**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].

$$
\nabla \cdot \overrightarrow{D} = \rho\_{\text{ext}} \tag{1}
$$

$$\nabla \cdot \overline{B} = \mathbf{0} \tag{2}$$

$$
\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \tag{3}
$$

$$
\nabla \times \overrightarrow{H} = \overrightarrow{J}\_{\text{ext}} + \frac{\partial \overrightarrow{D}}{\partial t} \tag{4}
$$

Here *D* ! , *E* ! , *H* ! and *B* ! are the dielectric displacement, the electric field, the magnetic field and the magnetic induction (or magnetic flux density), respectively; *ρext* and *J* ! *ext* are external charge and current densities, respectively. The four macroscopic fields are related by the following material Equations [11].

$$
\overrightarrow{D} = \varepsilon\_0 \epsilon \overrightarrow{E} \tag{5}
$$

$$
\overrightarrow{B} = \mu\_0 \mu \overrightarrow{H} \tag{6}
$$

where *ε*<sup>0</sup> and *μ*<sup>0</sup> are the electric permittivity and magnetic permeability of vacuum, respectively, *ε* is the relative permittivity and *μ* is the relative permeability of the medium, *μ* ¼ 1 for the nonmagnetic medium. The total charge and current densities *ρtot* and *J* ! *tot* consist of the external charge and current densities *ρext*, *J* ! *ext* and the internal ones *ρ*, *J* ! . They are given by [11]: *ρtot* ¼ *ρext* þ *ρ*, *J* ! *tot* ¼ *J* ! *ext* þ *J* ! . The internal current density has the form [11].

$$
\overrightarrow{J} = \sigma \overrightarrow{E} \tag{7}
$$

*Introductory Chapter: Nonlinear Optical Phenomena in Plasmonics, Nanophotonics… DOI: http://dx.doi.org/10.5772/intechopen.102498*

where *σ* 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].

$$
\overrightarrow{D}\left(\overrightarrow{r},t\right) = \varepsilon\_0 \int dt' d\overrightarrow{r}' \varepsilon\left(\overrightarrow{r} - \overrightarrow{r}', t - t'\right) \overrightarrow{E}\left(\overrightarrow{r}', t'\right) \tag{8}
$$

$$\overrightarrow{J}\left(\overrightarrow{r},t\right) = \varepsilon\_0 \int d t' d\overrightarrow{r}' \sigma \left(\overrightarrow{r} - \overrightarrow{r}', t - t'\right) \overrightarrow{E}\left(\overrightarrow{r}', t'\right) \tag{9}$$

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

$$
\overrightarrow{D}\left(\overrightarrow{k},\alpha\right) = \varepsilon\_0 \varepsilon\left(\overrightarrow{k},\alpha\right) \overrightarrow{E}\left(\overrightarrow{k},\alpha\right) \tag{10}
$$

$$
\overrightarrow{J}\left(\overrightarrow{k},\alpha\right) = \sigma\left(\overrightarrow{k},\alpha\right)\overrightarrow{E}\left(\overrightarrow{k},\alpha\right) \tag{11}
$$

where *k* ! and *ω* 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 *E* ! *r* !, *t* � � [1, 2, 11].

$$
\nabla \times \nabla \times \vec{E} = -\mu\_0 \frac{\partial^2 \vec{D}}{\partial t^2} \tag{12}
$$

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

$$
\overrightarrow{k}\left(\overrightarrow{k}\cdot\overrightarrow{E}\right) - k^2\overrightarrow{E} = -\varepsilon\left(\overrightarrow{k},\alpha\right)\frac{\alpha^2}{c^2}\overrightarrow{E}\tag{13}
$$

where *c* ¼ ffiffiffiffiffiffiffiffiffi *ε*0*μ*<sup>0</sup> � � p �<sup>1</sup> 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 *k* ! � *E* � �! ¼ 0 and the corresponding dispersion relation

$$k^2 = \varepsilon \left(\overrightarrow{k}, a\right) \frac{a^2}{c^2} \tag{14}$$

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

$$
\varepsilon \left( \overrightarrow{k}, \alpha \right) = \mathbf{0} \tag{15}
$$

It is seen from Eq. (15) that the longitudinal collective oscillations can exist only for the frequencies *ω* which correspond to the zeros of the dielectric function *ε k* ! , *ω* � � [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 *E* ! has the form [11].

$$m\frac{d^2\overrightarrow{\mathcal{X}}}{dt^2} + m\gamma\frac{d\overrightarrow{\mathcal{X}}}{dt} = -e\overrightarrow{E} \tag{16}$$

where *x* ! is the free electron displacement, *<sup>γ</sup>* <sup>¼</sup> *<sup>τ</sup>*�<sup>1</sup> is a characteristic collision frequency, *<sup>τ</sup>* is the free electron gas relaxation time. Typically, *<sup>τ</sup>* � <sup>10</sup>�<sup>14</sup> *<sup>s</sup>* at room temperature which yields *γ* ≈ 100*THz* [11]. For the external field *E* ! ðÞ¼ *t E* ! <sup>0</sup> exp ð Þ �*iωt* we obtain from Eq. (16) the following particular solution.

$$
\overrightarrow{\mathbf{x}}(t) = \overrightarrow{\mathbf{x}}\_0 \exp\left(-i\alpha t\right) = \frac{e}{m(\alpha^2 + i\gamma o)} \overrightarrow{E}(t) \tag{17}
$$

The macroscopic polarization *P* ! of the medium caused by the displaced electrons is given by [11].

$$
\overrightarrow{P} = -ne\overrightarrow{\mathbf{x}}\tag{18}
$$

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

$$\overrightarrow{P} = -\frac{ne^2}{m(\alpha^2 + i\gamma a)}\overrightarrow{E} \tag{19}$$

Taking into account that the dielectric displacement *D* ! in a medium is given by [1, 2, 11].

$$
\overrightarrow{D} = \varepsilon\_0 \overrightarrow{E} + \overrightarrow{P} \tag{20}
$$

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

$$
\overrightarrow{D} = \varepsilon\_0 \left( 1 - \frac{\alpha\_p^2}{\alpha^2 + i\gamma o} \right) \overrightarrow{E} \tag{21}
$$

where *<sup>ω</sup><sup>p</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ne*<sup>2</sup>*=*ð Þ *<sup>ε</sup>*0*<sup>m</sup>* <sup>p</sup> 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 *ε ω*ð Þ [4, 11].

$$\varepsilon(\alpha) = 1 - \frac{\alpha\_p^2}{\alpha^2 + i\gamma a} \tag{22}$$

The dielectric function *ε ω*ð Þ (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.

*Introductory Chapter: Nonlinear Optical Phenomena in Plasmonics, Nanophotonics… DOI: http://dx.doi.org/10.5772/intechopen.102498*

$$\operatorname{Re}\varepsilon(\boldsymbol{\omega}) = \mathbf{1} - \frac{\alpha\_p^2 \tau^2}{\mathbf{1} + \alpha^2 \tau^2}; \text{ Ime}(\boldsymbol{\omega}) = \frac{\alpha\_p^2 \tau}{\nu(\mathbf{1} + \alpha^2 \tau^2)}\tag{23}$$

Typically, the frequencies *ω*<*ω<sup>p</sup>* are considered where the metals have the pronounced metallic properties since in such a case Re *ε ω*ð Þ< 0 [11]. In the high frequencies limiting case *ωτ* ≫ 1 the imaginary part of the dielectric function (23) can be neglected while expression (22) takes the form [11].

$$\varepsilon(\alpha) \approx 1 - \frac{\alpha\_p^2}{\alpha^2} \tag{24}$$

Comparison of expressions (15) and (24) shows that the longitudinal waves can be excited at the plasma frequency *ω* ¼ *ωp*. 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].
