## 3 Boundary Conditions and Scattering Properties of Two-Dimensional Surfaces
### Relations between Scattering Properties and Sheet Impedances  
Since field relations across surfaces can be described by IBCs, their scattering coefficients can be determined from sheet impedances. By substituting the EM field expressions (Equations (13) and (14)) into IBCs (Equations (5) and (6)), \(\Gamma\) and \(T\) of isotropic EM surfaces can be written as functions of \(Z_{e}\) and \(Z_{m}\) as  
\[\Gamma^{\text{TE}}(\theta) =\frac{Z_{e}^{\text{TE}}(\theta)Z_{m}^{\text{TE}}(\theta)-(\eta_ {0}/\cos\theta)^{2}}{(2Z_{e}^{\text{TE}}(\theta)+\eta_{0}/\cos\theta)(\frac{ 1}{2}Z_{m}^{\text{TE}}(\theta)+\eta_{0}/\cos\theta)} \tag{17a}\] \[T^{\text{TE}}(\theta) =\frac{(\eta_{0}/\cos\theta)(2Z_{e}^{\text{TE}}(\theta)-\frac{ 1}{2}Z_{m}^{\text{TE}}(\theta))}{(2Z_{e}^{\text{TE}}(\theta)+\eta_{0}/\cos \theta)(\frac{1}{2}Z_{m}^{\text{TE}}(\theta)+\eta_{0}/\cos\theta)}, \tag{17b}\]  
\[\Gamma^{\text{TM}}(\theta) =\frac{Z_{e}^{\text{TM}}(\theta)Z_{m}^{\text{TM}}(\theta)-(\eta_ {0}\cos\theta)^{2}}{(2Z_{e}^{\text{TM}}(\theta)+\eta_{0}\cos\theta)(\frac{1}{ 2}Z_{m}^{\text{TM}}(\theta)+\eta_{0}\cos\theta)} \tag{18a}\] \[T^{\text{TM}}(\theta) =\frac{(\eta_{0}\cos\theta)(2Z_{e}^{\text{TM}}(\theta)-\frac{1}{ 2}Z_{m}^{\text{TM}}(\theta))}{(2Z_{e}^{\text{TM}}(\theta)+\eta_{0}\cos\theta)( \frac{1}{2}Z_{m}^{\text{TM}}(\theta)+\eta_{0}\cos\theta)}, \tag{18b}\]  
for TE and TM polarizations respectively. Through Equations (17) and (18), the scattering coefficients for arbitrary incident angles and polarizations can be readily computed if the corresponding sheet impedances are known. By rewriting Equations (17) and (18), \(Z_{e}\) and \(Z_{m}\) can be expressed in terms of \(\Gamma\) and \(T\), through which the sheet impedances of given surfaces can be extracted from the corresponding measured or simulated scattering coefficients. However, as mentioned before, the values of sheet impedances are related to the properties of applied fields. The values of \(Z_{e}\) and \(Z_{m}\) extracted from one situation cannot be utilized to compute scattering coefficients of others with different incident angle \(\theta\) or polarizations (TE/TM).  
As discussed before, for surfaces composed of zero-thickness PEC scatterers, the tangential electric fields are continuous and the magnetic sheet impedance \(Z_{m}\) is equal to zero. Then Equations (17) and (18) can be simplified as  
\[\Gamma_{e}^{\text{TE}}(\theta)=-\frac{\eta_{0}}{2Z_{e}^{\text{TE} }(\theta)\cos\theta}/(1+\frac{\eta_{0}}{2Z_{e}^{\text{TE}}(\theta)\cos\theta}) \tag{19a}\] \[T_{e}^{\text{TE}}(\theta)=1/(1+\frac{\eta_{0}}{2Z_{e}^{\text{TE} }(\theta)\cos\theta}), \tag{19b}\]and  
\[\Gamma^{\text{TM}}_{e}(\theta)=-\frac{\eta_{0}\cos\theta}{2\zeta^{ \text{TM}}_{e}(\theta)}/(1+\frac{\eta_{0}\cos\theta}{2\zeta^{\text{TM}}_{e}( \theta)}) \tag{20a}\] \[T^{\text{TM}}_{e}(\theta)=1/(1+\frac{\eta_{0}\cos\theta}{2\zeta^{ \text{TM}}_{e}(\theta)}), \tag{20b}\]  
which satisfy the relation shown in Equation (15). Similarly, expressions of \(\Gamma\) and \(T\) for surfaces composed of zero-thickness PMC scatterers can be derived by letting \(Z_{e}=\infty\).