## 2 Relevant Propagation Modeling Theory
### The Theory of the Three-Dimensional Vector PE
Replacing \(\Psi\) with \(e^{ik_{0}x}\Psi(x,y,z)\) and substituting \(e^{ik_{0}x}\Psi(x,y,z)\) into \(\
abla^{2}\Psi+k_{0}^{2}n^{2}\Psi=0\) yields  
\[\frac{\partial^{2}\Psi}{\partial x^{2}}+2 ik_{0}\frac{\partial\Psi}{\partial x} +\
abla_{t}^{2}\Psi+k_{0}^{2}(n^{2}-1)\Psi=0 \tag{5}\]  
where \(\
abla_{t}^{2}=\partial^{2}/\partial y^{2}+\partial^{2}/\partial z^{2}\) denotes the transverse Laplace operator. Equation (5) can be decomposed as  
\[[\partial/\partial x+ik_{0}(1-Q)][\partial/\partial x+ik_{0}(1+Q)]\Psi=0 \tag{6}\]  
where \(Q=\sqrt{\
abla_{t}^{2}/k_{0}^{2}+n^{2}}\) is referred to as the pseudodifferential operator. The approximation method of Feit-Fleck is \(Q=\sqrt{1+A+B}\approx\sqrt{1+A}+\sqrt{1+B}-1\). The forward and backward propagation (or scattering) of the wave is represented in Equation (6), respectively. If only the forward propagation is considered and the Feit-Fleckmethod is used for the approximate representation of the pseudodifferential operator \(\mathcal{Q}\), make \(A=\
abla_{t}^{2}/k_{0}^{2}\) and \(B=n^{2}-1\); then,  
\[Q=\sqrt{k_{0}^{2}+\
abla_{t}^{2}+k_{0}^{2}(n^{2}-1)}/k_{0}\approx\sqrt{\
abla_{ t}^{2}+k_{0}^{2}}/k_{0}+n-1 \tag{7}\]  
The forward three-dimensional wide-angle PE is then obtained as  
\[\frac{\partial\Psi(x,y,z)}{\partial x}=ik_{0}\bigg{(}\sqrt{k_{0}^{2}+\
abla_{ t}^{2}}/k_{0}+n-2\bigg{)}\Psi(x,y,z) \tag{8}\]  
That is,  
\[ \begin{cases}\frac{\partial\Psi_{x}(x,y,z)}{\partial x}=ik_{0}\bigg{(}\sqrt{k _{0}^{2}+\
abla_{t}^{2}}/k_{0}+n-2\bigg{)}\Psi_{x}(x,y,z)\\ \frac{\partial\Psi_{y}(x,y,z)}{\partial x}=ik_{0}\bigg{(}\sqrt{k_{0}^{2}+ \
abla_{t}^{2}}/k_{0}+n-2\bigg{)}\Psi_{y}(x,y,z)\\ \frac{\partial\Psi_{z}(x,y,z)}{\partial x}=ik_{0}\bigg{(}\sqrt{k_{0}^{2}+ \
abla_{t}^{2}}/k_{0}+n-2\bigg{)}\Psi_{z}(x,y,z)\end{cases}  \tag{9}\]  
In summary, the three-dimensional vector PE of the magnetic field (or electric field) is essentially a system of equations consisting of three-dimensional scalar PEs that can reflect three field components; thus, the three-dimensional vector and scalar PEs are in identical forms in the equations. Equation (8) is a first-order partial differential equation about \(x\). In the Cartesian coordinate system, we let the initial value of the electromagnetic field be \(\Psi(x_{0},y,z)\) and solve the partial differential equation on this basis. Therefore, the forward propagation equation can be solved as follows. \[\Psi(x,y,z)=\exp[ik_{0}\Delta x(\sqrt{k_{0}^{2}+\
abla_{t}^{2}}/k_{0}+n-2)] \Psi(x_{0},y,z) \tag{10}\]  
where \(\Delta x=x-x_{0}\) is the step size. According to Equation (10), the essence of solving the three-dimensional PE is that the wavefronts in the \((y,z)\) plane are stepped forward along the \(x\)-axis and the subsequent wavefront is determined by the previous wavefront. Separating the constant term and the refractive index term in Equation (10), we obtain  
\[\Psi(x,y,z)=\exp[ik_{0}\Delta x(n-2)]\exp[i\Delta x\sqrt{k_{0}^{2}+\
abla_{t}^ {2}}]\Psi(x_{0},y,z) \tag{11}\]  
In Equation (11), the refractive index term is first considered a constant, and a two-dimensional Fourier transform is applied to the remainder of \(\Psi(x,y,z)\) in the \((y,z)\) plane, where  
\[\mathcal{F}[\exp(i\Delta x\sqrt{k_{0}^{2}+\
abla_{t}^{2}})\Psi(x_{0},y,z)]= \exp(i\Delta x\sqrt{k_{0}^{2}-k_{y}^{2}-k_{z}^{2}})\mathcal{F}[\Psi(x_{0},y,z)] \tag{12}\]  
Therefore, the field in the next step can be found according to Equation (11) as  
\[\Psi(x+\Delta x,y,z)=\exp[ik_{0}\Delta x(n-2)]\mathcal{F}^{-1}\big{\{}\exp(ik_ {x}\Delta x)\mathcal{F}[\Psi(x_{0},k_{y},k_{z})]\big{\}} \tag{13}\]  
where \(k_{x}=\sqrt{k_{0}^{2}-k_{y}^{2}-k_{z}^{2}}\) denotes the component of \(k_{0}\) in the \(x\) direction.