**3. Surface Plasmon Polaritons (SPPs)**

Consider now the SPPs at the interface between the metal ð Þ *z*<0 and the dielectric ð Þ *z*>0 shown in **Figure 1**. The metal and the dielectric are characterized by the dielectric function *ε*1ð Þ *ω* defined by expression (22) and the real dielectric constant *ε*<sup>2</sup> >0, respectively. We consider the frequency range *ω*<*ω<sup>p</sup>* such that Re *ε*1ð Þ *ω* <0 . Wave Eq. (12) in such a case takes the form [11, 13].

$$
\nabla^2 \overrightarrow{E} - \frac{\varepsilon}{c^2} \frac{\partial^2 \overrightarrow{E}}{\partial t^2} = \mathbf{0} \tag{25}
$$

**Figure 1.** *Interface between a metal (z* <0*) and a dielectric (z*>0*).*

where it is assumed that the permittivity *ε* in the both media is constant over distances of the order of magnitude of an optical wavelength and ∇ � *D* ! ¼ 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].

$$\overrightarrow{E}(\mathbf{x}, z, t) = \overrightarrow{E}(\mathbf{z}) \exp i(\beta \mathbf{x} - a\mathbf{t})\tag{26}$$

Here *β* 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 *Ex*, *Ez*, *Hy*; the second solution is the transverse electric (TE) or s modes with the field components *Hx*, *Hz*, *Ey* [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].

$$z < 0 \quad H\_{1\circ} = A\_1 \exp k\_1 z \exp i(\beta \infty - \alpha t) \tag{27}$$

$$E\_{1\mathbf{x}} = -iA\_1 \frac{k\_1}{a\epsilon\_0 \varepsilon\_1(a)} \exp k\_1 z \exp i(\beta \mathbf{x} - a\mathbf{t})\tag{28}$$

$$E\_{1x} = -A\_1 \frac{\beta}{\alpha \varepsilon\_0 \varepsilon\_1(\alpha)} \exp k\_1 z \exp i(\beta x - \alpha t) \tag{29}$$

and

$$z > 0 \quad H\_{2\circ} = A\_2 \exp\left(-k\_2 z\right) \exp\left(\beta \mathbf{x} - \alpha \mathbf{t}\right) \tag{30}$$

$$E\_{2\mathbf{x}} = iA\_2 \frac{k\_1}{\alpha \epsilon\_0 \epsilon\_2} \exp\left(-k\_2 z\right) \exp i(\beta \mathbf{x} - \alpha t) \tag{31}$$

$$E\_{2x} = -A\_2 \frac{\beta}{a\varkappa\_0 \varepsilon\_2} \exp\left(-k\_2 x\right) \exp\left(\beta x - a\mathfrak{t}\right) \tag{32}$$

where *k*1,2 are the components of the wave vector in the Z-direction perpendicular to the interface. The value of 1*=*j j *k*1,2 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 *Hy* and *Ex* at the interface *z* ¼ 0 have the form.

$$H\_{\mathfrak{I}\mathfrak{y}}(\mathbf{z}=\mathbf{0}^{-})=H\_{\mathfrak{I}\mathfrak{y}}(\mathbf{z}=\mathbf{0}^{+});\ E\_{\mathfrak{L}\mathfrak{x}}(\mathbf{z}=\mathbf{0}^{-})=E\_{\mathfrak{L}\mathfrak{x}}(\mathbf{z}=\mathbf{0}^{+})\tag{33}$$

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

$$A\_1 = A\_2; \ \frac{k\_2}{k\_1} = -\frac{\varepsilon\_2}{\varepsilon\_1(a)}\tag{34}$$

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].

$$k\_1^2 = \beta^2 - k\_0^2 \varepsilon\_1(\alpha) \tag{35}$$

$$k\_2^2 = \beta^2 - k\_0^2 \varepsilon\_2 \tag{36}$$

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

Here *k*<sup>0</sup> ¼ *ω=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].

$$\beta = k\_0 \sqrt{\frac{\varepsilon\_1(o)\varepsilon\_2}{\varepsilon\_1(o) + \varepsilon\_2}}\tag{37}$$

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 *β* is close to *k*<sup>0</sup> [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 *ωsp* given by [11].

$$
\omega\_{sp} = \frac{\alpha\_p}{\sqrt{1 + \varepsilon\_2}} \tag{38}
$$

The surface plasmon is an electrostatic mode which is a limiting case of SPP for *β* ! ∞ [11]. In the general case of the essentially complex dielectric function *ε*1ð Þ *ω* (22) the SPP energy attenuation length, or the propagation length *<sup>L</sup>* <sup>¼</sup> ð Þ 2Im½ � *<sup>β</sup>* �<sup>1</sup> 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 *λ*<sup>0</sup> ¼ 450*nm* are characterized by the propagation length *L*≈16 *μm* and the evanescent decay length 1*=*j j *k*<sup>2</sup> ≈ 180*nm*; the corresponding values for *λ*<sup>0</sup> ≈1*:*5 *μm* are: *L*≈1080 *μm*, 1*=*j j *k*<sup>2</sup> ≈2*:*6*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 *λ*0*=*2 in the dielectric medium can be realized for optical frequencies *ω*≈ *ωsp* [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 threedimensional beams [11].
