## 3 Boundary Conditions and Scattering Properties of Two-Dimensional Surfaces
### Evolution of Boundary Conditions  
Conventional boundary conditions describe the relationship between field discontinuities across boundary and current sources distributed along boundary, which can be expressed as  
\[\hat{z}\times(\bar{H}_{2}-\bar{H}_{1})|_{z=0}=\bar{J}_{s,\varepsilon}|_{z=0} \tag{1}\]  
\[(\bar{E}_{2}-\bar{E}_{1})|_{z=0}\times\hat{z}=\bar{J}_{s,m}|_{z=0}, \tag{2}\]  
where \((\bar{E}_{1},\bar{H}_{1})\) and \((\bar{E}_{2},\bar{H}_{2})\) respectively represent the total fields in region \(z<0\) and \(z>0\) while \(\bar{J}_{s,\varepsilon}\) and \(\bar{J}_{s,m}\) are surface electric and magnetic currents along boundary. Here boundary is selected as the \(z=0\) plane where EM surfaces are located. If tangential components of \(\bar{E}\) are continuous along the boundary, the magnetic current \(\bar{J}_{s,m}\) in Equation (2) would become zero. Similarly, \(\bar{J}_{s,\varepsilon}\) in Equation (1) is equal to zero if tangential magnetic fields are continuous. For general surface problems, both \(\bar{J}_{s,\varepsilon}\) and \(\bar{J}_{s,m}\) are non-zero and their values depend on both surfaces' characteristic parameters and applied EM fields. The field discontinuities led by EM surfaces can be characterized by their effective sheet impedances, based on which IBCs are derived [15]. The electric sheet impedance \(\overline{\overline{Z}_{\varepsilon}}\) is defined  
Figure 9: Brief classifications of surface problems. as the ratio of average electric fields \(\vec{E}_{\text{av,tan}}\) along surfaces to surface electric current \(\vec{J}_{s,e}\) while magnetic sheet impedance \(\overline{\overline{Z}}_{m}\) is the ratio of \(\vec{J}_{s,m}\) to average magnetic fields \(\vec{H}_{\text{av,tan}}\)  
\[\vec{E}_{\text{av,tan}}|_{z=0}=\overline{\overline{Z}}_{e}\cdot\vec{J}_{s,e}|_{ z=0} \tag{3}\]  
\[\vec{H}_{\text{av,tan}}|_{z=0}=\overline{\overline{Z}}_{m}^{-1}\cdot\vec{J}_{s,m}|_{z=0}. \tag{4}\]  
Based on Equations (1) and (2), surface currents in Equations (3) and (4) can be replaced by the discontinuities of EM fields. Consequently, IBCs describe the relationship between average fields along surfaces and field discontinuities across surfaces  
\[\hat{z}\times(\vec{H}_{2}-\vec{H}_{1})|_{z=0}=\overline{\overline{Z}}_{e}^{-1 }\cdot\frac{1}{2}(\vec{E}_{1,\text{tan}}+\vec{E}_{2,\text{tan}})|_{z=0} \tag{5}\]  
\[(\vec{E}_{2}-\vec{E}_{1})|_{z=0}\times\hat{z}=\overline{\overline{Z}}_{m}\cdot \frac{1}{2}(\vec{H}_{1,\text{tan}}+\vec{H}_{2,\text{tan}})|_{z=0}. \tag{6}\]  
It is obvious that all field components that appear in IBCs are tangential ones. In general, \(\overline{\overline{Z}}_{e}\) and \(\overline{\overline{Z}}_{m}\) are two-dimensional tensors. If surfaces are isotropic, the tensorial sheet impedances can be simplified as scalars \(Z_{e}\) and \(Z_{m}\) for specific polarizations. If EM surfaces are modelled as two-port networks, then the tangential components of \(\vec{E}\) and \(\vec{H}\) can be equivalent to voltage \(V\) and current \(I\) in circuit theory. As a result, the field relations described by IBCs (Equations (5) and (6)) can be further represented by the bridged-T circuit model [28; 29] shown in Figure 10a. The impedance matrix \([Z]\)[10] of this two-port network can be written as  
\[ \begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix} = \begin{bmatrix}Z_{e}+Z_{m}/4&Z_{e}-Z_{m}/4\\ Z_{e}-Z_{m}/4&Z_{e}+Z_{m}/4\end{bmatrix}\begin{bmatrix}I_{1}\\ -I_{2}\end{bmatrix} , \tag{7}\]  
where \(Z_{e}\) and \(Z_{m}\) are the sheet impedances defined in Equations (5) and (6). Through the impedance matrix, the S-parameters of the two-port network can be readily computed [10], which stand for the scattering coefficients of EM surfaces that will be discussed in Section 3.2. Consider two special cases:  
* If the tangential electric fields are continuous across surfaces, the surface magnetic current \(\vec{J}_{s,m}\) becomes zero according to Equation (2). As a result, the magnetic impedance \(Z_{m}\) is equal to zero and the bridged-T circuit model can be simplified into a shunt impedance shown in Figure 10b.