## 2 Cold Plasma Characteristics  
A plasma is an ionized gas, macroscopically neutral. To generate a plasma, the neutral gas has to receive enough energy so that its ionization degree reaches a level that changes the gas macroscopic properties from an insulator to a conducting medium. Many structures can be used to couple the energy to the gas, but in this study we will focus on plasmas generated by applying a static electric field to the gas. Due to our conditions of excitation, the type of plasma we are dealing with in this study is defined as a non-equilibrium or cold plasma. One characteristic of a cold plasma is that the heavy particles keep their temperature close to the room temperature while the electrons are warmed enough to produce collisions. One way to describe a plasma medium is by considering its ionization degree \(\delta\), which is given by [26]:  
\[\delta=\frac{n_{e}}{n_{e}+n_{o}} \tag{1}\]  
with \(n_{e}\) the electron density and \(n_{o}\) the neutral density. The plasma used in this study exhibits an ionization degree \(\delta\) lower than \(10^{-4}\). A homogeneous cold plasma can be viewed as a dielectric medium whose complex relative permittivity \(\varepsilon_{p}\) is described by the [PERSON]'s model [26]  
\[\epsilon_{p}=1-\frac{{\omega_{p}}^{2}}{\omega^{2}-j\
u_{p}\omega} \tag{2}\]  
where \(\omega\) represents the angular frequency of the electromagnetic wave interacting with the plasma, \(\
u_{p}\) the plasma electron-neutral momentum transfer frequency, hereafter referred to as collision frequency, and \(\omega_{p}\) the plasma angular frequency defined as:  
\[\omega_{p}=\sqrt{\frac{n_{e}e^{2}}{\epsilon_{0}m_{e}}} \tag{3}\]with \(n_{e}\) the electron density, \(e\) the electron charge, \(e_{0}\) the vacuum permittivity, and \(m_{e}\) the electron mass. From Equations (2) and (3), we can see that the complex permittivity \(\varepsilon_{p}\) of a homogeneous cold plasma depends on \(n_{e}\) and \(v_{p}\). Actually, \(n_{e}\) is related to the power injected to the discharge, the type of gas, and its pressure, while \(v_{p}\) depends mainly on the type of gas and its pressure. We can theoretically control the permittivity of a plasma, and so modify the way it interacts with an incident electromagnetic wave. In a realistic system, however, the control of the power injected to the plasma discharge is the only one that can be easily implemented in real time. In practice, we can also describe the [PERSON]'s model as a function of the ratios \(\omega_{p}/\omega\) and \(v_{p}/\omega\), as shown in Equation (4):  
\[\varepsilon_{p}=\varepsilon^{\prime}-je^{\prime\prime}=\bigg{(}1-\frac{( \omega_{p}/\omega)^{2}}{1+(v_{p}/\omega)^{2}}\bigg{)}-j\bigg{(}\frac{(v_{p}/ \omega)(\omega_{p}/\omega)^{2}}{1+(v_{p}/\omega)^{2}}\bigg{)}. \tag{4}\]  
Figure 1 shows the values for the real and imaginary parts of the plasma permittivity as a function of the ratios \(\omega_{p}/\omega\) and \(v_{p}/\omega\). From Figure 1, we can see that the real part (\(\varepsilon^{\prime}\)) becomes negative when \(\omega_{p}>\omega\), while the imaginary part of the permittivity (\(\varepsilon^{\prime\prime}\)) is maximum when \(v_{p}/\omega\approx 1\), which means that we can expect additional losses due to the plasma medium.  
If the [PERSON]'s model helps in understanding the interaction between an incident electromagnetic wave and a homogeneous cold plasma, a real plasma remains an inhomogeneous medium. In fact, it exhibits a complex electron density distribution which is often difficult to assess. Nevertheless, for confined discharges, it can give a good estimation on the behavior when integrated into a high-frequency circuit [34].