## 2 Wave Field Transformation Theory Based on Fractional-Order Cole-Cole Model  
### Transformation from Real Diffusion Field to Fictitious Wave Field  
The Maxwell equation in the frequency domain of the real diffusion field is written as follows:  
\[-\
abla\times H(x,\omega)+\sigma(\omega)E(x,\omega)=-J(x,\omega), \tag{1}\]\[\
abla\times\mathrm{E}(x,\omega)-i\omega\mathrm{\mu H}(x,\omega)=-\mathrm{K}(x, \omega), \tag{2}\]  
where \(\mathrm{E}\) and \(\mathrm{H}\) are the electric and magnetic fields in the real diffusion field, respectively, \(\mathrm{J}\) denotes the current density, and \(\mathrm{K}\) denotes the magnetic current density. \(\omega\) is the angular frequency, \(\mu\) is permeability tensor, and \(\sigma\) is conductivity, while \(\mathrm{x}\) is the direction.  
The IP effect can be described by the Cole-Cole model [16].  
\[\sigma(\omega)=\sigma_{\infty}(1-\frac{\eta}{1+\left(-i\omega\tau\right)^{c}}), \tag{3}\]  
where \(\sigma_{\infty}\) is the conductivity corresponding to infinite frequency, \(\tau\) is the characteristic time constant, and \(\mathrm{c}\) and \(\eta\) are the frequency dependence and chargeability, respectively.  
Applying Equation (3) to the frequency-domain Maxwell equation (Equation (1)), we can get  
\[-\
abla\times\mathrm{H}(x,\omega)+\sigma_{\infty}\mathrm{E}(x,\omega)-\frac{ \sigma_{\infty}\eta}{1+\left(-i\omega\tau\right)^{c}}\mathrm{E}(x,\omega)=- \mathrm{J}(x,\omega). \tag{4}\]  
A fictitious dielectric constant is defined by the conductivity tensor ([PERSON], 2010).  
\[\sigma_{\infty}=2\omega_{0}\varepsilon_{\infty}{{}^{\prime}}, \tag{5}\]  
where \(\omega_{0}=2\pi\times f_{0}\) is the scaling parameter, \(f_{0}=1 Hz\), and \(\varepsilon_{\infty}{{}^{\prime}}\) is the fictitious dielectric permittivity tensor. Substituting Equation (5) into Equation (4), we can get  
\[-\
abla\times\mathrm{H}(x,\omega)+2\omega_{0}\varepsilon_{\infty}{{}^{\prime }}\mathrm{E}(x,\omega)-\frac{\eta}{1+\left(-i\omega\tau\right)^{c}}2\omega_{0} \varepsilon_{\infty}{{}^{\prime}}\mathrm{E}(x,\omega)=-\mathrm{J}(x,\omega). \tag{6}\]  
Multiplying both sides of Equation (6) by \(\sqrt{\frac{\mathrm{i}\omega}{2\omega_{0}}}\) gives  
\[-\
abla\times\left[\sqrt{\frac{\mathrm{i}\omega}{2\omega_{0}}}\mathrm{H}(x, \omega)\right]+\left[\sqrt{\frac{\mathrm{i}\omega}{2\omega_{0}}}2\omega_{0} \varepsilon_{\infty}{{}^{\prime}}\mathrm{E}(x,\omega)\right]-\left[\frac{ \eta}{1+\left(-i\omega\tau\right)^{c}}\sqrt{\frac{\mathrm{i}\omega}{2\omega_ {0}}}2\omega_{0}\varepsilon_{\infty}{{}^{\prime}}\mathrm{E}(x,\omega)\right]=- \sqrt{\frac{\mathrm{i}\omega}{2\omega_{0}}}\mathrm{J}(x,\omega). \tag{7}\]  
Equation (2) can be rewritten as  
\[\
abla\times E(x,\omega)-\sqrt{-2i\omega\omega_{0}}\mu\sqrt{\frac{-i\omega}{2 \omega_{0}}}H(x,\omega)=K(x,\omega). \tag{8}\]  
The relationship between the real diffusion field and fictitious wave field was given by Reference [23].  
\[-i\omega^{\prime}=\sqrt{-2i\omega\omega_{0}}, \tag{9}\]  
\[E^{\prime}(x,\omega^{\prime})=E(x,\omega), \tag{10}\]  
\[H^{\prime}(x,\omega^{\prime})=\sqrt{\frac{-i\omega}{2\omega_{0}}}H(x,\omega), \tag{11}\]  
\[J^{\prime}(x,\omega^{\prime})=\sqrt{\frac{-i\omega}{2\omega_{0}}}J(x,\omega), \tag{12}\]  
\[K^{\prime}(x,