2. Analysis of plasmon particle

A nanoparticle is generated by a set of atoms; the plasmon particle corresponds with the surface current distribution of the atoms. The analysis is implemented applying the electrostatic approximation given by

$$
\nabla^2 \phi = \mathbf{0},
\tag{1}
$$

where ϕ is the potential function. Using variable separation in Cartesian coordinates on the x � y plane, the equation acquires the form

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

Proposing the solution as

$$
\phi = X(\mathfrak{x})Y(\mathfrak{y}),
\tag{3}
$$

we obtain the equation system

$$
\ddot{X} - a^2 X = \mathbf{0} \tag{4a}
$$

$$
\ddot{Y} + a^2 Y = \mathbf{0},
\tag{4b}
$$

where the coupling constant α is a complex number having the form α ¼ a þ ib. This condition is necessary because perturbing the field, it must acquire a propagating behavior as it is shown below. Solving for X, we have

$$X = c\_1 e^{c\chi} e^{id\chi} + c\_2 e^{c\chi} e^{-id\chi},\tag{5}$$

and the solution for Y is given by

$$Y = D\_1 e^{i\gamma} e^{-d\gamma}.\tag{6}$$

Then, the complete solution ϕ acquires the form

$$
\phi = Ae^{c\chi}e^{id\chi}e^{-dy}e^{iy},
\tag{7}
$$

with c < 0 and d > 0. Eq. (7) representsthe boundary condition forthe plasmonic field.
