**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 [11–13]. 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 *P* ! *r* !, *t* � �which is given by [1–5].

$$P\_i(\overrightarrow{r},t) = \varepsilon\_0 \times \left[ \chi\_{\overrightarrow{ij}}^{(1)} E\_j(\overrightarrow{r},t) + \chi\_{\overrightarrow{ijk}}^{(2)} E\_j(\overrightarrow{r},t) E\_k(\overrightarrow{r},t) + \chi\_{\overrightarrow{ijk}}^{(3)} E\_j(\overrightarrow{r},t) E\_k(\overrightarrow{r},t) E\_l(\overrightarrow{r},t) + \dots \right] \tag{39}$$

Here *i*, *j*, *k*, *l* ¼ 1, 2, 3, *χ* ð Þ1 *ij* , *χ* ð Þ2 *ijk* , *χ* ð Þ3 *ijkl* are the linear, second-order and third-order optical susceptibilities, respectively [1–5]. In the general case, the optical susceptibility *<sup>χ</sup>*ð Þ*<sup>k</sup>* is the ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> th–rank tensor [1–5]. In the centrosymmetric media *<sup>χ</sup>*ð Þ<sup>2</sup> <sup>¼</sup> 0, in homogeneous and isotropic media optical susceptibilities are scalar quantities [1–5].

The second and third-order nonlinear optical phenomena are the most important for applications [1–5]. 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 *ω* gives rise to the term with the frequency 2*ω* [1–5]. The third-order response contains the terms with the incident frequency *ω*, the new harmonics with the combination frequencies, and the third harmonic with the frequency 3*ω* [1–5]. The term with the incident frequency *ω* is caused by the Kerr effect where the refractive index *n* is modified by the optical field [1–5]. It is given by [1–5].

$$n = n\_0 + n\_2 I \tag{40}$$

where *n*0, *n*<sup>2</sup> 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 *E* ! *loc ω*, *r* ! enhancement at the metal/dielectric interface caused by the SPPs or LSPs excitation is characterized by the frequency-dependent local-field factor *L ω*, *r* ! given by [3].

$$L\left(o,\overrightarrow{r}\right) = \left|E\_{\rm loc}\left(o,\overrightarrow{r}\right)/E\_0(o)\right|\tag{41}$$

where *ω* is the plasmonic excitation frequency and *E*0ð Þ *ω* 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, splitring 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 *ε*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 alloptical switching based on the plasmonic resonances of the split-ring resonators or nanorods and the electromagnetic coupling between these elements [3]. All-optical *Introductory Chapter: Nonlinear Optical Phenomena in Plasmonics, Nanophotonics… DOI: http://dx.doi.org/10.5772/intechopen.102498*

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* ≪ 1 of nano-spheres suspended in LC [8]. In such a case the effective optical dielectric constant *εcom* of the composite material is given by [8].

$$
\varepsilon\_{com} = \frac{1 + 2f\gamma}{1 - f\gamma} \varepsilon\_{host, \gamma} = \frac{\varepsilon\_{Au} - \varepsilon\_{host}}{\varepsilon\_{Au} + 2\varepsilon\_{host}} \tag{42}
$$

Here *εAu* is the permittivity of the Au nano-spheres described by the Drude model (22) and *εhost* is the NLC permittivity. It has been demonstrated experimentally that a 100 *μm* 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].
