## 2 The EHF Signal Propagation Model
The wave equation in magnetized plasma could be written in the form of:  
\[\
abla(\
abla\cdot\mathbf{E})-\
abla^{2}\mathbf{E}=-\frac{1}{c^{2}}\frac{\partial}{ \partial t}(\frac{\partial}{\partial t}\overset{\leftrightarrow}{\varepsilon }_{r}\cdot\mathbf{E}). \tag{10}\]  
Considering that \(\mathbf{k}\cdot\mathbf{E}=0\), where \(\mathbf{k}\) is the wave vector in plasma, the wave equation in frequency domain can be written as:  
\[k^{2}\mathbf{E}=k_{0}^{2}\overset{\leftrightarrow}{\varepsilon}_{r}\cdot\mathbf{E}. \tag{11}\]  
The wave number for the right-hand circular (RHC) and left-hand circular (LHC) modes are defined as \(k_{\pm}=k_{0}\sqrt{\varepsilon_{1}\pm\varepsilon_{2}}\), where \(+\) and \(-\) correspond to the RHC and LHC modes, respectively. The signal attenuation for the LHC mode in magnetized plasma is obviously lower than the RHC mode [16]. Hence, the present study will mainly concern the LHC mode. For the LHC mode, \(E_{x}=-iE_{y}\). By substituting that relation into (6), the following equation is obtained:  
\[\mathbf{r_{0}}=\frac{q}{m}\frac{1}{\omega_{0}}\frac{1}{iv-\omega_{c}-\omega_{0}}E. \tag{12}\]  
According to (12), it can be estimated that \(|\mathbf{r_{0}}|\approx 10^{-11}\)\(E\), which meets the weak inhomogeneity condition (2). \(\mathbf{r_{1}}\) can be regarded as a constant relative to a cyclotron period. By substituting \(\mathbf{r_{0}}\) into (5) and averaging it in each single cyclotron period, \(\left\langle\frac{d\mathbf{r_{1}}}{dt}\times\mathbf{B}\right\rangle\approx\frac{d\bm {r_{1}}}{dt}\times\left\langle\mathbf{B}\right\rangle=0\). Then, the electron motion equation in an inhomogeneous magnetic field can be written as:  
\[\frac{d^{2}\mathbf{r_{1}}}{dt^{2}}=\frac{q}{m}[\langle\frac{d\mathbf{r_{0}}}{dt} \times(\mathbf{B_{1}}+\mathbf{B})\rangle+\langle\frac{d\mathbf{r_{1}}}{dt}\times\mathbf{B_{0 }}\rangle]-v\frac{d\mathbf{r_{1}}}{dt}, \tag{13}\]  
where \(q(\frac{d\mathbf{r_{0}}}{dt}\times\mathbf{B_{1}})\) reflects the gradient drift. \[ \begin{split} q(\frac{d\mathbf{r_{0}}}{dt}\times\mathbf{B_{1}})& =q[(\mathbf{\omega_{0}}\cdot\mathbf{B_{1}})\mathbf{r_{0}}-(\mathbf{r_{0}}\cdot\mathbf{B_ {1}})\mathbf{\omega_{0}}]\\ &=q\omega_{0}|\mathbf{r_{0}}|^{2}\frac{\partial B_{r}}{\partial r} \bigg{|}_{0}\mathbf{e_{z}},\end{split}  \tag{14}\]  
where \(\mathbf{\omega_{0}}\) is a pseudo-vector whose value is equal to the frequency of the EM wave corresponding to \(k_{-}\), which points to the negative direction of the z-axis. The ponderomotive force \(q(\frac{d\mathbf{r_{0}}}{dt}\times\mathbf{B})\) cannot be obtained directly in the frequency domain. Thus, it is solved in the time domain as follows. By substituting \(k_{-}=\beta-i\alpha\) into (11), the oscillating electric field after entering the plasma can be expressed as:  
\[\mathbf{E}=E_{0}e^{-\alpha z}[e^{i\theta_{z}}\mathbf{e_{x}}+e^{i\left(\theta_{x}+\frac {\alpha}{2}\right)}\mathbf{e_{y}}]e^{i\left(\omega_{0}t-\beta z\right)}, \tag{15}\]  
where \(\alpha\) is the attenuation coefficient and \(\beta\) is the phase coefficient. [PERSON]'s law in the frequency domain could be expressed as:  
\[\mathbf{B}=\frac{k_{-}}{\omega_{0}}(\mathbf{e_{z}}\times\mathbf{E}). \tag{16}\]According to (12), (15) and (16), the following relation for the ponderomotive force could be obtained:  
\[q\langle\frac{d\mathbf{r_{0}}}{dt}\times\mathbf{B}\rangle=\frac{q^{2}|\sqrt{\epsilon_{1 }-\epsilon_{2}}|E^{2}{\sin\theta}}{mc|iv-\omega_{c}-\omega_{0}|}\mathbf{e}_{z}, \tag{17}\]  
where \(\theta=\theta_{0}+\theta_{1}\), \(\sqrt{\epsilon_{1}-\epsilon_{2}}=|\sqrt{\epsilon_{1}-\epsilon_{2}}|e^{i\theta _{0}}\), and \(iv-\omega_{c}-\omega_{0}=|iv-\omega_{c}-\omega_{0}|e^{i\theta_{1}}\). According to (14) and (17), it can be seen that the gradient drift of electrons in the direction perpendicular to the wave vector is absent. In such a case, \(\mathbf{r_{1}}=r_{1}\mathbf{e}_{z}\), which indicates that \(\mathbf{r_{1}}\) is always parallel to the z-axis.