## 2 Relevant Propagation Modeling Theory  
### The Theory of the Three-Dimensional Vector PE  
The three-dimensional PE is derived from the electromagnetic wave equation. The scalar three-dimensional PE can only express the propagation or scattering characteristics of a component of the electromagnetic wave in three-dimensional space and can be solved according to the characteristics of a component of the electromagnetic field. Therefore, the scalar three-dimensional PE cannot resolve phenomena such as the depolarization of electromagnetic waves as caused by impedance boundaries. The three-dimensional vector PE differs from the three-dimensional scalar PE. The three-dimensional vector PE is an equation set consisting of three-dimensional scalar PEsthat reflect the electromagnetic field components. Therefore, the three-dimensional vector PE can resolve phenomena such as the depolarization of electromagnetic waves as caused by the impedance boundaries. The three-dimensional vector PE is derived from the vector wave equation. Considering only forward propagation, the expressions for the electric and magnetic vectors are shown below. \[\overset{\rightarrow}{E}=(E_{x},E_{y},E_{z})\qquad\overset{\rightarrow}{H}=(H_{ x},H_{y},H_{z}) \tag{1}\]  
In the uniform passive region, \(E\) and \(H\) satisfy [PERSON]'s equations. \[\left\{ \begin{aligned} \
abla\cdot D&=0\\ \
abla\cdot B&=0\end{aligned} \right.\qquad\left\{ \begin{aligned} \
abla\times H&=-i\omega D\\ \
abla\times E&=i\omega B\end{aligned} \right. \tag{2}\]  
where the free-space wavenumber is \(k_{0}=\omega\sqrt{\eta_{0}\varepsilon_{0}}\) and \(\mu_{0}\) and \(\omega\) are the magnetic permeability in a vacuum and the angular frequency of the electromagnetic wave, respectively. Wave impedance in a vacuum is \(\eta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}\approx 120\pi\). The refractive index of the medium is \(n=\sqrt{\varepsilon_{r}}\). The dielectric constant of the medium is \(\varepsilon=\varepsilon_{r}\varepsilon_{0}\). The magnetic permeability of the medium is \(\mu\approx\mu_{0}\), \(D=\varepsilon E\), and \(B=\mu H\). When \(\overset{\rightarrow}{F}=0\) and \(\overset{\rightarrow}{A}\
eq 0\), an arbitrary transverse magnetic (TM) field is represented by \(\Psi^{e}\). Let \(\overset{\rightarrow}{A}=\sharp\Psi^{e}\) denote the TM field for \(z\). Then, the relationship between \(\Psi^{e}\) and each component of the electromagnetic field is  
\[\left\{ \begin{aligned} & E_{z}^{e}=\frac{i\eta_{0}}{n^{2}k_{0}}\left( \frac{\partial^{2}}{\partial x^{2}}+k_{0}^{2}\right)\Psi^{e}\\ & E_{y}^{e}=\frac{i\eta_{0}}{n^{2}k_{0}}\frac{\partial^{2}\Psi^{e }}{\partial y\partial z}\\ & E_{x}^{e}=\frac{i\eta_{0}}{n^{2}k_{0}}\frac{\partial^{2}\Psi^{ e}}{\partial x\partial z}\end{aligned} \right.\qquad\left\{ \begin{aligned} & H_{z}^{e}=0\\ & H_{y}^{e}=-\frac{\partial^{2}\Psi^{e}}{\partial x}\\ & H_{x}^{e}=\frac{\partial^{2}\Psi^{e}}{\partial y}\end{aligned} \right. \tag{3}\]  
When \(\overset{\rightarrow}{F}\
eq 0\) and \(\overset{\rightarrow}{A}=0\), an arbitrary electric magnetic field is represented by \(\Psi^{e}\). Let \(\overset{\rightarrow}{F}=\sharp\Psi^{m}\) denote the TE field for z. Then, the relationship between \(\Psi^{e}\) and each component of the electromagnetic field is  
\[\left\{ \begin{aligned} & E_{z}^{m}=0\\ & E_{y}^{m}=\frac{\partial^{2}\Psi^{m}}{\partial x}\\ & E_{x}^{e}=-\frac{\partial^{2}\Psi^{m}}{\partial y}\end{aligned} \right. \tag{4}\]  
In the homogeneous passive region, any scalar electromagnetic field component can be set as \(\Psi=\Psi^{e}+\Psi^{m}\), which satisfies the homogeneous scalar wave equation \(\
abla^{2}\Psi+k_{0}^{2}n^{2}\Psi=0\) in the passive region. \(\
abla^{2}=\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}+\partial^{ 2}/\partial z^{2}\) is the Laplace operator.