# Issues with Modeling a Tunnel Communication Channel through a Plasma Sheath
## 2 Resonant Tunneling at Oblique Incidence  
Preliminary calculations [12] have shown that by changing the angle of incidence of electromagnetic waves on the plasma surface, one can reduce the transmission of radio signals through the tunnel communication channel by almost an order of magnitude. Here, we plan to discuss the possibility to obtain a more stable signal for different angles of incidence. Let us consider the oblique incidence of the electromagnetic wave on the vehicle surface. In this case, one can distinguish two types of electromagnetic waves: transverse electric (TE), when the electric field vector is parallel to the vehicle surface covered by the plasma sheath, and transverse magnetic (TM), when the same is valid for the magnetic field vector (see Figure 1). Definitely, for the normal wave incidence, the results for these two cases coincide with each other. We start with the case of the TE wave. Here, we assume that all the layers are lying in the \(xy\)-plane, the wave vector lies in the \(xz\)-plane, and \(\theta\) is the angle of incidence of the electromagnetic wave measured from the \(z\)-axis (Figure 1). In this case, the electric field has the only a tangential \(y\)-component, while for the magnetic field, both \(x\)- and \(z\)-components are non-zero. For the electric field component, one should solve the following equation [15]:  
\[\frac{\partial^{2}E_{\omega}}{\partial x^{2}}+\frac{\partial^{2}E_{\omega}}{ \partial z^{2}}+\varepsilon(z)\frac{\omega^{2}}{c^{2}}E_{\omega}=0 \tag{1}\]  
where \(\varepsilon(z)\) represents the permittivity profile of considered structure:  
\[\varepsilon(z)=\left\{ \begin{array}{c}\varepsilon_{d},\quad 0\leq z\leq a\\ \varepsilon_{p},\ a<z\leq a+d\\ \varepsilon_{air},\ z>a+d\end{array} \right. \tag{2}\]  
Here, \(a\) is the thickness of the dielectric layer (acting as a resonator) with permittivity \(\varepsilon_{d}>>1\), \(d\) and \(\varepsilon_{d}\) are the same parameters for the plasma sheath (typically \(d=5-10\) cm, plasma density \(n_{e}=10^{10}\div 10^{11}\) cm\({}^{-3}\)) and \(\varepsilon_{air}=1\) is the permittivity of the atmospheric air. Figure 1: Schematic illustration of the oblique incidence of the electromagnetic wave components (TE and TM waves) on the spacecraft surface covered by a plasma sheath. As concerns the TM polarization of waves, it turns out to be more convenient to consider the Helmholtz equation for the magnetic field component:  
\[\frac{\partial^{2}H_{\omega}}{\partial x^{2}}+\frac{\partial^{2}H_{\omega}}{ \partial z^{2}}+\varepsilon(z)\frac{\omega^{2}}{c^{2}}H_{\omega}=0 \tag{3}\]  
Obviously, (3) provides the solutions for the \(H\)-field component in the TM wave similar to those for \(E\)-field component in the TE wave. Then, the structure of TM wave electric field can be found from [PERSON]'s equations:  
\[\overrightarrow{E}_{\omega}=\frac{ic}{\omega\varepsilon(z)}rot\quad \overrightarrow{H}_{\omega}=\frac{ic}{\omega\varepsilon(z)}\bigg{[} \overrightarrow{e}_{x}\bigg{(}-\frac{\partial H_{\omega}}{\partial z}\bigg{)} +\overrightarrow{e}_{z}\bigg{(}\frac{\partial H_{\omega}}{\partial x}\bigg{)} \bigg{]} \tag{4}\]  
One should notice the specifics of the above expression consisting of the presence of \(\varepsilon(z)\) in the denominator, which in the case of the considered dielectric medium, leads to differences in representing the boundary conditions for TM and TE waves. We are going to show (in discussion) that such a feature leads to the difference in TE and TM wave tunneling for both polarizations. This effect is beyond the scope of the optical-mechanical analogy used in [12] to analyze the radio-communication blackout problem. In turn, in the case of a magnetic medium, such specificity will be observed for the TE wave.