyzt. SmALC can be described by the one-dimensional periodic density wave due to its layered structure.

Smectic layer oscillations uxyztin the external electric field Exyztare shown in Figure 2. Hydrodynamics of SmALC in general case is very complicated because SmALC is a strongly anisotropic viscous liquid including the layer oscillations, the mass density, and the elongated molecule orientation variations [29, 30, 31]. However, the elastic constant related to the SmALC bulk compression is much larger than the elastic constant B106107Jm3related to the smectic layer compression [29, 30, 31]. The layers can oscillate without the change of the mass density [29, 30, 31]. For this reason two uncoupled acoustic modes can propagate in SmALC: the ordinary longitudinal sound wave caused by the mass density variation and the second-sound (SS) wave caused by the layer oscillations [29, 30, 31]. SS wave is characterized by strongly anisotropic dispersion relation being neither purely transverse nor longitudinal. It propagates in the direction oblique to the layer plane and vanishes for the wave vector kSperpendicular or parallel to the layer plane [29]. SmALC is characterized by the complex order parameter, and SS represents the oscillations of the order parameter phase [29]. SS in SmALC has been observed experimentally by different methods [32, 33, 34]. The system of hydrodynamic equations for the incompressible SmALC under the constant temperature far from the phase transition has the form [29, 30, 31]

Figure 2.

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

divv=0E1
ρvit=∂Πxi+Λi+σikxkE2
Λi=δFδuiE3
σik=α0δikAll+α1δizAzz+α4Aik+α56δizAzk+δkzAzi+α7δizδkzAllE4
Aik=12vixk+vkxiE5
vz=utE6

Here, vis the hydrodynamic velocity, ρ103kgm3is the SmALC mass density, Πis the pressure, Λis the generalized force density, σikis the viscous stress tensor, αi101kgsm1are the viscosity Leslie coefficients, δik=1,i=k;δik=0,ik, and Fis the free energy density of SmALC. Typically, SmALC is supposed to be an incompressible liquid according to Equation (1) [29]. For this reason, we assume that the pressure Π=0and the SmALC free energy density Fdo not depend on the bulk compression [29, 30, 31]. We are interested in the SS propagation and neglect the ordinary sound mode. The normal layer displacement uxyztby definition has only one component along the Z axis. In such a case, the generalized force density has only the Z component according to Eq. (3): Λ=00Λz. Eq. (6) is specific for SmALC since it determines the condition of the smectic layer continuity [29, 30, 31]. The SmALC free energy density F in the presence of the external electric field Exyzthas the form [29, 30, 31]

F=12Buz2+12K2ux2+2uy2212ε0εikEiEkE7

Here K1011Nis the Frank elastic constant associated with the SmALC orientational energy inside layers, ε0is the free space permittivity, and εikis the SmALC permittivity tensor including the terms defined by the smectic layer strains. The purely orientational second term in the free energy density F(7) can be neglected since for the typical values of the elastic constants Band KKkS2Bwhere kS, the SS wave vector component is parallel to the layer plane. The permittivity tensor εikis given by [30]

εxx=εyy=ε+auz;εzz=ε+auz;εxz=εzx=εaux;εyz=εzy=εauy;εa=εεE8

where ε,εare the diagonal components of the permittivity tensor εikalong and perpendicular to the optical axis and a1,a1are the phenomenological dimensionless coefficients [29, 30]. SmALC is an optically uniaxial medium with the optical Z axis perpendicular to the smectic layer plane [29, 30, 31]. Combining Eqs. (1)(8), we obtain the equation of motion for the smectic layer normal displacement uxyztin the electric field Exyzt[16, 17]:

ρ22ut2+α122z2+12α4+α5622ut+B22uz2=ε022zaEx2+Ey2+aEz22εaxExEz+yEyEzE9

Here 2u=2u/x2+2u/y2. In the absence of the external electric field, the homogeneous solution of the equation of motion (9) represents the SS wave with the dispersion relation [29]:

ΩS=s0kSkzSkS; s0=BρE10

Here, kS2=kxS2+kyS2and ΩSand s0 are the SS frequency and velocity, respectively [29]. It is seen from Eq. (10) that the SS frequency ΩS=0for the propagation direction along the smectic layer plane and perpendicular to it. The decay constant Γis given by

Γ=12ρα1kS2kzS2kS2+12α4+α56kS2E11

If the viscosity terms responsible for the SS wave decay can be neglected, then the homogeneous part of Eq. (9) reduces to the SS wave equation with the dispersion relation (10) [29, 30, 31]:

ρ22ut2=B22uz2

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

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 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, phase-shifters, 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, 37, 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=±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=uxzt, 2u=2u/x2+2u/z2, 2u=2u/x2, kS2=kxS2, and the SmALC permittivity tensor (8) takes the form

Figure 3.

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

εxx=ε+auz;εzz=ε+auz;εxz=εzx=εaux;εa=εεE12

The permittivity εmωof the metal claddings is described by the Drude model [7, 8]:

εmω=1ωp2ω2+/τE13

where ωp=n0e2/ε0mis the plasma frequency of the free electron gas; n0 is the free electron density in the metal;e, m are the electron charge and mass, respectively; and ω,τare the SPP angular frequency and lifetime, respectively [7, 8]. The electric field Exztof the optical wave propagating in a nonlinear medium is described by the following wave equation including the nonlinear part of the electric induction DNL[6]:

curlcurlE+μ02DLt2=μ02DNLt2E14

Here μ0 is the free space permeability and DLis 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 ETM=Ex0Ez; HTM=0Hy0[7]. In such a case, we obtain for DLand DNLin SmALC using Eq. (12)

DxL=ε0εEx; DzL=ε0εEzE15
DxNL=ε0auzExεauxEz; DzNL=ε0auzEzεauxExE16

The linear part DmLof the electric induction in the metal claddings has the form: DmL=ε0εmωE[7]. The SPP TM mode electric and magnetic fields for z>din the metal claddings H1,2xzt,E1,2xztand for zdin SmALC HSAxzt,ESAxzthave the form [24, 26]

H1,2xzt=12ayH1,20expkzmz+ikxxiωt+c.c.,z>dE17
E1,2xzt=12axE1,2x0+azE1,2z0expkzmz+ikxxiωt+c.c.,z>dE18
HSAxzt=12ayAexpkzSz+BexpkzSzexpikxxiωt+c.c.,zdE19
ESAxzt=12{axkzSiωε0εAexpkzSzBexpkzSzazkxωε0εAexpkzSz+BexpkzSz}expikxxωt+c.c.;zdE20

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 (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 kzmand kzS[24, 26]:

kzm=kx2εmωω2/c2E21
kzS=kx2ε/εω2ε/c2E22

where c is the free space light velocity. The boundary conditions for the fields (17)(20) at the interfaces z=±dhave the form [7, 8]

H1yz=d=HSAyz=d;H2yz=d=HSAyz=dE23
E1xz=d=ESAxz=d;E2xz=d=ESAxz=dE24

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]

exp4kzSd=kzmεmω+kzSε2kzmεmωkzSε2E25

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 ωcorresponding to the optical wavelength range λopt11.6μmand 2d1μmare presented in Figures 4 and 5.

Figure 4.

The spectral dependence of Re k z S (a) and Im k z S (b).

Figure 5.

The spectral dependence of Re k x (a) and Im k x (b).

These results show that RekzS106m1ImkzS104m1and Rekx107m1Imkx103m1[24, 26]. In such a case, the SPP oscillation length in the Z direction is defined by the relationship 2πImkzS1104md106m, and ImkzScan be neglected inside the MIM waveguide, and kzSRekzS[24, 26]. The SPP propagation length in the Xdirection LSPP=Imkx1104103mλSPP=2πRekx1<106mwhere λSPPis the SPP wavelength. Hence, at the optical wavelength-scale distances, Imkxcan be neglected, and kxRekx[24, 26]. 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 ESAxztgiven by [24, 26]

ESA=E0axcoshRekzSzazikxεkzSεsinhRekzSz×expiRekxxωt+c.c.E26

The numerical estimations show that for the SPP modes with the close optical frequencies ω1,21015s1and the frequency difference Δω=ω1ω2108s1ω1, the wave numbers of the both SPPs kz1,2Sand kx1,2are 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

ESA1,2=ESA1,20axcoshRekzSzazikxεkzSεsinhRekzSz×exp±iRekxxiω1,2t+c.c.E27

Substituting expression (27) into equation of motion (9), we obtain the expression of the smectic layer displacement localized dynamic grating uxzt:

uxzt=U0sinh2RekzSzexpi2RekxxΔωt+c.c.E28

Here

U0=4ε0ESA10ESA20GkxkzSΔωRekx2RekzSh;h=aakx2ε2kzS2ε22εaRekx2εRekzS2εE29
GkxkzSΔω=4ρΔω2Rekx2+RekzS2iΔω{α12Rekx22RekzS2+12α4+α562Rekx2+2RekzS22}B2Rekx22RekzS2E30

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 Δω108s1, and it can be neglected. The normalized smectic layer displacement uxzt=t0/U0for the optical wavelength λopt=1.6μmis 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.

Figure 6.

The normalized smectic layer displacement u x z t = t 0 for the optical wavelength λ opt = 1.6 μm .

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 counter-propagating SPPs (27), we should solve wave Eq. (14) including the nonlinear term DNL. We use the SVA approximation for the SPP electric field amplitudes ESA1,20t=ESA1,20texpiθSA1,2twhere ESA1,20tand θSA1,2tare the SVA magnitudes and phases, respectively [6]. For the distances of the order of magnitude of the SPP wavelength λSPP<1μm, the dependence of SAVs on the x coordinate can be neglected. We assume according to the SVA approximation that

2ESA1,20t2ω1ESA1,20tE31

Substituting expressions (27) and (28) into Eqs. (16), we evaluate the nonlinear part DNLof the electric induction in SmALC. Then, substituting relationships (15), (16), and (27) into wave equation (14), taking into account the dispersion relation (22), neglecting the terms 2ESA1,20/t2according to condition (31), combining the phase-matched terms with the frequencies ω1,2, and dividing the real and imaginary parts, we derive the equations for the SVA magnitudes ESA1,20tand phases θSA1,2t. They have the form

1ω1,2ESA1,20t2tF1z=8ε0ImGkxkzSΔωRekx2hESA10t2ESA20t2εGkxkzSΔω2F2zE32
1ω1,2θ1,2tF1z=4ε0ReGkxkzSΔωESA2,10t2Rekx2hεGkxkzSΔω2F2zE33

Here we assumed that the factor exp±Imkxx1for the distances xImkx1. The functions F1,2zdescribing the SPP mode localization inside the MIM waveguide are given by

F1z=cosh2kzSz+kx2kzS2sinh2kzSzE34
F2z=cosh2kzSzcosh2kzSzakzS2+εaεεkx2εaεεkx2kx2sinh2kzSzcosh2kzSzaεεεaεaE35

Here we neglected the small quantities Imkxand ImkzSassuming that for xImkx1and zd, we may use the relationships kxRekxand kzSRekzS. We integrate both parts of Eqs. (32) and (33) over the MIM waveguide thickness dzd[40]. After the integration, Eqs. (32) and (33) take the form

1ω1,2ESA1,20t2t=8ε0ImGkxkzSΔωkx2hkzS2εGkxkzSΔω2×ESA10t2ESA20t2FNkxkzSE36
1ω1,2θ1,2t=4ε0ReGkxkzSΔωkx2hkzS2εGkxkzSΔω2ESA2,10t2FNkxkzSE37

where the localization factor FNkxkzSis given by

FNkxkzS={sinh4kzSd+4kzSd4aaεkx2εkzS2+εakx2kzS21+εε+sinh2kzSda+aεkx2εkzS22kzSdεakx2kzS21+εε}×sinh2kzSd1+kx2kzS2+2kzSd1kx2kzS21E38

The spectral dependence of the localization factor FNkxkzSis presented in Figure 7.

Figure 7.

The spectral dependence of the localization factor F N k x k z S .

It is seen from Figure 7 that FNkxkzSis 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]:

tESA10t2ω1+ESA20t2ω2=0E39

We obtain from Eq. (39) the Manley-Rowe relation for the SVA magnitudes ESA1,20t2[6]:

ESA10t2ω1+ESA20t2ω2=const=I0E40

We introduce the dimensionless quantities

I1,2t=ESA1,20t2ω1,2I0;I1t+I2t=1E41

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

I1,2t=gI1I2E42

where the gain g has the form

g=8ε0ImGkxkzSΔωkx2hkzS2ω1ω2I0εGkxkzSΔω2FNkxkzSE43

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

Figure 8.

The spectral dependence of the gain g for the SPP electric field amplitude E SA 10 t = 10 6 V / m .

I1t=I10expgt1I101expgtE44
I2t=1I101I101expgtE45

It is easy to see from Eqs. (44) and (45) that the solutions I1,2tsatisfy the Manley-Rowe relation (40). Expressions (44) and (45) describe the energy exchange between the SPPs interfering on the smectic layer dynamic grating. Indeed, I10and I21for g>0and t. Actually, the SLS of the orientational type takes place [6]. The SPP1 with the normalized intensity I1plays a role of the pumping wave, while the SPP2is a signal wave. The temporal dependence of I1,2tfor the pumping wave amplitude ESA10t=106V/mis shown in Figure 9.

Figure 9.

The temporal dependence of the SPP normalized intensities I 1 , 2 t for pumping wave amplitude E SA 10 t = 10 6 V / m .

It is seen form Figure 6 that for I10>I20, the characteristic time t0exists when I1t0=I2t0. Using expressions (44) and (45), we obtain

t0=1glnI10I20E46

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

I1,2t=121tanhg2tt0E47

The time duration of the energy exchange between the SPPs is about 1nsas it is seen from Figure 9. Substituting relationships (43)(45) into Eq. (37), we evaluate the phases θSA1,2t. They are given by the following expressions:

θSA1tθSA10=ReGkxkzSΔω2ImGkxkzSΔω×lnI20expgt+I10E48
θSA2tθSA20=ReGkxkzSΔω2ImGkxkzSΔω×ln1I10+I10expgtE49

The temporal dependence of the SPP SVA phases θSA1,2tis shown in Figure 10.

Figure 10.

The temporal dependence of the SPP SVA phases θ SA 1 t (a) and θ SA 2 t (b).

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:

θSA1tθSA10ReGkxkzSΔω2ImGkxkzSΔωgtE50

Such a behavior corresponds to the rapid oscillations of the depleted pumping wave amplitude. The signal wave phase for ttends to a constant value:

θSA2tθSA20ReGkxkzSΔω2ImGkxkzSΔωln1I10E51

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

U0=2ε0I0kx2kzSω1ω2hGkxkzSΔωcoshg2tt0expiθSA1θSA2E52

The temporal dependence of the amplitude (52) normalized absolute value U0/U0maxis presented in Figure 11. Here

Figure 11.

The temporal dependence of the dynamic grating amplitude normalized absolute value U 0 / U 0 max .

U0max=2ε0I0kx2kzSω1ω2hGkxkzSΔωE53

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

vzxzt=iΔωU0sinh2kzSzexpi2kxxΔωt+c.c.E54
vxxzt=ΔωU0kzSkxcosh2kzSzexpi2kxxΔωt+c.c.E55

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

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 10−9 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.

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Boris I. Lembrikov, David Ianetz and Yossef Ben Ezra (October 30th 2019). Stimulated Scattering of Surface Plasmon Polaritons in a Plasmonic Waveguide with a Smectic A Liquid Crystalline Core [Online First], IntechOpen, DOI: 10.5772/intechopen.89483. Available from:

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