## 3 Surface Plasmon Polaritons and the Extraordinary Optical Transmission  
In 1998, [PERSON] reported that nanostructures, composed by periodic circular holes, can amplify and concentrate electromagnetic radiation [5]. These structures, also known as nanoantennas, allow an unexpected transmission spectrum. This phenomenon was called extraordinary optical transmission and [PERSON] stated that its main agents are the surface plasmon polaritons [1; 5]. Classical theories cannot predict this phenomenon, since it is possible to amplify light using a structure with nano holes. Two conditions are fundamental to the excitation of polaritons, considering that the metal has a complex dielectric function given by \(\bar{e}_{1}=e_{1}^{\prime}+je_{1}^{\prime\prime}\) and the dielectric medium is characterized by a real function \(\epsilon_{2}\): (i) \(\epsilon_{1}^{\prime}<0\) and \(|\epsilon_{1}^{\prime}|>\epsilon_{2}\); (ii) a match between the wavevector of the polariton and the parallel component one of the incident light with the interface [1; 2; 6; 30; 31; 32; 33].  
Years later, [PERSON] did an analytical study from [PERSON]'s equations that proved the propagation of surface plasmon polaritons, as a fundamental stage for this phenomenon, as well as its combination with a creeping wave propagation [33].  
In the 1950s, the interest in plasmons received a great impulse with the works of [PERSON] and [PERSON]. Later, in 1956 [PERSON] introduces the concept of polariton and in 1957 [PERSON] presented the first theoretical description of surface plasmons. Surface plasmon polaritons TM propagation equations are presented on expression (8), for a surface at \(z=0\), where there is metal/plasma for \(z<0\) and dielectric for \(z>0\)[1; 6; 30; 31; 32]. Thus, the plane is extended along the \(y\)-axis and there are oscillations along \(x\) and an exponential decay over the \(z\)-direction.  
A plasmon is the redistribution of the plasma's free electrons, a quantum of a plasma oscillation [1; 6; 30; 31; 32; 33]. Light is also an oscillation of particles, in this case the photons. Both oscillations can couple and in this case the polariton is created. A polariton, a quasiparticle as the plasmon is, is the result of a strong coupling between an electromagnetic wave (light in this case), and an electric or magnetic dipole (the nanoantenna as verified by [PERSON]'s and [PERSON]'s theories) [1; 6; 28; 30; 31; 32]. This energy exchange is a resonant behavior, capable to amplify and concentrate electromagnetic energy near the surfaces.  
\[\left\{\begin{aligned} \bar{E}_{x,m}(x,y,z,t)&=E_{0}e^{j( \bar{k}_{x}x+\bar{k}_{z,m}|z|-\omega t)}\\ \bar{E}_{z,m}(x,y,z,t)&=\pm\frac{\bar{k}_{x}}{\bar{k }_{z,m}}E_{0}e^{j(\bar{k}_{x}x+\bar{k}_{z,m}|z|-\omega t)}\\ \bar{H}_{y,m}(x,y,z,t)&=H_{0}e^{j(\bar{k}_{x}x+\bar{ k}_{z,m}|z|-\omega t)}\end{aligned}\right. \tag{8}\]  
When extraordinary optical transmission was discovered by [PERSON], he sooner reported that this resonant behavior can be useful to develop a new type of miniaturized photonic devices, as optical sensors [1; 2; 5].  
However, new studies reported EOT at frequencies where the surface plasmon polaritons cannot be generated or in materials that do not fulfill the conditions to excite these polaritons [33; 34; 35; 36; 37]. Thus, their role, in other frequencies out of the visible and near-infrared spectral regions or for othermaterials than metals, is taken by other waves propagation, as creeping and leaky waves [33; 38; 39].  
The behavior of nanoantennas has already been modeled using electronic circuits, based on frequency-selective surfaces, waveguides and transmission lines theories [37]. These models give an interpretation of EOT in terms of impedance matching [37]. As any equivalent circuit, they have some advantages since they are simpler to study and understand.