## 2 The EHF Signal Propagation Model  
As shown in Figure 1, the plasma sheath is magnetized. The external magnetic field is inhomogeneous and generated by a circular coil. Therefore, the magnetic field involved in the present study is a dipole field. The incident wave is meant to have the form of:  
\[\mathbf{E}=(\hat{E_{x}}\mathbf{e_{x}}+\hat{E_{y}}\mathbf{e_{y}})e^{i(\omega_{0}t-k_{0}z)}, \tag{1}\]  
where \(\omega_{0}\) is the angular frequency for the wave, and \(\mathbf{k_{0}}=k_{0}\mathbf{e_{z}}\) is the wave vector propagating in free space. High directive antennas used to be employed by EHF communication systems, i.e., the beam width would be very small. In the present study, the beam direction is supposed to be perpendicular to the coil in order to make the magnetic field direction in the center of the coil parallel to the wave vector. The wave-particle interactions occur in the coverage of antenna beam only. In such a case, the guiding center approximation is adopted, as it is in the present study. The displacement vector for an electron could be expressed as \(\mathbf{r}=\mathbf{r_{0}}+\mathbf{r_{1}}\), \(\mathbf{r_{0}}\) is the cyclotron movement around the guiding center, and \(\mathbf{r_{1}}\) is the relatively slow movement of the guiding center. The magnetic field is a weakly inhomogeneous, which satisfies the following relation:  
\[|(\mathbf{r_{0}}\cdot\
abla)\mathbf{B_{e}}|_{0}\ll|\mathbf{B_{0}}|, \tag{2}\]  
where \(\mathbf{B_{0}}\) is the magnetic field strength at the guiding center. The magnetic field generated by the coil in a cylindrical coordinate system could be expressed as:  
\[\mathbf{B_{e}}=B_{z}(r,z)\mathbf{e_{z}}+B_{r}(r,z)\mathbf{e_{r}}\approx\mathbf{B_{0}}+[(\mathbf{r_ {0}}\cdot\
abla)\mathbf{B_{e}}]_{0}=\mathbf{B_{0}}+\mathbf{B_{1}}, \tag{3}\]  
where:  
\[\left\{ \begin{array}{l}B_{r}=\frac{\mu_{0}laz}{4\pi}\int_{0}^{2\pi}\frac{ \cos\phi}{(r^{2}+a^{2}+z^{2}-2\arccos\phi)^{3/2}}d\phi\\ B_{z}=\frac{\mu_{0}l}{4\pi}\int_{0}^{2\pi}\frac{a^{2}-\arccos\phi}{(r^{2}+a^{2} +z^{2}-2\arccos\phi)^{3/2}}d\phi\\.\end{array} \right. \tag{4}\]  
\(\mu_{0}\) is the magnetic permeability in the vacuum, \(a\) is the radius of the coil, and \(I\) is the current intensity. The electron motion equation is given below:  
\[\frac{d^{2}\mathbf{r}}{dt^{2}}=\frac{q}{m}[\mathbf{E}+\frac{d\mathbf{r}}{dt}\times(\mathbf{B_{0 }}+\mathbf{B_{1}})+\frac{d\mathbf{r}}{dt}\times\mathbf{B}]-v\frac{d\mathbf{r}}{dt}, \tag{5}\]  
where \(q\) is the value of electron charge, \(m\) is the mass of electron, \(\mathbf{B}\) is the magnetic field component of the incident wave, and \(v\) is the collision frequency between electrons and neutral particles. The electron motion equation in a homogeneous magnetic field is:  
\[\frac{d^{2}\mathbf{r_{0}}}{dt^{2}}=\frac{q}{m}\mathbf{E}+\frac{q}{m}\frac{d\mathbf{r_{0}}} {dt}\times\mathbf{B_{0}}-v\frac{d\mathbf{r_{0}}}{dt}. \tag{6}\]  
From (6), the dielectric constant tensor in the sheath can be solved:  
\[\overset{\leftrightarrow}{\varepsilon_{r}}=\left( \begin{array}{ccc} \varepsilon_{1}&i\varepsilon_{2}&0\\ -i\varepsilon_{2}&\varepsilon_{1}&0\\ 0&0&\varepsilon_{3}\\,\end{array} \right) \tag{7}\]  
\[\varepsilon_{1}=1+\frac{\omega_{pe}^{2}(iv-\omega_{0})}{\omega_{0}[(-i\omega _{c})^{2}+(iv-\omega_{0})^{2}]}, \tag{8}\]  
Figure 1: The model for the wave propagation in an inhomogeneous plasma sheath. The external magnetic field is generated by a coil. \[\varepsilon_{2}=-\frac{\omega_{pe}^{2}\omega_{e}}{\omega_{0}\big{[}(-i\omega_{c})^{ 2}+(iv\!-\!\omega_{0})^{2}\big{]}}, \tag{9}\]  
where \(\omega_{pe}\) is the plasma frequency, which has the form of \(\omega_{pe}=\sqrt{\frac{N_{e}d^{2}}{\varepsilon_{0}m}}\), \(\varepsilon_{0}\) is the dielectric constant in vacuum, \(N_{e}\) is the electron density, \(\omega_{c}=-\frac{qB_{0}}{m}\) is the electron cyclotron frequency, and \(i=\sqrt{-1}\).