## 2 Wave Field Transformation Theory Based on Fractional-Order Cole-Cole Model
### Loading Complex Frequency Shifted Perfectly Matched Layer (CFS-PML) in Fictitious Wave Field  
[PERSON] (2000) proposed CFS-PML (complex frequency shifted perfectly matched layer) boundary conditions, which significantly saved on memory [27]. The CFS-PML boundary conditions are introduced into the fictitious wave field to improve the absorption effect of the boundary on the low-frequency wave [28].  
In x-components, [PERSON]'s curl equations are as follows:  
\[\
abla_{\mathrm{s}}\times H^{\prime}(\omega^{\prime})+i\omega^{\prime}\varepsilon ^{\prime}E^{\prime}(\omega^{\prime})=0, \tag{29}\]  
\[\
abla_{\mathrm{s}}=\frac{1}{\hat{\mathrm{s}}_{x}}\frac{\partial}{\partial \mathrm{x}}\mathrm{i}+\frac{1}{\hat{\mathrm{s}}_{y}}\frac{\partial}{\partial \mathrm{y}}\mathrm{j}+\frac{1}{\hat{\mathrm{s}}_{z}}\frac{\partial}{\partial \mathrm{z}}\mathrm{k}. \tag{30}\]  
It should be noted that \(i\) is the imaginary unit in Maxwell equations, and \(\mathrm{i}\), \(\mathrm{j}\), and \(\mathrm{k}\) are the three direction vectors of the coordinate system. \(\widehat{\mathrm{s}}_{x}\), \(\widehat{\mathrm{s}}_{y}\), and \(\widehat{\mathrm{s}}_{z}\) are the coordinate stretching factors, \(\widehat{\mathrm{s}}_{(x^{\prime}y^{\prime}z)}=k_{(x,y,z)}+\frac{\partial(x,y, z)}{\alpha_{(x,y,z)}+i\omega^{\prime}}\)\(\alpha_{(x,y,z)}>0\), \(\sigma_{(x,y,z)}>0\), and \(k_{(x,y,z)}\geq 1\). By setting \(\alpha_{(x,y,z)}=\sigma_{(x,y,z)}=0\), \(k_{(x,y,z)}=1\), we can get the original stretched coordinate metrics, which is the PML boundary condition.  
Taking \(\mathrm{E}^{\prime}_{x}\) as an example, and using Equations (29) and (30), we can get  
\[i\omega^{\prime}\varepsilon^{\prime}E^{\prime}_{\mathrm{x}}(\omega^{\prime})= -\left(\frac{1}{\hat{s}_{y}}\frac{\partial H^{\prime}z(\omega^{\prime})}{ \partial y}-\frac{1}{\hat{s}_{z}}\frac{\partial H^{\prime}y(\omega^{\prime})} {\partial z}\right). \tag{31}\]  
Figure 1: Loading mode of fictitious wave field long wire source.  
Applying Laplace transform to (31) to the time domain, the iterative solution can be obtained.  
\[E^{rn+1}_{\pi_{i+1/2,j\pm}} =E^{rn}_{\pi_{i+1/2,j\pm}}+\Delta t^{\prime}\frac{2\omega_{0}}{ \sigma}[\partial_{y}^{-1}H^{rn+1/2}_{\pi_{i+1/2,j+1/2}\pm}-\partial_{z}^{-1}H^{ rn+1/2}_{\pi_{i+1/2,j\pm}\pm 1/2}]+\Delta t^{\prime}\frac{2\omega_{0}}{\sigma}(\Psi^{n+1/2}_{ \varepsilon_{\pi_{i+1/2,j\pm}}}-\Psi^{n+1/2}_{\varepsilon_{\pi_{i+1/2,j\pm}}}), \tag{32}\] \[\Psi^{n+1/2}_{\varepsilon_{\pi_{i+1/2,j\pm}}} =e^{-(\frac{\sigma_{y}}{\tau_{y}}+\alpha_{y})\frac{\Delta t^{ \prime}}{\sigma}}\Psi^{n-1/2}_{\varepsilon_{\pi_{i+1/2,j\pm}}}+\frac{\sigma_{ y}}{(\sigma_{y}k_{y}^{+}\delta_{y})}(e^{-(\frac{\sigma_{y}}{\tau_{y}}+\alpha_{y}) \frac{\Delta t^{\prime}}{\sigma^{\prime}}}-1)(H^{rn+1/2}_{\pi_{i+1/2,j+1/2}}- H^{rn+1/2}_{\pi_{i+1/2,j+1/2}})/\Delta_{y},\] (33) \[\Psi^{n+1/2}_{\varepsilon_{\pi_{i+1/2,j\pm}}} =e^{-(\frac{\sigma_{y}}{\tau_{y}}+\alpha_{y})\frac{\Delta t^{ \prime}}{\sigma}}\Psi^{n-1/2}_{\varepsilon_{\pi_{i+1/2,j\pm}}}+\frac{\sigma_{ x}}{(\sigma_{x}k_{y}^{+}\delta_{x})}(e^{-(\frac{\sigma_{y}}{\tau_{x}}+\alpha_{y}) \frac{\Delta t^{\prime}}{\sigma^{\prime}}}-1)(H^{rn+1/2}_{\pi_{i+1/2,j+1/2}}- H^{rn+1/2}_{\pi_{i+1/2,j-1/2}})/\Delta_{z}. \tag{34}\]  
On the boundary parameters, \(\sigma_{i}\) and \(k_{i}\), which depend on the relative positions of nodes and target areas, are not fixed, whereas \(\Psi^{n+1/2}_{\varepsilon_{\pi_{i+1/2,j\pm}}}\) and \(\Psi^{n+1/2}_{\varepsilon_{\pi_{i+1/2,j\pm}}}\) are two discrete variables.  
\[\sigma_{i}=\sigma_{\max}\frac{|k-k_{0}|^{m}}{d^{m}},(i=x,y,z), \tag{35}\]  
\[k_{i}=1+(k_{\max}-1)\frac{|k-k_{0}|^{m}}{d^{m}},(i=x,y,z), \tag{36}\]  
where \(k_{0}\) indicates the intersection of the boundary and the target area boundary, \(d\) is the boundary thickness, and \(m\) is the polynomial parameter.