## 2 The EHF Signal Propagation Model
In addition, it can be seen that the movement of electrons in an inhomogeneous magnetic field can still be regarded as a cyclotron movement around the guiding center, yet the guiding center is now moving along the magnetic field line at a speed of \(\frac{d\mathbf{r_{1}}}{dt}\). In the coordinate system with the guidance center as the origin, the centrifugal force on an electron is:  
\[\mathbf{\mathcal{F}_{BC}}=-(m\frac{d\mathbf{r_{1}}}{dt}\cdot\frac{d\mathbf{r_{1}}}{dt})[( \frac{\mathbf{B_{e}}}{B_{e}}\cdot\
abla)\frac{\mathbf{B_{e}}}{B_{e}}]_{0}. \tag{18}\]  
Hence:  
\[[(\frac{\mathbf{B_{e}}}{B_{e}}\cdot\
abla)\frac{\mathbf{B_{e}}}{B_{e}}]_{0}=[\frac{ \
abla B_{e}}{B_{e}}-\frac{\mathbf{B_{e}}}{B_{e}^{2}}(\frac{\mathbf{B_{e}}}{B_{e}} \cdot\
abla)B_{e}]_{0}=0. \tag{19}\]  
Equation (19) indicates that it is not necessary to consider the curvature drift of the electrons. Obviously, \((q\frac{d\mathbf{r_{1}}}{dt}\times\mathbf{B})\cdot\frac{d\mathbf{r_{1}}}{dt}\) originates from the incident wave. Thus:  
\[(q\frac{d\mathbf{r_{1}}}{dt}\times\mathbf{B_{e}})\cdot\frac{d\mathbf{r_{0}}}{dt}+(q\frac {d\mathbf{r_{0}}}{dt}\times\mathbf{B_{e}})\cdot\frac{d\mathbf{r_{1}}}{dt}=0. \tag{20}\]  
The first term on the left side indicates that \(\mathbf{B_{e}}\) reduces \(\mathbf{r_{0}}\), which results in the reduction of kinetic energy of the electron cyclotron motion, while the second term indicates that \(\mathbf{B_{e}}\) enlarges \(\mathbf{r_{1}}\), i.e., it increases the kinetic energy of drift. The two terms are of the same magnitude with opposite signs. Therefore, \(\mathbf{B_{e}}\) makes an ignorable contribution to the total kinetic energy. The kinetic energy of electrons along the z-axis originates from the incident wave. The electrons are accelerated by the incident wave in the XY plane. Meanwhile, the inhomogeneous magnetic field transfers the kinetic energy of electrons in the XY plane into the Z direction. \(\mathbf{\mathcal{F}}\) contains the contributions of the dipole magnetic field inhomogeneities and the electromagnetic wave magnetic field components, and is defined as the following equation:  
\[\mathbf{\mathcal{F}}=q\langle\frac{d\mathbf{r_{0}}}{dt}\times\mathbf{B}\rangle+q\langle \frac{d\mathbf{r_{0}}}{dt}\times\mathbf{B_{1}}\rangle. \tag{21}\]  
Then, (13) can be written in the form of:  
\[\frac{d^{2}\mathbf{r_{1}}}{dt^{2}}=\frac{\mathbf{\mathcal{F}}}{m}-v\frac{d\mathbf{r_{1}}}{ dt}. \tag{22}\]  
It can be solved such that \(\mathbf{r_{1}}=\frac{\mathbf{\mathcal{F}}}{mv}t\). Therefore:  
\[\mathbf{\mathcal{F}}\cdot\frac{d\mathbf{r_{1}}}{dt}=mv\frac{d\mathbf{r_{1}}}{dt}\cdot \frac{d\mathbf{r_{1}}}{dt}+\frac{d}{dt}(\frac{1}{2}m\frac{d\mathbf{r_{1}}}{dt}\cdot \frac{d\mathbf{r_{1}}}{dt})=\frac{\mathcal{F}^{2}}{mv}. \tag{23}\]  
This part of the electron movement yields dissipation of transmitting signals only, i.e., it will not transmit a receivable signal. The total signal attenuation [20] is defined as:  
\[Att=\frac{|E_{0}|^{2}}{|E_{t}|^{2}}=e^{\int_{0}^{lmax}2\mathrm{d}ldl}, \tag{24}\]where \(L_{max}\) is the total length of wave propagation path within the plasma sheath. \(|E_{t}|\) and \(|E_{0}|\) are the amplitudes of the electric fields of the transmitted wave and incident wave, respectively. If the whole propagation path is divided into \(N_{L}\) layers, the thickness of each layer is \(L=L_{max}/N_{L}\). The propagation time of the EM wave in the layer of thickness \(L\) is about \(\tau=L\beta/\omega_{0}\). If \(N_{L}\) is large enough, \(\left.\frac{\partial B_{t}}{\partial t}\right|_{0}\), \(B_{0}\), \(N\), \(E\), and \(v\) can be treated as constants in each single layer. Hence:  
\[\mathbf{B}=\frac{k_{-}}{\omega_{0}}(\mathbf{e_{z}}\times\mathbf{E}). \tag{16}\]0  
\[\mathbf{B}=\frac{k_{-}}{\omega_{0}}(\mathbf{e_{z}}\times\mathbf{E}). \tag{16}\]1  
where \(\alpha|_{(n-1)L}\) is the the attenuation coefficient at \(l=(n-1)L\), and \(W\) is the energy dissipated by collision damping.