2.1 Description for the interaction between plasmon particles

The model is extended to describe the propagation of the electric field. For this, we propose that the electrostatic approximation is no longer fulfilled, acquiring the form of the Helmholtz equation having the form

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

$$
\nabla^2 \phi + k^2 \phi = \mathbf{0}.\tag{8}
$$

Looking for propagation along the x-coordinate, the equation acquires the form

$$\frac{\partial^2 \phi}{\partial \mathbf{x}^2} + \frac{\partial^2 \phi}{\partial \mathbf{y}^2} + k^2 \phi = \mathbf{0},\tag{9}$$

where k is the complex wave number k ¼ k<sup>1</sup> þ ik2. Proposing a solution of the form ϕ ¼ X xð ÞY yð Þ, we obtain the equation system given by

$$
\ddot{X} + (k^2 - h^2)X = \mathbf{0} \tag{10a}
$$

$$Y + a^2 Y = \mathbf{0},\tag{10b}$$

whose solution acquires the form

$$
\phi\_p = M e^{r\chi} e^{i\Omega \mathbf{x}} e^{-dy} e^{icy},\tag{11}
$$

this equation must recover the structure of the electrostatic approximation for a single nanoparticle.

From the previous solution, it is easy to identify its behavior. Along the y-coordinate the field is bounded by the exponential term, which remains unperturbed by the presence of a second particle; the interaction occurs mainly in the x-coordinate. This behavior may be generalized acquiring a wave effect. A balance relation between the complex wave number k and the constant coupling α can be predicted; this interaction decreases the evanescent term, and the propagating term becomes dominant. This interaction is sketched in Figure 1.

In Figure 1a, the electrostatic approximation is valid for a single nanoparticle; the wave behavior is generated by another set of particles interacting shown in Figure 1c.

Until this point we have described the generation of a wave propagating in the x-coordinate; this analysis can be extended to the propagation in the x � y plane, which is analyzed in the following section.

## Figure 1.

(a) Localized electric field for a plasmon particle. (b) Interaction between two plasmon particles. (c) Sketch to describe the generation of a plasmon field in an array of nanoparticles.
