## 2 Principle Analysis
### Scattered Electromagnetic Field from the Target  
The incident illumination light on the target surface is scattered and forms a superimposed field with the incident field and a scattered field. The superimposed field is defined as follows [14]:  
\[\mathbf{E}(\mathbf{r})=\mathbf{E}^{(inc)}(\mathbf{r})+\mathbf{E}^{(sca)}(\mathbf{r}), \tag{2}\]  
\[\mathbf{H}(\mathbf{r})=\mathbf{H}^{(inc)}(\mathbf{r})+\mathbf{H}^{(sca)}(\mathbf{r}), \tag{3}\]  
where the superscript \((inc)\) is the incident field, and the superscript \((sca)\) denotes the scattered field. Combined with Maxwell equations, Equations (2) and (3) can be rewritten as follows:  
\[\mathbf{E}(\mathbf{r})=\mathbf{E}^{(inc)}(\mathbf{r})+\
abla\!\cdot\!\
abla\!\cdot\!\Pi_{c}( \mathbf{r})+\hat{i}\mathbf{k}\
abla\!\cdot\!\Pi_{m}(\mathbf{r})-4\pi\mathbf{P}(\mathbf{r}), \tag{4}\]  
\[\mathbf{H}(\mathbf{r})=\mathbf{H}^{(inc)}(\mathbf{r})+\
abla\!\cdot\!\
abla\!\cdot\!\Pi_{m}( \mathbf{r})-\hat{i}\mathbf{k}\
abla\!\cdot\!\Pi_{c}(\mathbf{r})-4\pi\mathbf{M}(\mathbf{r}), \tag{5}\]where \(\mathbf{P}(\mathbf{r})\) and \(\mathbf{M}(\mathbf{r})\) are the incident polarized intensity of the electric and magnetic vectors in the target, respectively. \[\mathbf{P}(\mathbf{r})=\eta(\mathbf{r})\mathbf{E}(\mathbf{r}), \tag{6}\]  
\[\mathbf{M}(\mathbf{r})=\chi(\mathbf{r})\mathbf{H}(\mathbf{r}), \tag{7}\]  
where \(\eta(\mathbf{r})\) and \(\chi(\mathbf{r})\) are dielectric and magnetic susceptibilities, respectively. \(\Pi_{e}(\mathbf{r})\) and \(\Pi_{m}(\mathbf{r})\) are electric and magnetic Hertz potentials, respectively, as expressed in the following equations:  
\[\Pi_{e}(\mathbf{r})=\int_{V}\mathbf{P}\big{(}\mathbf{r}^{\prime}\big{)}\frac{e^{i\mathbf{k}| \mathbf{r}-\mathbf{r}^{\prime}|}}{|\mathbf{r}-\mathbf{r}^{\prime}|}d^{3}\mathbf{r}^{\prime}, \tag{8}\]  
\[\Pi_{m}(\mathbf{r})=\int_{V}\mathbf{M}\big{(}\mathbf{r}^{\prime}\big{)}\frac{e^{i\mathbf{k}| \mathbf{r}-\mathbf{r}^{\prime}|}}{|\mathbf{r}-\mathbf{r}^{\prime}|}d^{3}\mathbf{r}^{\prime}, \tag{9}\]  
where \(V\) is the target's volume in space. Equations (6), (7), (8), and (9) are substituted into Equations (4) and (5). The total field can be described by integral-differential equations in which each component of the electric and magnetic fields is coupled with others. The effect of coupling is negligible because the dimension of the target is much smaller than the wavelength. Therefore, this scattering should not be described with a simple scalar equation. According to the relationship between \(\eta(\mathbf{r})\) and relative permittivity \(\varepsilon_{r}(\mathbf{r})\),  
\[\varepsilon_{r}(\mathbf{r})=1+\eta(\mathbf{r}), \tag{10}\]  
the relationship between \(\varepsilon_{r}(\mathbf{r})\) and refractive index \(n(\mathbf{r})\),  
\[\varepsilon_{r}(\mathbf{r})\!\cdot\!\varepsilon_{0}=n^{2}(\mathbf{r}), \tag{11}\]  
and the equivalent transformation from the topographic structure to the refractive index around the surface, the effect of illumination polarization and target structure on the spatial distribution of the scattered light is provided as follows:  
\[ \begin{array}{c}\mathbf{E}(\mathbf{r})=\mathbf{E}^{(inc)}(\mathbf{r})+\
abla\!\cdot\!\
abla \!\cdot\!\int_{V}\mathbf{P}^{\prime}(\mathbf{r}^{\prime})d^{3}\mathbf{r}^{\prime}+\hat{i} k\
abla\!\cdot\!\int_{V}\mathbf{M}^{\prime}(\mathbf{r}^{\prime})d^{3}\mathbf{r}^{\prime}\\ -4\pi\Big{(}\frac{n^{2}(\mathbf{r})}{\varepsilon_{0}}-1\Big{)}\mathbf{E}(\mathbf{r}),\end{array}