# Issues with Modeling a Tunnel Communication Channel through a Plasma Sheath
## 2 Resonant Tunneling at Oblique Incidence
For a quantitative description of the electromagnetic signal tunneling through the plasma sheath, the filling factor \(F(\omega)\) was introduced in [12]:  
\[F(\omega)=max\bigg{\{}\big{|}E_{\omega}^{(d)}\big{|}^{2}/\big{|}E_{\omega}^{(0 )}\big{|}^{2}\bigg{\}} \tag{5}\]  
In (5), \(E_{\omega}^{(0)}\) is the given amplitude of the incident wave field, and \(E_{\omega}^{(d)}\) is the amplitude of the wave field stored in the dielectric resonator. As mentioned above, this dependence is characterized by the sharp maxima at the resonant frequencies. A similar definition for the filling factor is used for the TE and TM waves. Generally, for wave incidents at some angle to the vehicle surface, besides the energy flux perpendicular to the surface, there will be a flux propagating alongside it. The latter reduces the effectiveness of the resonant transmission and shifts the resonance position to higher frequencies according to the expression:  
\[\omega_{n}\approx\frac{\pi c}{a\sqrt{\varepsilon_{d}}cos\phi}n \tag{6}\]  
where \(\varphi\) is the angle of the slope of the electromagnetic wave in the dielectric layer: \(\sin\varphi=\sin\theta\sqrt{\varepsilon_{air}/\varepsilon_{d}}\). It can be seen that embedded dielectric with a high value of permittivity provides resonant frequency stabilization: the greater the ratio \(\varepsilon_{d}/\varepsilon_{air}\), the closer the angle \(\varphi\) to the zero value. The wave Equations (1) and (3) were solved numerically as described in [12]. The plasma permittivity was chosen in the form  
\[\varepsilon_{p}=1-\frac{\omega_{p}^{2}}{\omega^{2}+v_{tr}^{2}}+i\frac{\omega_ {p}^{2}v_{tr}}{\big{(}\omega^{2}+v_{tr}^{2}\big{)}\omega} \tag{7}\]  
where \(\omega_{p}^{2}=4\pi e^{2}n_{e}/m\) is the plasma frequency squared and \(v_{tr}\) is the transport collisional frequency. Below, we provide the results of the calculations of the filling factor for the cases of TE, TM, and arbitrary (half the sum of TE and TM) wave polarizations. To demonstrate the specific features of TE and TM wave tunneling, we assume that the thickness \(d\) of a plasma layer is 10 cm with electron density \(n_{e}=10^{11}\) cm\({}^{-3}\) (the plasma frequency is \(\omega_{p}\approx 1.78\times 10^{10}\) s\({}^{-1}\)) and \(v_{tr}=10^{8}\) s\({}^{-1}\). The surface of the re-entry spacecraft is covered by the dielectric layer of thickness \(a=0.50\) cm and permittivity \(\varepsilon_{d}=700\), which correspond to novel materials, in particular to barium titanate [16; 17]. For tunnel radio communication channel, the telemetric signal frequency should coincide with one of the eigen-frequencies of the resonator (dielectric layer). In nonresonant cases, the wave field is reflected from the plasma layer and filling is negligible. We have already shown [12] that for the analysis of really important channel features we can consider that the plasma layer is characterized by a rectangular profile of the electron density and the spacecraft surface is ideally conductive. The obtained data for the filling factor in dependence on \(f=\omega/2\pi\) near one of the resonances are presented in Figure 2. First, it can be seen that for considered dielectric parameters, the resonance shifting is rather small (within the resonance bandwidth). Further, the decreasing of its maximum value is observed. The last circumstance appears due to the decrease of the normal component of the incoming electromagnetic wave flux. As a result, the greater the angle of incidence, the greater the resonance shifting (see Equation (6)). Nevertheless, the dielectric layer with a high value of permittivity plays the role of a <<stabilizer>>, preventing significant displacement of resonant peaks. For \(\varepsilon_{d}=700\), which is within the range of particular interest for the problem under discussion, the resonance shifting is negligible. For almost any angle of incidence, the wave in the resonator propagates nearly normally to the vehicle surface.