## 3 Boundary Conditions and Scattering Properties of Two-Dimensional Surfaces
### Relations between Scattering Properties and Surface Susceptibilities  
The scattering coefficients of EM surfaces can also be determined from surface susceptibilities through the field relations defined by GSTCs. By substituting EM field expressions (Equations (13) and (14)) into GSTCs (Equations (8) and (9)), \(\Gamma\) and \(T\) can be written as functions of \(\overline{\overline{\chi}}_{ee}\) and \(\overline{\overline{\chi}}_{mm}\)[7]  
\[\Gamma^{\text{TE}}(\theta)=\frac{-j\frac{k_{0}}{2\cos\theta}( \chi^{yy}_{ee}-\chi^{xx}_{mm}\cos^{2}\theta+\chi^{zz}_{mm}\sin^{2}\theta)}{1-( \frac{k_{0}}{2})^{2}\chi^{xx}_{mm}(\chi^{yy}_{ee}+\chi^{zz}_{mm}\sin^{2}\theta )+j\frac{k_{0}}{2\cos\theta}(\chi^{yy}_{ee}+\chi^{xx}_{mm}\cos^{2}\theta+\chi^ {zz}_{mm}\sin^{2}\theta)} \tag{21a}\] \[T^{\text{TE}}(\theta)=\frac{1+(\frac{k_{0}}{2})^{2}\chi^{xx}_{mm }(\chi^{yy}_{ee}+\chi^{zz}_{mm}\sin^{2}\theta)}{1-(\frac{k_{0}}{2})^{2}\chi^{ xx}_{mm}(\chi^{yy}_{ee}+\chi^{zz}_{mm}\sin^{2}\theta)+j\frac{k_{0}}{2\cos\theta}( \chi^{yy}_{ee}+\chi^{xx}_{mm}\cos^{2}\theta+\chi^{zz}_{mm}\sin^{2}\theta)}, \tag{21b}\]  
and  
\[\Gamma^{\text{TM}}(\theta)=\frac{-j\frac{k_{0}}{2\cos\theta}(- \chi^{yy}_{mm}+\chi^{xx}_{ee}\cos^{2}\theta-\chi^{zz}_{ee}\sin^{2}\theta)}{1-( \frac{k_{0}}{2})^{2}\chi^{xx}_{ee}(\chi^{yy}_{mm}+\chi^{zz}_{ee}\sin^{2}\theta )+j\frac{k_{0}}{2\cos\theta}(\chi^{yy}_{mm}+\chi^{xx}_{ee}\cos^{2}\theta+\chi^ {zz}_{ee}\sin^{2}\theta)} \tag{22a}\] \[T^{\text{TM}}(\theta)=\frac{1+(\frac{k_{0}}{2})^{2}\chi^{xx}_{ee }(\chi^{yy}_{mm}+\chi^{zz}_{ee}\sin^{2}\theta)}{1-(\frac{k_{0}}{2})^{2}\chi^{ xx}_{ee}(\chi^{yy}_{mm}+\chi^{zz}_{ee}\sin^{2}\theta)+j\frac{k_{0}}{2\cos\theta}( \chi^{yy}_{mm}+\chi^{xx}_{ee}\cos^{2}\theta+\chi^{zz}_{ee}\sin^{2}\theta)}, \tag{22b}\]  
for TE and TM polarizations respectively. Through Equations (21) and (22), \(\Gamma\) and \(T\) for arbitrary incident conditions can be analytically computed once \(\overline{\overline{\chi}}_{ee}\) and \(\overline{\overline{\chi}}_{mm}\) are determined. Different from sheet impedances, surface susceptibilities are characteristic parameters of surfaces that do not vary with properties of applied fields. Thus, \(\overline{\overline{\chi}}_{ee}\) and \(\overline{\overline{\chi}}_{mm}\) extracted from one situation can be directly utilized in the computation of others. The extraction method of surface susceptibilities from limited sets of full-wave simulations will be discussed in Section 4.1.  
Consider surfaces composed of isotropic zero-thickness PEC scatterers again. Since transverse magnetic current cannot exist along PEC surfaces, the values of \(\chi^{xx}_{mm}\) and \(\chi^{yy}_{mm}\) should be equal to zero. Meanwhile, \(\chi^{zz}_{ee}\) is also equal to zero because there is no electric current along \(z\)-direction due to zero thickness. \(\chi^{xx}_{ee}\) and \(\chi^{yy}_{ee}\) are non-zero due to the induced electric currents along PEC surface. Also, there are electric current loops formed along surfaces when the incident angle \(\theta\) is not equal to zero, leading to the non-zero \(\chi^{zz}_{mm}\). Therefore, for this special case, surface susceptibilities can be simplified as  
\[\overline{\overline{\chi}}_{ee}=\begin{bmatrix}\chi^{xx}_{ee}&0&0\\ 0&\chi^{yy}_{ee}&0\\ 0&0&0\end{bmatrix},\overline{\overline{\chi}}_{mm}=\begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&\chi^{zz}_{mm}\end{bmatrix}, \tag{23}\]  
and  
\[\Gamma_{e}^{\rm TM}(\theta) = \frac{-j\frac{k_{0}}{2}\chi_{ee}^{xx}\cos\theta}{1+j\frac{k_{0}}{ 2}\chi_{ee}^{xx}\cos\theta} \tag{25a}\] \[T_{e}^{\rm TM}(\theta) = \frac{1}{1+j\frac{k_{0}}{2}\chi_{ee}^{xx}\cos\theta}. \tag{25b}\]  
The corresponding results for zero-thickness surfaces composed of PMC scatterers can be derived using a similar procedure by letting \(\chi_{ee}^{xx}\), \(\chi_{ee}^{yy}\) and \(\chi_{mm}^{zz}\) equal to zero.