## 3 Boundary Conditions and Scattering Properties of Two-Dimensional Surfaces
### Scattering Properties of Two-Dimensional Surfaces  
Scattering properties of periodic EM surfaces are usually characterized by reflection and transmission coefficients, which describe the field relations between incidence, reflection and transmission, under plane-wave incidence. Here it is assumed that EM surfaces are located along \(z=0\) in free space and incident plane wave illuminates the surfaces from region \(z<0\). Thus \((\vec{E}_{1},\vec{H}_{1})\) in region \(z<0\) are the superposition of incident field \((\vec{E}_{i},\vec{H}_{i})\) and reflected fields \((\vec{E}_{r},\vec{H}_{r})\) while \((\vec{E}_{2},\vec{H}_{2})\) in region \(z>0\) are equal to transmitted fields \((\vec{E}_{t},\vec{H}_{t})\) only. Since it is free space on both sides of surfaces and the phase discontinuities led by periodic structures are same everywhere, the incident, reflected and transmitted angles should be equal to each other according to phase matching. Then the wave vectors of incidence, reflection and transmission can be respectively expressed as  
\[\vec{k}_{i} =k_{0}(\hat{x}\sin\theta\cos\varphi+\hat{y}\sin\theta\sin\varphi+ \hat{z}\cos\theta) \tag{12a}\] \[\vec{k}_{r} =k_{0}(\hat{x}\sin\theta\cos\varphi+\hat{y}\sin\theta\sin\varphi- \hat{z}\cos\theta)\] (12b) \[\vec{k}_{t} =k_{0}(\hat{x}\sin\theta\cos\varphi+\hat{y}\sin\theta\sin\varphi+ \hat{z}\cos\theta), \tag{12c}\]  
where \(k_{0}=\omega\sqrt{\varepsilon_{0}\mu_{0}}\) is the wavenumber in free space. For simplicity, only isotropic surfaces are discussed here. As mentioned in Section 2.3, scattering properties of isotropic surfaces are independent of the azimuth angle \(\varphi\). Thus \(\varphi\) is set as \(0^{\circ}\) in the following derivations. With the assumption of isotropy, the TE-polarized fields can be expressed as [22]  
\[\vec{E}_{i}^{\text{TE}} =\hat{y}E_{0}e^{-\hat{y}_{i}\cdot\vec{r}} \tag{13a}\] \[\vec{E}_{r}^{\text{TE}} =\hat{y}\Gamma^{\text{TE}}E_{0}e^{-\hat{y}_{r}\cdot\vec{r}}\] (13b) \[\vec{E}_{t}^{\text{TE}} =\hat{y}\Gamma^{\text{TE}}E_{0}e^{-\hat{y}_{t}\cdot\vec{r}}\] (13c) \[\vec{H}_{i}^{\text{TE}} =(-\hat{x}\cos\theta+\hat{z}\sin\theta)\Gamma^{\text{TE}}E_{0}e^{ -\hat{y}_{i}\cdot\vec{r}}/\eta_{0}\] (13d) \[\vec{H}_{r}^{\text{TE}} =(\hat{x}\cos\theta+\hat{z}\sin\theta)\Gamma^{\text{TE}}E_{0}e^{ -\hat{y}_{r}\cdot\vec{r}}/\eta_{0}\] (13e) \[\vec{H}_{t}^{\text{TE}} =(-\hat{x}\cos\theta+\hat{z}\sin\theta)\Gamma^{\text{TE}}E_{0}e^{ -\hat{y}_{t}\cdot\vec{r}}/\eta_{0}, \tag{13f}\]  
and for TM polarization  
\[\vec{E}_{i}^{\text{TM}} =(-\hat{x}\cos\theta+\hat{z}\sin\theta)E_{0}e^{-\hat{y}_{i}\cdot \vec{r}} \tag{14a}\] \[\vec{k}_{r} =k_{0}(\hat{x}\sin\theta\cos\varphi+\hat{y}\sin\theta\sin\varphi- \hat{z}\cos\theta)\]0 (14b) \[\vec{k}_{r} =k_{0}(\hat{x}\sin\theta\cos\varphi+\hat{y}\sin\theta\sin\varphi- \hat{z}\cos\theta)\]1 (14c) \[\vec{k}_{r} =k_{0}(\hat{x}\sin\theta\cos\varphi+\hat{y}\sin\theta\sin\varphi- \hat{z}\cos\theta)\]2 (14d) \[\vec{k}_{r} =k_{0}(\hat{x}\sin\theta\cos\varphi+\hat{y}\sin\theta\sin\varphi- \hat{z}\cos\theta)\]3 (14e) \[\vec{k}_{r} =k_{0}(\hat{x}\sin\theta\cos\varphi+\hat{y}\sin\theta\sin\varphi- \hat{z}\cos\theta)\]4where \(\vec{r}=x\hat{x}+y\hat{y}+z\hat{z}\) and \(\eta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}\) is the wave impedance in free space. \(\Gamma\) and \(T\) represent reflection and transmission coefficients respectively. According to Equations (13) and (14), if the tangential components of \(\vec{E}\) are continuous along surfaces, for instance, surfaces composed of zero-thickness PEC scatterers, then \(\Gamma\) and \(T\) satisfy that  
\[\vec{k}_{r} =k_{0}(\hat{x}\sin\theta\cos\varphi+\hat{y}\sin\theta\sin\varphi- \hat{z}\cos\theta)\]5  
Under this condition, once either \(\Gamma\) or \(T\) is determined, the other one can be easily computed through Equation (15).