# Issues with Modeling a Tunnel Communication Channel through a Plasma Sheath
## 3 Accounting for a Protective Layer and Antenna Elements  
In this chapter, we will make our study more specific. First, as the dielectric resonator is of a rather small size, we will study the bolometric signal detection. In this case, the thin conducting layer inside the resonator serves as a part of the receiving antenna. Second, we will take into account the additional heat-resistant dielectric layer that protects the resonator from plasma heating. Both additional layers can contribute to the eigen-mode frequencies and field distribution in the structure. The characteristics of the resonator considered above give rise to the possibility to study the case of the normal incidence of the electromagnetic wave only. Our structure under investigation is shown in Figure 3.  
Figure 2: Filling factor for the first resonance at normal and angular incidence for TE (**a**), TM (**b**), and mixed wave polarization (**c**). Dielectric layer parameters: \(\varepsilon_{d}=700\), \(a=0.5\) cm; plasma sheath parameters: \(n_{e}=10^{11}\) cm\({}^{-3}\), \(\
u_{tr}=10^{8}\) s\({}^{-1}\), \(d=10\) cm. The following parameters were introduced in the model: \(a\) and \(\epsilon_{d}\) are the total thickness and permittivity of the dielectric layer, \(b\) and \(z_{res}\) are the thickness and position of the conducting layer, \(N_{e}\) and \(\
u\) are the electronic concentration and transport frequency in the conducting layer, and \(c\) and \(\epsilon_{c}\) are the thickness and permittivity of the protecting layer. Further, we will neglect the possible electromagnetic loss in all the dielectric layers. For considered permittivity and the desired frequency range, one can obtain the possible range of the dielectric layer thickness. The thickness of the protective layer was chosen for reasons of compactness and efficiency, and the materials were selected from among those already in operation for these purposes. As the first step, we investigated field modes inside the dielectric layer without a metal layer. For the given parameters of the structure, there were three eigen-frequencies in the resonator below the plasma frequency. It can be seen that due to a wide protecting layer, the distribution of the field in the modes behaves as if in a shallow well of width \(a\). Further, we mainly analyze the tunneling channel at the frequency of the third mode of the resonator (\(f=2.47\) GHz). In this case, the electromagnetic field packet passing through the plasma undergoes the fastest decay [14]. The fraction of the flux absorbed by the thin conductive layer can be calculated as follows [18]:  
\[\eta_{\omega}=\frac{1}{cI_{\omega}^{2}}\frac{\omega_{p}^{2}\
u}{\omega^{2}+v^ {2}}\int E_{\omega}^{2}(z)dz \tag{8}\]  
where \(I_{\omega}\) is the intensity of the incident wave of frequency \(\omega\), \(E_{\omega}\) is the spatial field distribution, and the integral is taken over the area inside the conducting layer, \(z\in(z_{res},b+z_{res})\). Numerical calculations have shown that the metal layer of the parametric antenna in a relatively thin dielectric resonator changes the structure of the field, and for the optimal absorption, we chose \(z_{res}=0.115\). The final absorption weakly depends on the value of \(z_{res}\). It is much more interesting to consider the dependence of the absorption on the width of the metal layer, \(b\). An example for \(z_{res}=0.115\) cm is shown in Figure 4. There are two local maxima (see Figure 5): the first one corresponds to the value \(b=0.4\) nm with \(\eta_{\omega}\approx 0.44\) and the second to the value \(b=318\) nm with \(\eta_{\omega}\approx 0.64\). The reason for this non-monotonic dependence is that the metal layer divides the resonator into two parts with its own standing waves. By gradually changing the value of \(b\), we change the boundary conditions for each of the modes, obtaining the maximum absorption at the \"new\" resonance.