## 3 Boundary Conditions and Scattering Properties of Two-Dimensional Surfaces
### Evolution of Boundary Conditions
A specific example is surfaces composed of zero-thickness scatterers made of perfect electric conductor (PEC). * If the tangential magnetic fields are continuous, then the surface electric current \(\vec{J}_{s,e}\) is equal to zero based on Equation (1). Consequently, the electric impedance \(Z_{e}\) would become infinity and the circuit model is simplified as a series impedance shown in Figure 10c. Surfaces made of zero-thickness perfect magnetic conductor (PMC) can be characterized by this model. The equivalent circuit model can be also utilized to characterize advanced EM surfaces such as perfect electromagnetic conductor (PEMC), which is a generalization of both PEC and PMC with similar properties [30]. With IBCs, field discontinuities led by EM surfaces can be determined once their sheet impedances \(Z_{e}\) and \(Z_{m}\) are known. However, since IBCs only describe the relationship between tangential field components, the values of \(Z_{e}\) and \(Z_{m}\) are sometimes related to the propagation directions and polarizations of applied fields. Thus, they are not characteristic parameters of EM surfaces and their values for different situations cannot be readily determined. As mentioned in Section 1, the characteristic parameters of two-dimensional surfaces are effective surface susceptibilities, whose values are independent of applied fields. Based on surface susceptibilities, GSTCs are initially derived for metafilms [18], the isolated-scatterer type of metasurfaces [7]. Ignoring the mutually-polarized effects between \((\vec{P}_{s},\vec{M}_{s})\) and \((\vec{E},\vec{H})\)[31], where \(\vec{P}_{s}\) and \(\vec{M}_{s}\) are the surface electric and magnetic surface polarization densities respectively, GSTCs for metafilms located in free space can be expressed as  
\[\hat{z}\times(\vec{H}_{2}-\vec{H}_{1})|_{z=0}=j\omega e_{0}\overline{\overline{ \chi}}_{ee}\cdot\frac{1}{2}(\vec{E}_{1,\text{tan}}+\vec{E}_{2,\text{tan}})|_{z =0}-\hat{z}\times\
abla_{t}(\chi_{mm}^{zz}\frac{1}{2}(H_{1,z}+H_{2,z}))|_{z=0} \tag{8}\]  
\[(\vec{E}_{2}-\vec{E}_{1})|_{z=0}\times\hat{z}=j\omega\mu_{0}\overline{\overline {\chi}}_{mm}\cdot\frac{1}{2}(\vec{H}_{1,\text{tan}}+\vec{H}_{2,\text{tan}})|_{ z=0}+\hat{z}\times\
abla_{t}(\chi_{ee}^{zz}\frac{1}{2}(E_{1,z}+E_{2,z}))|_{z=0}, \tag{9}\]  
where three-dimensional tensors \(\overline{\overline{\chi}}_{ee}\) and \(\overline{\overline{\chi}}_{mm}\) are effective surface electric and magnetic susceptibilities respectively. They are related to the polarizability densities of scatterers per unit area and have unit of meters [7]. For isotropic metafilms, \(\overline{\overline{\chi}}_{ee}\) and \(\overline{\overline{\chi}}_{mm}\) become diagonal matrices  
\[\overline{\overline{\chi}}_{ee}= \begin{bmatrix}\chi_{ee}^{xx}&0&0\\ 0&\chi_{ee}^{yy}&0\\ 0&0&\chi_{ee}^{zz}\end{bmatrix} ,\overline{\overline{\chi}}_{mm}= \begin{bmatrix} \chi_{mm}^{xx}&0&0\\ 0&\chi_{mm}^{yy}&0\\ 0&0&\chi_{mm}^{zz}\end{bmatrix} , \tag{10}\]  
with \(\chi_{ee}^{xx}=\chi_{ee}^{yy}\) and \(\chi_{mm}^{xx}=\chi_{mm}^{yy}\). Thus, there are four unknown terms in total in surface susceptibilities of isotropic problems. Through GSTCs, field relations across metafilms can be analytically determined once \(\overline{\overline{\chi}}_{ee}\) and \(\overline{\overline{\chi}}_{mm}\) are known. An overview of characterizing metafilms using GSTCs and its related applications in metasurfaces design is given in [7], including waveguide design [32] and guided wave characterization [27]. Numerical methods based on GSTCs are also derived from Equations (8) and (9) [33; 34; 35]. However, the surface characterization method based on surface susceptibilities and GSTCs are only appropriate for single-layer metafilms [18]. For EM surfaces with other types of features like isolated aperture or multiple layers, the corresponding characterization methods remain to be explored.