Introductory Chapter: Nonlinear Optical Phenomena in Plasmonics, Nanophotonics and Metamaterials

1. Introduction

The observation of nonlinear optical effects started with the discovery of the laser in 1960 as a source of the high-intensity coherent radiation [1, 2]. Nonlinear optics is the study of the nonlinear light and matter interaction effects such as the light self-focusing and self-trapping, second harmonic generation (SHG), third-harmonic generation (THG), four-wave mixing (FWM), parametric processes, different types of stimulated light scattering (SLS), soliton generation and propagation [1, 2]. The nonlinear optical phenomena are widely used in modern communication systems for such applications as the generation of ultra-short pulses, all-optical signal processing and ultrafast switching [2, 3]. Recently, new fields of nonlinear optics emerged such as strong-field nano-optics, nonlinear plasmonics, and nonlinear metamaterials [3, 4, 5, 6, 7, 8].

In the case of the strong-field nano-optics the optical field interacts with matter at the wavelength or even at the subwavelength scale [5]. Strong electromagnetic fields of a light wave with the wavelength $\lambda \sim 1\mathit{\mu m}$ interacting with electrons are confined to a few nanometers which result in a substantial enhancement of the local electric field [4]. Typically, an increase of the nonlinear optical response at a nanoscale is caused by the plasmonic effects, i.e., the coherent oscillations of the conduction electrons near the surface of noble-metal structures [3, 4, 5, 6]. At the extended metal surfaces, the surface plasmon polaritons (SPPs) can occur [3]. SPP is a surface electromagnetic wave propagating at the metal-dielectric interface [3, 4, 5]. In the case of the metal nanoparticles there exist localized surface plasmons (LSPs) with resonances depending on the nanoparticle size and shape [3]. Nonlinear optical effects can be significantly enhanced by plasmonic excitations in two ways: (i) the plasmonic structures provide the optical field enhancement near the metal-dielectric interface due to SPPs or LSPs; (ii) the SPP and LSP parameters are very sensitive to refractive indices of the metal and the surrounding dielectric medium [3, 4, 5, 6]. Plasmonic excitations are characterized by the timescale of several femtoseconds which makes it possible the ultrafast optical signal processing [3].

Metamaterials are artificial materials with desirable properties [7]. For this reason, the magnitude of the metamaterial optical nonlinearity can be substantially increased [7]. Generally, all fundamental nonlinear optical phenomena such as self-action effects, wavelength conversion, nonlinear surface waves, nonlinear guided waves and solitons are possible in nonlinear metamaterials [7]. The nonlinear metamaterials are closely related to plasmonics, active media, and nonlinearity based on liquid crystals (LCs) [7, 8]. LCs possess simultaneously the properties of a liquid and a crystal [8, 9, 10]. Thermotropic LCs self-assemble in a different ordered arrangements of their crystalline axis depending on the temperature [8, 9, 10]. The orientational order of LC may be changed by a moderate external electric field [8, 9, 10]. For this reason, liquid crystalline media are characterized by strong optical nonlinearity [8, 9, 10]. Nematic LCs (NLCs) characterized by the ordering of molecular elongated axes are mainly used in applications [8].

In this chapter, we discuss the fundamentals of plasmonics and the specific features of the nonlinear optical effects in plasmonic nanostructures. The chapter is organized as follows. In Section 2 the interaction of the electromagnetic field with the free electrons in metals is described based on Maxwell’s equations and equation of motion for a free electron in the external electric field. The dielectric function of the free electron gas is obtained based on the Drude model. In Section 3 SPPs at the interface of a metal and a dielectric are investigated. In Section 4 the nonlinear phenomena in the plasmonic structures are briefly discussed. The details of this topic may be found in the corresponding references. Conclusions are presented in Section 5.

2. Interaction between electromagnetic field and free electrons in metals

The interaction of electromagnetic fields with metals is described by Maxwell’s equations of macroscopic electromagnetism [11]. This approach is valid also for metallic nanostructures characterized by the sizes of several nanometers [11]. The Maxwell’s equations have the form [1, 2, 11].

$$\nabla \cdot \overrightarrow{D}={\rho }_{\mathit{ext}}$$E1

$$\nabla \cdot \overrightarrow{B}=0$$E2

$$\nabla \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$$E3

$$\nabla \times \overrightarrow{H}={\overrightarrow{J}}_{\mathit{ext}}+\frac{\partial \overrightarrow{D}}{\partial t}$$E4

Here $\overrightarrow{D},\overrightarrow{E},\overrightarrow{H}$ and $\overrightarrow{B}$ are the dielectric displacement, the electric field, the magnetic field and the magnetic induction (or magnetic flux density), respectively; ${\rho }_{\mathit{ext}}$ and ${\overrightarrow{J}}_{\mathit{ext}}$ are external charge and current densities, respectively. The four macroscopic fields are related by the following material Equations [11].

$$\overrightarrow{D}={\epsilon }_0\epsilon \overrightarrow{E}$$E5

$$\overrightarrow{B}={\mu }_0\mu \overrightarrow{H}$$E6

where ${\epsilon }_0$ and ${\mu }_0$ are the electric permittivity and magnetic permeability of vacuum, respectively, $\epsilon$ is the relative permittivity and $\mu$ is the relative permeability of the medium, $\mu =1$ for the nonmagnetic medium. The total charge and current densities ${\rho }_{\mathit{tot}}$ and ${\overrightarrow{J}}_{\mathit{tot}}$ consist of the external charge and current densities ${\rho }_{\mathit{ext}}$, ${\overrightarrow{J}}_{\mathit{ext}}$ and the internal ones $\rho ,\overrightarrow{J}$. They are given by [11]: ${\rho }_{\mathit{tot}}={\rho }_{\mathit{ext}}+\rho ,{\overrightarrow{J}}_{\mathit{tot}}={\overrightarrow{J}}_{\mathit{ext}}+\overrightarrow{J}$. The internal current density has the form [11].

$$\overrightarrow{J}=\sigma \overrightarrow{E}$$E7

where $\sigma$ is the conductivity of the medium. The optical response of metals typically depends on frequency and wave vector in the case of a spatial dispersion [11]. In such a case, Eqs. (5) and (7) can be generalized to the following relationships taking into account the non-locality in time and space [11].

$$\overrightarrow{D}\left(\overrightarrow{r},t\right)={\epsilon }_0\int d{t}^{\prime }d{\overrightarrow{r}}^{\prime }\epsilon \left(\overrightarrow{r}-{\overrightarrow{r}}^{\prime },t-t^{\prime }\right)\overrightarrow{E}\left({\overrightarrow{r}}^{\prime },t^{\prime }\right)$$E8

$$\overrightarrow{J}\left(\overrightarrow{r},t\right)={\epsilon }_0\int d{t}^{\prime }d{\overrightarrow{r}}^{\prime }\sigma \left(\overrightarrow{r}-{\overrightarrow{r}}^{\prime },t-t^{\prime }\right)\overrightarrow{E}\left({\overrightarrow{r}}^{\prime },t^{\prime }\right)$$E9

The Fourier transform of expressions (8) and (9) results in the following relationships in the Fourier domain [11].

$$\overrightarrow{D}\left(\overrightarrow{k},\omega \right)={\epsilon }_0\epsilon \left(\overrightarrow{k},\omega \right)\overrightarrow{E}\left(\overrightarrow{k},\omega \right)$$E10

$$\overrightarrow{J}\left(\overrightarrow{k},\omega \right)=\sigma \left(\overrightarrow{k},\omega \right)\overrightarrow{E}\left(\overrightarrow{k},\omega \right)$$E11

where $\overrightarrow{k}$ and $\omega$ are the wave vector and angular frequency of the field plane-wave components, respectively.

Combining Eqs. (3) and (4) we obtain the wave equation for the electric field $\overrightarrow{E}\left(\overrightarrow{r},t\right)$ [1, 2, 11].

$$\nabla \times \nabla \times \overrightarrow{E}=-{\mu }_0\frac{{\partial }^2\overrightarrow{D}}{\partial t^2}$$E12

which takes the form in the frequency domain [11].

$$\overrightarrow{k}\left(\overrightarrow{k}\cdot \overrightarrow{E}\right)-k^2\overrightarrow{E}=-\epsilon \left(\overrightarrow{k},\omega \right)\frac{{\omega }^2}{c^2}\overrightarrow{E}$$E13

where $c={\left(\sqrt{{\epsilon }_0{\mu }_0}\right)}^{-1}$ is the speed of light in vacuum. There are two types of solutions of Eq. (13): the transverse waves where the electric field vector is perpendicular to the wave propagation direction, and the longitudinal waves where the electric field vector is parallel to the wave propagation direction. The transverse waves are characterized by the condition $\left(\overrightarrow{k}\cdot \overrightarrow{E}\right)=0$ and the corresponding dispersion relation

$$k^2=\epsilon \left(\overrightarrow{k},\omega \right)\frac{{\omega }^2}{c^2}$$E14

The dispersion relation for the longitudinal waves can be obtained from Eq. (13).

$$\epsilon \left(\overrightarrow{k},\omega \right)=0$$E15

It is seen from Eq. (15) that the longitudinal collective oscillations can exist only for the frequencies $\omega$ which correspond to the zeros of the dielectric function $\epsilon \left(\overrightarrow{k},\omega \right)$ [11].

Consider a plasma model that explains the optical properties of alkali metals and noble metals for the frequencies up to the ultraviolet ones and to the visible ones, respectively [11, 12]. In the framework of the phenomenological plasma model the metal crystal lattice potential and electron–electron interactions are not taken into account [11]. It is assumed that the details of the metal energy band structure are included into the effective optical mass $m$ of each electron [11]. In such a model, the equation of motion for an electron with the charge $e$ and the effective mass $m$ of the plasma sea in the external electric field $\overrightarrow{E}$ has the form [11].

$$m\frac{d^2\overrightarrow{x}}{{\mathit{dt}}^2}+\mathit{m\gamma }\frac{d\overrightarrow{x}}{\mathit{dt}}=-e\overrightarrow{E}$$E16

where $\overrightarrow{x}$ is the free electron displacement, $\gamma ={\tau }^{-1}$ is a characteristic collision frequency, $\tau$ is the free electron gas relaxation time. Typically, $\tau \sim {10}^{-14}s$ at room temperature which yields $\gamma \approx 100\mathit{THz}$ [11]. For the external field $\overrightarrow{E}\left(t\right)={\overrightarrow{E}}_{0}exp\left(-\mathit{i\omega t}\right)$ we obtain from Eq. (16) the following particular solution.

$$\overrightarrow{x}\left(t\right)={\overrightarrow{x}}_{0}exp\left(-\mathit{i\omega t}\right)=\frac{e}{m\left({\omega }^2+\mathit{i\gamma \omega }\right)}\overrightarrow{E}\left(t\right)$$E17

The macroscopic polarization $\overrightarrow{P}$ of the medium caused by the displaced electrons is given by [11].

$$\overrightarrow{P}=-\mathit{ne}\overrightarrow{x}$$E18

where $n$ is the free electrons concentration. Substituting expression (17) into Eq. (18) we obtain.

$$\overrightarrow{P}=-\frac{{\mathit{ne}}^2}{m\left({\omega }^2+\mathit{i\gamma \omega }\right)}\overrightarrow{E}$$E19

Taking into account that the dielectric displacement $\overrightarrow{D}$ in a medium is given by [1, 2, 11].

$$\overrightarrow{D}={\epsilon }_0\overrightarrow{E}+\overrightarrow{P}$$E20

and substituting expression (19) into Eq. (20) we obtain.

$$\overrightarrow{D}={\epsilon }_0\left(1-\frac{{\omega }_p^2}{{\omega }^2+\mathit{i\gamma \omega }}\right)\overrightarrow{E}$$E21

where ${\omega }_p=\sqrt{{\mathit{ne}}^2/\left({\epsilon }_{0}m\right)}$ is the plasma frequency of the free electron gas [4, 11]. Comparing relationships (5), (10) and (21) we obtain the expression of the free electron gas dielectric function $\epsilon \left(\omega \right)$ [4, 11].

$$\epsilon \left(\omega \right)=1-\frac{{\omega }_p^2}{{\omega }^2+\mathit{i\gamma \omega }}$$E22

The dielectric function $\epsilon \left(\omega \right)$(22) is known as the Drude model of the optical response of metals [11]. Expression (22) can be divided into real and imaginary components as follows.

$$Re\epsilon \left(\omega \right)=1-\frac{{\omega }_p^2{\tau }^2}{1+{\omega }^2{\tau }^2};Im\epsilon \left(\omega \right)=\frac{{\omega }_p^2\tau }{\omega \left(1+{\omega }^2{\tau }^2\right)}$$E23

Typically, the frequencies $\omega <{\omega }_p$ are considered where the metals have the pronounced metallic properties since in such a case $Re\epsilon \left(\omega \right)<0$ [11]. In the high frequencies limiting case $\mathit{\omega \tau }\gg 1$ the imaginary part of the dielectric function (23) can be neglected while expression (22) takes the form [11].

$$\epsilon \left(\omega \right)\approx 1-\frac{{\omega }_p^2}{{\omega }^2}$$E24

Comparison of expressions (15) and (24) shows that the longitudinal waves can be excited at the plasma frequency $\omega ={\omega }_p$. These longitudinal oscillations are called the volume plasma oscillations, and the quasi-particles of these oscillations are called the volume plasmons [11]. The volume plasmons cannot interact with the transverse electromagnetic waves and can be excited by particle impact [11].

3. Surface Plasmon Polaritons (SPPs)

Consider now the SPPs at the interface between the metal $\left(z<0\right)$ and the dielectric $\left(z>0\right)$ shown in Figure 1. The metal and the dielectric are characterized by the dielectric function ${\epsilon }_1\left(\omega \right)$ defined by expression (22) and the real dielectric constant ${\epsilon }_2>0$, respectively. We consider the frequency range $\omega <{\omega }_p$ such that $Re{\epsilon }_1\left(\omega \right)<0$ . Wave Eq. (12) in such a case takes the form [11, 13].

Figure 1.

Interface between a metal ($z<0$) and a dielectric ($z>0$).

$${\nabla }^2\overrightarrow{E}-\frac{\epsilon }{c^2}\frac{{\partial }^2\overrightarrow{E}}{\partial t^2}=0$$E25

where it is assumed that the permittivity $\epsilon$ in the both media is constant over distances of the order of magnitude of an optical wavelength and $\nabla \cdot \overrightarrow{D}=0$ since the free external charges are absent [11].

The solution of the wave Eq. (25) in such a case is sought to be {11, 13].

$$\overrightarrow{E}\left(x,z,t\right)=\overrightarrow{E}\left(z\right)expi\left(\mathit{\beta x}-\mathit{\omega t}\right)$$E26

Here $\beta$ is the propagation constant which is complex in the general cases. It can be shown that there exist two types of solution of the Maxwell’s Eqs. (1)-(4) with different polarizations of the propagating waves [11, 13]. The first solution is the transverse magnetic (TM) or p modes with the field components $E_x,E_z,H_y$; the second solution is the transverse electric (TE) or s modes with the field components $H_x,H_z,E_y$ [11, 13]. We start with the TM modes. The TM electric and magnetic fields in the regions $z<0$ and $z>0$ have the form, respectively [11, 13].

$$z<0H_{1y}=A_{1}exp{k}_{1}zexpi\left(\mathit{\beta x}-\mathit{\omega t}\right)$$E27

$$E_{1x}=-{\mathit{iA}}_1\frac{k_1}{{\mathit{\omega \epsilon }}_0{\epsilon }_1\left(\omega \right)}exp{k}_{1}zexpi\left(\mathit{\beta x}-\mathit{\omega t}\right)$$E28

$$E_{1z}=-A_1\frac{\beta }{{\mathit{\omega \epsilon }}_0{\epsilon }_1\left(\omega \right)}exp{k}_{1}zexpi\left(\mathit{\beta x}-\mathit{\omega t}\right)$$E29

and

$$z>0H_{2y}=A_{2}exp\left(-k_{2}z\right)expi\left(\mathit{\beta x}-\mathit{\omega t}\right)$$E30

$$E_{2x}={\mathit{iA}}_2\frac{k_1}{{\mathit{\omega \epsilon }}_0{\epsilon }_2}exp\left(-k_{2}z\right)expi\left(\mathit{\beta x}-\mathit{\omega t}\right)$$E31

$$E_{2z}=-A_2\frac{\beta }{{\mathit{\omega \epsilon }}_0{\epsilon }_2}exp\left(-k_{2}z\right)expi\left(\mathit{\beta x}-\mathit{\omega t}\right)$$E32

where $k_{1,2}$ are the components of the wave vector in the Z-direction perpendicular to the interface. The value of $1/\left|k_{1,2}\right|$ is the evanescent decay length of the fields (27)-(32) in the direction perpendicular to the interface. It defines the confinement of the TM waves. The boundary conditions of the continuity of the tangential field components $H_y$ and $E_x$ at the interface $z=0$ have the form.

$$H_{1y}\left(z=0^{-}\right)=H_{2y}\left(z=0^{+}\right);E_{1x}\left(z=0^{-}\right)=E_{2x}\left(z=0^{+}\right)$$E33

Substituting expressions (27), (28), (30) and (31) into Eqs. (33) we obtain [11].

$$A_1=A_2;\frac{k_2}{k_1}=-\frac{{\epsilon }_2}{{\epsilon }_1\left(\omega \right)}$$E34

On the other hand, substituting expressions (28), (29) and (31), (32) into wave Eq. (25) for the medium 1 and medium 2, respectively, we obtain the following dispersion relations [11, 13].

$$k_1^2={\beta }^2-k_0^2{\epsilon }_1\left(\omega \right)$$E35

$$k_2^2={\beta }^2-k_0^2{\epsilon }_2$$E36

Here $k_0=\omega /c$ is the absolute value of the wave vector of the propagating wave in vacuum. Substituting the second expression (34) into Eqs. (35), (36) we obtain the SPP dispersion relation [11, 13].

$$\beta =k_0\sqrt{\frac{{\epsilon }_1\left(\omega \right){\epsilon }_2}{{\epsilon }_1\left(\omega \right)+{\epsilon }_2}}$$E37

Analysis of TE modes shows that they cannot exist in the form of the surface modes [11, 13]. SPPs can exist only for the TM polarization [11, 13].

Analysis of dispersion relation (37) with the Drude dielectric function (22) shows that in the case of the mid-infrared or lower frequencies the SPP propagation constant $\beta$ is close to $k_0$ [11]. As a result, SPPs can propagate over many wavelengths into the dielectric medium $z>0$ [11]. These SPPs are called Sommerfeld – Zenneck waves.

In the case of large wave vectors, the SPP frequency tends to the surface plasmon frequency ${\omega }_{\mathit{sp}}$ given by [11].

$${\omega }_{\mathit{sp}}=\frac{{\omega }_p}{\sqrt{1+{\epsilon }_2}}$$E38

The surface plasmon is an electrostatic mode which is a limiting case of SPP for $\beta \to \infty$ [11]. In the general case of the essentially complex dielectric function ${\epsilon }_1\left(\omega \right)$(22) the SPP energy attenuation length, or the propagation length $L={\left(2Im\left[\beta \right]\right)}^{-1}$ which is usually between 10 and 100 μm for visible range optical frequencies and different types of the metal/ dielectric interfaces [11]. For instance, the SPPs at a silver/air interface for a vacuum wavelength ${\lambda }_0=450\mathit{nm}$ are characterized by the propagation length $L\approx 16\mathit{\mu m}$ and the evanescent decay length $1/\left|k_2\right|\approx 180\mathit{nm}$; the corresponding values for ${\lambda }_0\approx 1.5\mathit{\mu m}$ are: $L\approx 1080\mathit{\mu m}$, $1/\left|k_2\right|\approx 2.6\mathit{nm}$ [11]. Generally speaking, the lower SPP propagation length corresponds to the better SPP confinement [11]. The confinement of the optical field below the diffraction limit of ${\lambda }_0/2$ in the dielectric medium can be realized for optical frequencies $\omega \approx {\omega }_{\mathit{sp}}$ [11].

The SPP excitation at the metal/dielectric interface can be achieved by using the special phase-matching techniques such as a grating or prism coupling for the three-dimensional beams [11].

4. Nonlinear optical phenomena in Plasmonic nanostructures

In this section, we briefly discuss the peculiarities of the nonlinear optical effects in plasmonic structures. Plasmonics is a new field of photonics related to the interaction of light with matter in metallic nanostructures [12]. Plasmonics combines the capacity of photonics and the miniaturization of electronics because SPs and SPPs can confine light to subwavelength dimensions as it was mentioned above [11, 12, 13]. As a result, the effective nonlinear optical response can be enhanced significantly [3]. SPP crystals and waveguides, nano-antennas and plasmonic metamaterials can be used for the creation of optical responses by using the resonances of the individual units and their electromagnetic coupling [3]. The response of materials to an optical field is determined by its polarization $\overrightarrow{P}\left(\overrightarrow{r},t\right)$which is given by [1, 2, 3, 4, 5].

$$\begin{array}{l}{P}_i\left(\overrightarrow{r},t\right)={\epsilon }_0\times \left[{\chi }_{\mathit{ij}}^{\left(1\right)}{E}_j\left(\overrightarrow{r},t\right)+{\chi }_{\mathit{ijk}}^{\left(2\right)}{E}_j\left(\overrightarrow{r},t\right)E_k\left(\overrightarrow{r},t\right)+{\chi }_{\mathit{\text{ijkl}}}^{\left(3\right)}{E}_j\left(\overrightarrow{r},t\right)E_k\left(\overrightarrow{r},t\right)E_l\left(\overrightarrow{r},t\right)+\dots \right]\end{array}$$E39

Here $i,j,k,l=1,2,3$, ${\chi }_{\mathit{ij}}^{\left(1\right)},{\chi }_{\mathit{ijk}}^{\left(2\right)},{\chi }_{\mathit{\text{ijkl}}}^{\left(3\right)}$ are the linear, second-order and third-order optical susceptibilities, respectively [1, 2, 3, 4, 5]. In the general case, the optical susceptibility ${\chi }^{\left(k\right)}$ is the $\left(k+1\right)$ th–rank tensor [1, 2, 3, 4, 5]. In the centrosymmetric media ${\chi }^{\left(2\right)}=0$, in homogeneous and isotropic media optical susceptibilities are scalar quantities [1, 2, 3, 4, 5].

The second and third-order nonlinear optical phenomena are the most important for applications [1, 2, 3, 4, 5]. The second-order response results in the wave-mixing effects such as the sum and difference frequency generation and second harmonic generation (SHG) where the incident frequency $\omega$ gives rise to the term with the frequency $2\omega$ [1, 2, 3, 4, 5]. The third-order response contains the terms with the incident frequency $\omega$, the new harmonics with the combination frequencies, and the third harmonic with the frequency $3\omega$ [1, 2, 3, 4, 5]. The term with the incident frequency $\omega$ is caused by the Kerr effect where the refractive index $n$ is modified by the optical field [1, 2, 3, 4, 5]. It is given by [1, 2, 3, 4, 5].

$$n=n_0+n_{2}I$$E40

where $n_0,n_2$ are linear and nonlinear refractive indices of the medium, respectively, and $I$ is the optical field intensity. The optical Kerr effect results in all-optical switching and light modulation [3]. The local field ${\overrightarrow{E}}_{\mathit{loc}}\left(\omega ,\overrightarrow{r}\right)$ enhancement at the metal/dielectric interface caused by the SPPs or LSPs excitation is characterized by the frequency-dependent local-field factor $L\left(\omega ,\overrightarrow{r}\right)$ given by [3].

$$L\left(\omega ,\overrightarrow{r}\right)=\left|E_{\mathit{loc}}\left(\omega ,\overrightarrow{r}\right)/E_0\left(\omega \right)\right|$$E41

where $\omega$ is the plasmonic excitation frequency and $E_0\left(\omega \right)$is the incident field.

The surface–enhanced SHG, third-harmonic generation and FWM in nanoplasmonic structures have been demonstrated experimentally [3]. It has been shown that the SHG signal at the electrochemically roughened silver surfaces was increased by four orders of magnitude for a flat reference surface [3]. However, the SHG in such a case is diffused and incoherent being enhanced by LSP resonances of the nanoscale surface features [3].

The structured plasmonic surfaces for enhanced nonlinear effects represent metal gratings, arrays of different types of nanoparticles and nano-apertures, split-ring resonators [3]. For instance, FWM from a gold grating was 2000 times stronger than FWM from a flat film [3].

Controlling light with light in plasmonic nanostructures is based on LSP and SPP nonlinear effects. In such a case plasmonic structures are used for all-optical modulation or switching due to the enhanced Kerr-type nonlinearities [3]. For instance, the plasmon-enhanced nonlinear materials are metal nanoparticles and bulk materials doped with such nanoparticles [3]. Propagating SPPs can serve as the signal carrier [3]. The modulation and switching of SPPs in plasmonic waveguides can be achieved by changing of the real or imaginary part of the permittivity ${\epsilon }_{1,2}$ by controlling light in the metal or dielectric at the interface shown in Figure 1 [3]. Optically controlled dispersion in plasmonic waveguides can be realized by the Kerr nonlinearity in dielectric materials or by using the nonlinearity in a metal [3]. It has been shown that under certain conditions the bistability and modulational instability are possible in nonlinear plasmonic waveguides [3]. Plasmonic soliton-like excitation is also possible if the gain and nonlinearity are balanced correspondingly [3].

Consider now the plasmonic metamaterials [3, 7, 8]. They may be used for all-optical switching based on the plasmonic resonances of the split-ring resonators or nanorods and the electromagnetic coupling between these elements [3]. All-optical modulation in metamaterials can be realized by controlling the coupling strength between molecular excitons and plasmonic excitations [3].

Tunable and nonlinear metamaterials can be created by inserting of LCs into a metamaterial structure [7]. The optical fields and the bias electric field are simultaneously applied to a metamaterial with LC influence on LC reorientation [7]. Their interplay demonstrates a specific mechanism of electrically controlled optical nonlinearity in metamaterials [7]. For example, all-optical control of metamaterials with E7 NLC at the telecom wavelength of 1550 nm was investigated experimentally [7]. The integration of highly nonlinear LCs with plasmonic and metamaterials enables active switching and tuning of optical signals with a very low threshold [8]. Three general methods are combining the responses of highly nonlinear NLCs and plasmonics: (i) dissolving of nanoparticles in a bulk NLC cell; (ii) incorporating LC into nanostructures; (iii) chemical synthesis of nanoparticles with LC molecules [8]. Consider the case of a small volume fraction $f\ll 1$ of nano-spheres suspended in LC [8]. In such a case the effective optical dielectric constant ${\epsilon }_{\mathit{com}}$ of the composite material is given by [8].

$${\epsilon }_{\mathit{com}}=\frac{1+2\mathit{f\gamma }}{1-\mathit{f\gamma }}{\epsilon }_{\mathit{\text{host}},}\gamma =\frac{{\epsilon }_{\mathit{Au}}-{\epsilon }_{\mathit{\text{host}}}}{{\epsilon }_{\mathit{Au}}+2{\epsilon }_{\mathit{\text{host}}}}$$E42

Here ${\epsilon }_{\mathit{Au}}$ is the permittivity of the Au nano-spheres described by the Drude model (22) and ${\epsilon }_{\mathit{\text{host}}}$ is the NLC permittivity. It has been demonstrated experimentally that a 100 $\mathit{\mu m}$ thick sample of NLC L34 containing 0.5% Au nanoparticles provides the enhancement of the nonlinear absorption coefficients by about 250% or even more due to the inclusion of these plasmonic nano-particles into NLC cells [8].

5. Conclusions

We considered the interaction of the optical waves with free electrons in metals. The theoretical analysis in such a case is based on Maxwell’s equations and the equation of motion of a free electron in an external electric field. As a result, plasmonic excitations emerge. The dielectric function of a free electron gas with plasmonic oscillations is described by the dielectric function (22) known as the Drude model. The longitudinal plasma oscillations excited at the plasma frequency $\omega ={\omega }_p$ are called the volume plasma oscillations, and the quasi-particles related to these oscillations are called the volume plasmons. SPPs may exist at the interface of the metal described by the Drude dielectric function and the dielectric medium. SPPs are the TM waves. They are characterized by the limited propagation length in the dielectric and the spatial confinement in the direction perpendicular to the propagation direction. The subwavelength confinement of the optical field is possible for SPPs. The nonlinear optical effects in plasmonic structures are significantly enhanced. The nonlinear optical phenomena such as modulation, switching, harmonic generation, FWM are possible in plasmonic nano-structures. Highly efficient all-optical switching and all-optical modulation can be realized in nonlinear plasmonic metamaterials based on nano-particles, split-ring resonators, and LCs.

Plasmonics plays an essential role in modern nonlinear optics. The nonlinear optical effects can be achieved at the reduced optical power due to enhanced effective nonlinearity in plasmonic nanostructures. The size of the nonlinear components can be scaled down which is important for the development of functional nanophotonic circuits. The ultrafast optical signal processing on the femtosecond scale can be realized due to the small response time of the plasmonic excitations.