4. Propagation in a tandem array of thin metal films

The natural extension of the analysis presented is the transfer of the plasmonic mode to a tandem array of thin metal surface, shown in Figure 6. This is possible using the evanescent behavior along the x-axis of the curved surface plasmon field. This behavior has been implemented to generate an optical field redistribution propagating along an optical waveguide array [11]. In this model, the evanescent character is used to tunnel the optical field.

The transmission of the plasmonic mode satisfies the following system of differential equations:

$$i\frac{dE\_n}{dz} + \beta E\_n + C\_{n+1}E\_{n+1} + C\_{n-1}E\_{n-1} = 0\tag{21a}$$

$$n = \mathbf{1}, \mathbf{2}, \mathbf{3}...,\tag{21b}$$

where β is the dispersion relation function and Ci represents the coupling constant, which depends on the relative separation between neighborhood surfaces [12]. The solution of the previous equation is similar to that presented in [11]; however, to associate a physical meaning to the coupling constant Ci, we present the analysis of two thin metal films.

The simplest case occurs when the system is formed by two thin metal films separated by a dielectric medium whose thickness must be less than 50 nm. The evanescent decay depends on the modulus of the permittivity quotient [13], and at this thickness is possible to generate tunneling effects [11]. Subsequently, the system of Eq. (21a) acquires the simple form

Figure 6.

Tandem array to propagate the plasmon field: the width of the metal is 20 - 40 nm and that dielectric film is 20 - 40 nm.

Synthesis of Curved Surface Plasmon Fields through Thin Metal Films in a Tandem Array DOI: http://dx.doi.org/10.5772/intechopen.81931

$$i\frac{dE\_1}{dz} + \beta E\_1 + C\_2 E\_2 = 0,\tag{22a}$$

$$i\frac{dE\_2}{dz} + \beta E\_2 + C\_1 E\_1 = 0.\tag{22b}$$

Rewriting it in matrix form, we obtain

$$i\left(\frac{dE\_1}{dz}\right) = -\begin{pmatrix} \beta & c\_2\\ c\_1 & \beta \end{pmatrix}.\tag{23}$$

It can be deduced that, as a consequence of the energy conservation, the matrix structure must be symmetric. This indicates that c<sup>1</sup> ¼ c<sup>2</sup> ¼ c, and the general solution is

$$
\begin{pmatrix} E\_1 \\ E\_2 \end{pmatrix} = d\_1 \begin{pmatrix} \xi\_1 \\ \xi\_2 \end{pmatrix} \exp\left(\lambda\_1 z\right) + d\_2 \begin{pmatrix} \eta\_1 \\ \eta\_2 \end{pmatrix} \exp\left(\lambda\_2 z\right), \tag{24}
$$

where di represents arbitrary constants and ξ1, <sup>2</sup> and η1, <sup>2</sup> represent the eigenvectors with eigenvalues λ1,<sup>2</sup> satisfying the characteristic equation depending on the coupling constant:

$$
\lambda\_{1,2} = \beta \pm c.\tag{25}
$$

Moreover, it is known that the eigenvectors must be complex [14]. Subsequently, without loss of generality, the solution can be rewritten as

$$
\begin{pmatrix} E\_1 \\ E\_2 \end{pmatrix} = d\_1 \begin{pmatrix} 1 \\ i \end{pmatrix} \exp\left(\lambda\_1 \mathbf{z}\right), \tag{26}
$$

which indicates that the shift generated between each plasmon mode presents similar features as the coupling mode theory [12]. This analysis leads to the expression for the plasmonic mode as

$$E\_1 = A \stackrel{\cdot}{\cdot} \exp\left(-|a\mathbf{x}|\right) \exp\left(i\beta\mathbf{s}\right) \tag{27a}$$

$$E\_2 = iA \stackrel{\rightarrow}{\xi} \exp\left(-|a\mathbf{x}|\right) \exp\left(i\beta\mathbf{s}\right),\tag{27b}$$

! where ξ is a unit vector tangent to the correlation curve and s is the arc length on the same curve; we remark that the correlation trajectory is given by Eq. (20).

Eq. (24) describes the evanescent coupling through a tandem array of thin metal films. Notably, the boundary conditions of the electric field indicate that the geometry of the plasmon field generated in the first thin metal film must be preserved in all the surfaces. This shows that the transmission of the curved plasmonic mode allows inducing magnetic properties in the system [15–18].
