## 2 Evolution of the Theoretical Foundations  
Since the XVII century, based on experimental works, theories have been proposed to explain light behavior, namely when interacting with matter. [1; 3; 4]. In this century, [PERSON] stated that each unobstructed point from a plane wave front can be analyzed as a punctual source, emitting a spherical wave with the same characteristics from the main (real), wave [1; 3; 4]. However, only in the XVIII century, [PERSON] reformulated the [PERSON] principle in order to explain the rectilinear propagation of light and diffraction effects. This was a huge progress, because it aims on the study of the near-field regime. To assume a plane wave, the distance between the source and the target, where the analysis is performed, should be very large and, in this condition, geometrical optics methods can aim to analyze the wave propagation [1; 3; 4]. In the beginning of the XIX century, [PERSON] proved the wave characteristic of light, using a double slit experiments and the Huygens-Fresnel principle [1; 3; 4]. [PERSON] also used the [PERSON] principle to compose his diffraction theory, that relates the irradiance at a certain angle of diffraction \(\theta\) and the main lobe one (\(\theta=0^{\circ}\)), after the wave propagates though a slit in a metallic plane. Nonetheless, as presented in expression (1), for N slits, the relation is between the irradiance at a certain angle and an irradiance \(I_{0}\), that is the contribution of each equal slit for the overall irradiance [1; 3; 4; 6]. However, he assumed a plane and monochromatic wave, characterized by an incident wavelength \(\lambda\) and a wavevector \(\vec{\textbf{k}}\), with perpendicular incidence to a perfect conductive metal, with N slits of diameter \(a\) and spaced by a distance \(d\)[1; 3; 4; 6]. The analysis target is over the slits plane, at a distance \(D\) and the distance between the plane position given by \(\theta\) and the main lobe one is given by y [1; 3; 4; 6]. [PERSON] determine the maximum and the minimum intensity positions, y, on the target, regarding the constructive and destructive interference among different rays propagating from the slits plane [1; 3; 4; 6]. Expressions (2) and (3) are obtained using these distance definitions and under the far-field assumption (\(D\gg y\)) [1; 3; 4; 6]. \[I(\theta)=I_{0}\left(\frac{sin(\beta)}{\beta}\right)^{2}\left( \frac{sin(N\alpha)}{\alpha}\right)^{2} \tag{1}\] \[\beta=\frac{ka}{2}sin(\theta)=\frac{\pi a}{\lambda}sin(\theta)= \frac{\pi ay}{\lambda D}\] (2) \[\alpha=\frac{kd}{2}sin(\theta)=\frac{\pi d}{\lambda}sin(\theta)= \frac{\pi dy}{\lambda D} \tag{3}\]After that, [PERSON] composed his theory, that includes as limit cases the [PERSON] and [PERSON]'s theories [1; 6; 28]. He assumed the propagation of a plane and scalar wave along the \(x\) direction and an opaque plane perpendicular to it [1; 6; 28]. The opaque plane has an aperture and the transmitted wave propagation is determined by expression (4), that degenerates expression (5), using [PERSON]'s theorem [1; 28]. The scalar wave is \(\bar{u}\) and the integration coordinates should be \(y^{\prime}\) and \(z^{\prime}\), such as \(x^{\prime}\), \(y^{\prime}\) and \(z^{\prime}\) are the coordinates of a point \(\mathbf{r^{\prime}}\) before or over the metal surface. Moreover, \(\mathbf{\phi}\) is the [PERSON]'s function, for a monochromatic wave at a point \(\mathbf{r}\) after the plane, given by \(r^{2}=x^{2}+y^{2}+z^{2}\). \[\
abla^{2}\,\bar{u}+k_{0}^{2}\,\bar{u}=0 \tag{4}\]  
\[u(\mathbf{r})=\frac{1}{4\pi}\int\left[-\frac{\partial u}{\partial x^{\prime}}(\bm {r^{\prime}})\mathbf{\phi}(|\mathbf{r}-\mathbf{r^{\prime}}|)+u(\mathbf{r^{\prime}})\frac{ \partial\mathbf{\phi}}{\partial x^{\prime}}\right]d\sigma \tag{5}\]  
However, boundary conditions are not verified at the metal surface on [PERSON]'s theory, since he assumed an opaque and finite plane [1; 6; 28]. As the incident and reflected waves do not exist over the surface, these conditions are not satisfied, especially on the surfaces near the slits [1; 28].