## 2 Formulation
### Algorithm and Its Optimization  
The iterative equations of the magnetic field and the electric field in [PERSON]'s equations are shown in Equation (20) and Equation (21) as follows. \[\
abla\times E=-\mu_{0}\frac{\partial H}{\partial t} \tag{20}\]  
\[\
abla\times H=\varepsilon_{0}\frac{\partial E}{\partial t}+J \tag{21}\]  
In the formula, \(\vec{H}\) is the magnetic-field intensity (A/m) ; \(\vec{E}\) is the electric-field intensity (V/m); \(\vec{j}\) is the current density (A\({}^{2}\)/m); \(\varepsilon_{0}\) is the dielectric constant in vacuum (F/m); and \(\mu_{0}\) is the magnetic permeability in vacuum (H/m). In the spherical coordinate system,the electric-field vector \(\vec{E}\) and the magnetic-field vector \(\vec{H}\) can both be split into scalars in three directions, corresponding to the radial direction r perpendicular to the ground, the electromagnetic-wave propagation direction \(\theta\) along the ground, and the \(\varphi\) direction perpendicular to the wave vector direction, respectively. The expansion in the spherical coordinate system is as follows,  
\[r\sin\theta\mu_{0}\frac{\partial H_{r}}{\partial t}=-\frac{\partial\big{(}\sin \theta E_{\varphi}\big{)}}{\partial\theta} \tag{22}\]  
\[r\mu_{0}\frac{\partial H_{\theta}}{\partial t}=\frac{\partial\big{(}rE_{ \varphi}\big{)}}{\partial r} \tag{23}\]  
\[r\mu_{0}\frac{\partial H_{\varphi}}{\partial t}=-\frac{\partial(rE_{\theta})} {\partial r}+\frac{\partial E_{r}}{\partial\theta} \tag{24}\]  
The components are differentiated at the nth moment and organized to obtain Equations as follows. The magnetic field \(H\) at the \(n-1/2\) moment and the electric field \(E\) at the nth moment can deduce the magnetic field \(H\) at the \(n+1/2\) moment, which is the difference formula for the advancement and iteration of H in the calculation region over time. For the r-direction component, \(\theta-\)direction component, and \(\varphi-\)direction component of the magnetic field, at the \(n+1/2\) moment, the iteration equations at \((i+1/2,j)\), \((i,j+1/2)\), and \((i+1/2,j+1/2)\), respectively, are as follows,  
\[\frac{\mu_{0}}{\Delta t}H_{r,i+1/2,j}^{n+1/2}=H_{r,i+1/2,j}^{n-1/2}+\frac{1}{ \Delta\theta}(E_{\varphi,ij}^{n}+E_{\varphi,i+1,j}^{n}) \tag{25}\]  
\[\frac{\mu_{0}}{\Delta t}H_{\theta,i+1/2,j+1/2}^{n+1/2}=H_{\theta,i+1/2,j+1/2}^ {n-1/2}+\frac{1}{\Delta r}(E_{\varphi,ij,i+1}^{n}-E_{\varphi,i,j}^{n}) \tag{26}\]  
\[ \begin{array}{ll}\frac{\mu_{0}}{\Delta t}H_{\varphi,i+1/2,j+1/2}^{n+1/2}=H_ {\varphi,i+1/2,j+1/2}^{n-1/2}&+\frac{1}{\Delta\theta}(E_{r,i+1,j+1/2}^{n}-E_ {r,i,j+1/2}^{n})\\ &-\frac{1}{\Delta t}(E_{\theta,i+1/2,j+1}^{n}-E_{\theta,i+1/2,j}^{n})\end{array}  \tag{27}\]  
The iterative equation of the electric field  
\[\varepsilon_{0}\frac{\partial\vec{E}}{\partial t}+\vec{J}=\
abla\times\vec{H} \tag{28}\]  
The iterative equation of the electric field is combined with the current-density equation. When the current density is discretized at the \(n+1/2\) moment, and the electric field is discretized at the \(n\) moment, but the current density and the electric field are at the same position, the iterative equation between the current density and the electric field can be obtained. For the iterative equation of the electric field in the Equation (28), it can be simply discretized as  
\[\varepsilon_{0}\Big{(}\vec{E}^{n+1}-\vec{E}^{n}\Big{)}=\Delta t\
abla\times \vec{H}^{n+1/2}-\Delta t\vec{J}^{n+1/2} \tag{29}\]  
Suppose the electric field \(E\) is known at the nth moment; then, the current density \(J\) can be iterated from the value at the \(n-1/2\) moment to obtain the value at the \(n+1/2\) moment. The electric field \(E\) can be iterated from the \(n\) th moment to obtain the value at the \(n+1\) moment. Thus, cyclic iterations can be performed to obtain all the field quantities. When combined with the iterative equation of the magnetic field, a complete iterative equation can be obtained.