## 3 Boundary Conditions and Scattering Properties of Two-Dimensional Surfaces
### Relations between Sheet Impedances and Surface Susceptibilities  
Two-dimensional periodic surfaces can be represented by either sheet impedances or surface susceptibilities, both of which can be utilized to characterize scattering properties of surfaces analytically. Using sheet impedances, EM surfaces are equivalent to the circuit models shown in Figure 10, based on which more complicated surface problems like multiple layers can be solved through cascading network. However, the values of sheet impedances are related to the properties of applied fields, which makes it difficult to determine \(Z_{e}\) and \(Z_{m}\) for arbitrary incident angles and polarizations. Surface susceptibilities are characteristic parameters that do not vary with the applied field, thus their values can be determined through limited sets of \((\Gamma,T)\) and then be utilized to compute scattering coefficients for arbitrary situations through the field relations described by GSTCs. But GSTCs are only appropriate for metasurfaces with single-layer structures, implying that it cannot be directly utilized to solve problems of multiple layers. Thus, here it is proposed to combine sheet impedances and surface susceptibilities together in order to solve more complicated surface problems. Once surface susceptibilities of given surfaces are determined, the corresponding sheet impedances for arbitrary incident angles and polarizations can be readily computed and utilized for surface characterizations.  
Take the TE polarization as an example. Based on the EM field expressions of TE polarization (Equation (13)), GSTCs can be rewritten as  
\[\hat{z}\times(\vec{H}_{2}-\vec{H}_{1})|_{z=0}=j\frac{k_{0}}{\eta_{0}}(\chi_{ee }^{yy}+\chi_{mm}^{zz}\sin^{2}\theta)\frac{1}{2}(\vec{E}_{1,\tan}+\vec{E}_{2, \tan})|_{z=0} \tag{26}\]  
\[(\vec{E}_{2}-\vec{E}_{1})|_{z=0}\times\hat{z}=j\eta_{0}k_{0}\chi_{mm}^{xx} \frac{1}{2}(\vec{H}_{1,\tan}+\vec{H}_{2,\tan})|_{z=0}. \tag{27}\]  
By comparing Equations (26) and (27) with IBCs (Equations (5) and (6)), relations between sheet impedances and surface susceptibilities can be written as  
\[Z_{e}^{\rm TE}(\theta)=-\frac{j\eta_{0}}{k_{0}(\chi_{ee}^{yy}+\chi_{mm}^{zz} \sin^{2}\theta)} \tag{28}\]  
\[Z_{m}^{\rm TE}(\theta)=j\eta_{0}k_{0}\chi_{mm}^{xx}, \tag{29}\]  
where \(Z_{e}\) and \(Z_{m}\) are the impedance shown in Figure 10a. It is obvious that for TE polarization \(Z_{e}\) is related to the incident angle \(\theta\) while \(Z_{m}\) is \(\theta\)-independent. Similarly, for TM polarization, the corresponding relations can be written as \[Z_{e}^{\text{TM}}(\theta)=-\frac{j\eta_{0}}{k_{0}\chi_{ee}^{xx}} \tag{30}\]  
\[Z_{m}^{\text{TM}}(\theta)=j\eta_{0}k_{0}(\chi_{mm}^{yy}+\chi_{ee}^{zz}\sin^{2} \theta), \tag{31}\]  
where \(Z_{e}\) is \(\theta\)-independent but \(Z_{m}\) is not. Therefore, the values of \(Z_{e}\) and \(Z_{m}\) for arbitrary incident angles and polarizations can be readily computed through Equations (28)-(31), once \(\overline{\overline{\chi}}_{ee}\) and \(\overline{\overline{\chi}}_{mm}\) of given surfaces are determined.  
For arbitrary single-layer isotropic metafilms, their surface susceptibilities can be extracted using the method that will be introduced in Section 4.1. With surface susceptibilities determined, the scattering coefficients for arbitrary incident angles and polarizations can be readily computed through Equations (21) and (22). Meanwhile, the corresponding sheet impedances can be determined through Equations (28)-(31), which can be further utilized to solve more complicated problems with equivalent circuit model and impedance matrix. While the characterization method described above is only valid for metafilms with isolated-scatterer topological structures, the corresponding method for metascreens composed of periodic isolated apertures will be derived in Section 5. Also, the combination of sheet impedances and surface susceptibilities will be utilized in Section 6 for characterizing surface-wave mode of multiple-layer impenetrable surfaces.