# Electromagnetic Imaging of Uniaxial Objects by Two-Step Neural Network
## 2 Formula
We define the incident wave as:  
\[E_{x}^{i}\left(\overset{\rightharpoonup}{r}\right)=-sin\varnothing e^{-jk_{0}( xcos\varnothing+ysin\varnothing)} \tag{5}\]  
\[E_{y}^{i}\left(\overset{\rightharpoonup}{r}\right)=cos\varnothing e^{-jk_{0}( xcos\varnothing+ysin\varnothing)} \tag{6}\]  
The total field \(\overline{E}\left(\overset{\rightharpoonup}{r}\right)=E_{x}\left(\overset{ \rightharpoonup}{r}\right)\hat{x}+E_{y}\left(\overset{\rightharpoonup}{r} \right)\hat{y}\) and the external scattered field \(\overline{E}^{s}(\overline{r})=E_{x}^{s}(\overline{r})\hat{x}+E_{y}^{s}( \overline{r})\hat{y}\) can be defined as follows:  
\[E_{x}\left(\overset{\rightharpoonup}{r}\right)=\left(\frac{\partial^{2}}{ \partial x^{2}}+k_{0}^{2}\right)\left[\int_{s}G\left(\overset{\rightharpoonup}{r},\overset{\rightharpoonup}{r}^{\prime}\right)\left(\varepsilon_{x}\left( \overset{\rightharpoonup}{r}^{\prime}\right)-1\right)E_{x}\left(\overset{ \rightharpoonup}{r}^{\prime}\right)ds^{\prime}\right]+\frac{\partial^{2}}{ \partial x\partial y}\left[\int_{s}G\left(\overset{\rightharpoonup}{r}, \overset{\rightharpoonup}{r}^{\prime}\right)\left(\varepsilon_{y}\left( \overset{\rightharpoonup}{r}^{\prime}\right)-1\right)E_{y}\left(\overset{ \rightharpoonup}{r}^{\prime}\right)ds^{\prime}\right]+\overset{\rightharpoonup }{E_{x}}\left(\overset{\rightharpoonup}{r}\right) \tag{7}\]  
\[E_{y}\left(\overset{\rightharpoonup}{r}\right)=\frac{\partial^{2}}{\partial x \partial y}\left[\int_{s}G\left(\overset{\rightharpoonup}{r},\overset{ \rightharpoonup}{r}^{\prime}\right)\left(\varepsilon_{x}\left(\overset{ \rightharpoonup}{r}^{\prime}\right)-1\right)E_{x}\left(\overset{ \rightharpoonup}{r}^{\prime}\right)ds^{\prime}\right]+\left(\frac{\partial^{2} }{\partial y^{2}}+k_{0}^{2}\right)\left[\int_{s}G\left(\overset{\rightharpoonup}{r},\overset{\rightharpoonup}{r}^{\prime}\right)\left(\varepsilon_{y}\left( \overset{\rightharpoonup}{r}^{\prime}\right)-1\right)E_{y}\left(\overset{ \rightharpoonup}{r}^{\prime}\right)ds^{\prime}\right]+\frac{\partial^{2}}{ \partial y}\left[\int_{s}G\left(\overset{\rightharpoonup}{r},\overset{ \rightharpoonup}{r}^{\prime}\right)\left(\varepsilon_{y}\left(\overset{ \rightharpoonup}{r}^{\prime}\right)-1\right)E_{y}\left(\overset{ \rightharpoonup}{r}^{\prime}\right)ds^{\prime}\right] \tag{8}\]  
\[E_{x}^{s}\left(\overset{\rightharpoonup}{r}\right)=\left(\frac{\partial^{2}}{ \partial x^{2}}+k_{0}^{2}\right)\left[\int_{s}G\left(\overset{\rightharpoonup}{r},\overset{\rightharpoonup}{r}^{\prime}\right)\left(\varepsilon_{x}\left( \overset{\rightharpoonup}{r}^{\prime}\right)-1\right)E_{x}\left(\overset{ \rightharpoonup}{r}^{\prime}\right)ds^{\prime}\right]+\frac{\partial^{2}}{ \partial x\partial y}\left[\int_{s}G\left(\overset{\rightharpoonup}{r}, \overset{\rightharpoonup}{r}^{\prime}\right)\left(\varepsilon_{y}\left( \overset{\rightharpoonup}{r}^{\prime}\right)-1\right)E_{y}\left(\overset{ \rightharpoonup}{r}^{\prime}\right)ds^{\prime}\right] \tag{9}\]  
\[E_{y}^{s}\left(\overset{\rightharpoonup}{r}\right)=\frac{\partial^{2}}{ \partial x\partial y}\left[\int_{s}G\left(\overset{\rightharpoonup}{r}, \overset{\rightharpoonup}{r}^{\prime}\right)\left(\varepsilon_{x}\left( \overset{\rightharpoonup}{r}^{\prime}\right)-1\right)E\left(\overset{ \rightharpoonup}{r}^{\prime}\right)ds^{\prime}\right]+\left(\frac{\partial^{2} }{\partial y^{2}}+k_{0}^{2}\right)\left[\int_{s}G\left(\overset{\rightharpoonup}{r},\overset{\rightharpoonup}{r}^{\prime}\right)\left(\varepsilon_{y}\left( \overset{\rightharpoonup}{r}^{\prime}\right)-1\right)E\left(\overset{ \rightharpoonup}{r}^{\prime}\right)ds^{\prime}\right] \tag{10}\]  
By first solving the total electric field \(E\) using Equations (3), (7), and (8) with the tensor distribution information, we can subsequently calculate the scattered field \(E^{s}\) from Equations (4), (9), and (10). Further information on the Green's functions can be found in [30]. In the numerical computations for the direct problem, we begin by dividing both the scatterer and the closed surface S into N small segments.