## 5 Characterization of Isolated-Aperture Surfaces
### Scattering Coefficients of Metascreens and Surface Susceptibilities of Complementary Metafilms  
For the metafilm shown in Figure 15b, its scattering coefficients can be analytically computed from its own surface susceptibilities through GSTCs for metafilms. For metafilms composed of zero-thickness PMC scatterers, the transmission coefficients for TE and TM polarizations can be respectively written as  
\[T_{m}^{\text{film,TE}}(\theta)=\frac{1}{1+j\frac{k_{0}}{2}\chi_{mm}^{xx}\cos \theta}, \tag{35}\]  
and  
\[T_{m}^{\text{film,TM}}(\theta)=\frac{1}{1+j\frac{k_{0}}{2\cos\theta}(\chi_{mm }^{yy}+\chi_{ee}^{zz}\sin^{2}\theta)}. \tag{36}\]  
By substituting Equations (35) and (36) into [PERSON]'s principle (Equation (34)), the transmission coefficients of zero-thickness PEC metascreen can be written as  
\[T_{e}^{\text{screen,TE}}(\theta)=\frac{j\frac{k_{0}}{2}\chi_{mm}^{xx}\cos \theta}{1+j\frac{k_{0}}{2}\chi_{mm}^{xx}\cos\theta}, \tag{37}\]  
and  
\[T_{e}^{\text{screen,TM}}(\theta)=\frac{j\frac{k_{0}}{2\cos\theta}(\chi_{mm}^{ yy}+\chi_{ee}^{zz}\sin^{2}\theta)}{1+j\frac{k_{0}}{2\cos\theta}(\chi_{mm}^{yy}+ \chi_{ee}^{zz}\sin^{2}\theta)}, \tag{38}\]  
where \(\chi_{mm}^{xx}\), \(\chi_{mm}^{yy}\) and \(\chi_{ee}^{zz}\) are the surface susceptibilities of its complementary PMC metafilm. For metascreen composed of zero-thickness PEC scatterers, the tangential components of \(\vec{E}\) are continuous everywhere along surfaces. Therefore, their reflection coefficients \(\Gamma_{e}^{\text{screen}}\) can be derived through the relations defined by Equation (15). In order to characterize scattering coefficients of zero-thickness metascreens based on surface susceptibilities, there are three unknowns to be determined: \(\chi_{mm}^{xx}\), \(\chi_{mm}^{yy}\) and \(\chi_{ee}^{zz}\). Since the discussed surfaces are assumed to be isotropic, the first two quantities are equal to each other. As shown in Section 4, the surface susceptibilities of complementary PMC metafilms can be determined through full-wave simulations of their unit cells. For convenience, they can also be extracted from simulated scattering coefficients of the given metascreens, based on the relations described by Equation (34). Derived from Equations (37) and (38), the unknowns \(\chi_{mm}^{xx}=\chi_{mm}^{yy}\) and \(\chi_{ee}^{zz}\) can be determined through  
\[\chi_{mm}^{xx}=\chi_{mm}^{yy}=\frac{T_{e}^{\text{screen}}(0^{\circ})}{j\frac{ k_{0}}{2}(1-T_{e}^{\text{screen}}(0^{\circ}))} \tag{39a}\] \[\chi_{ee}^{zz}=\frac{T_{e}^{\text{screen,TM}}(\theta_{0})-j\frac{ k_{0}}{2\cos\theta_{0}}\chi_{mm}^{yy}(1-T_{e}^{\text{screen,TM}}(\theta_{0}))}{j \frac{k_{0}}{2\cos\theta_{0}}\sin^{2}\theta_{0}(1-T_{e}^{\text{screen,TM}}( \theta_{0}))}, \tag{39b}\]  
where \(T_{e}^{\text{screen}}\) is the transmission coefficient of the given zero-thickness PEC metascreens. According to Equation (39), only two sets of simulations of given metascreens are required to determine the surface susceptibilities of complementary metafilms:  
1. one simulation of normal incidence with \(\theta=0^{\circ}\) to obtain \(T_{e}^{\text{screen}}(0^{\circ})\);
2. one simulation of TM-polarized oblique incidence with \(\theta=\theta_{0}\) to obtain \(T_{e}^{\text{screen,TM}}(\theta)\). Once the values of complementary metafilms' surface susceptibilities are determined, scattering coefficients of given metascreens for arbitrary \(\theta\) and polarizations can be readily computed. The procedure of characterizing isotropic zero-thickness PEC metascreens based on surface susceptibilities is listed below:  
1. carry out two sets of unit cell simulations of given metascreens, including \(\theta=0^{\circ}\) and \(\theta=\theta_{0}\) for only TM polarization;
2. extract \(\overline{\chi}_{ee}\) and \(\overline{\overline{\chi}}_{mm}\) from simulated scattering coefficients through Equation (39);compute \((\Gamma,T)\) for arbitrary \(\theta\) and polarizations using Equations (37) and (38). An example of metascreen characterization based on this procedure will be given in Section 5.3.