# Wave Diffraction by Metamaterial-Coated Wedges: The UAPO Solution for Skew Incidence
## 2 PO Equivalent Surface Currents  
A DNG MTM uniform layer with thickness \(d\), permittivity \(\varepsilon=\varepsilon_{0}\varepsilon_{r}=-\varepsilon_{0}(e^{\prime}+je^{ \prime\prime})\) and permeability \(\mu=\mu_{0}\mu_{r}=-\mu_{0}(\mu^{\prime}+j\mu^{\prime\prime})\), where \(\varepsilon^{\prime},\varepsilon^{\prime\prime},\mu^{\prime},\mu^{\prime\prime}\) are positive quantities and \(\varepsilon_{0},\mu_{0}\) are related to the free space, covers an arbitrary-angled metallic wedge that is assumed to be PEC at the working frequencies and to have an apex angle \(\alpha=(2-n)\pi\).  
According to Figure 1, the external surface \(S=S_{0}\cup S_{n\pi}\) of the wedge is lit by an incident plane wave with the electric field \(\overset{i}{-}=\overset{i}{-0}\exp(-jk^{i}\cdot\overset{r}{-})\), where \(k^{i}=k_{0}\overset{i}{k}\) (\(k_{0}\) is the free-space propagation constant) and the vector \(\overset{r}{-}\) locates the observation point \(P\). If the PO approximation holds, such a field generates the following electric \(\left(\overset{J}{-_{s}}\right)\) and magnetic \(\left(\overset{M}{-_{s}}\right)\) equivalent surface currents on \(S_{0}\) and \(S_{n\pi}\):\[\begin{split}&\left.\begin{array}{l}J\\ -s_{0,n\pi}\end{array}=\hat{n}_{0,n\pi}\times\left.\begin{array}{l}(H^{i}+H^{ r})\\ -\end{array}\right|_{\mathcal{S}_{0,n\pi}}=\\ &=\left[(1-R_{\perp_{0,n\pi}})E^{i}_{\perp_{0,n\pi}}\cos\theta^{i}_{0,n\pi} \hat{n}_{\perp_{0,n\pi}}+(1+R_{\parallel_{0,n\pi}})E^{i}_{\parallel_{0,n\pi}} (\hat{n}_{0,n\pi}\times\hat{n}_{\perp_{0,n\pi}})\right]\frac{\exp(-jk^{i},r^{ \prime}}{\zeta_{0}})\\ &\left.\begin{array}{l}M\\ -s_{0,n\pi}\end{array}\right.=\left.\begin{array}{l}(E^{i}+E^{r}_{-})\\ \end{array}\right|_{\mathcal{S}_{0,n\pi}}\times\hat{n}_{0,n\pi}=\\ &=\left[(1-R_{\parallel_{0,n\pi}})E^{i}_{\parallel_{0,n\pi}}\cos\theta^{i}_{0, n\pi}\hat{n}_{\perp_{0,n\pi}}-(1+R_{\perp_{0,n\pi}})E^{i}_{\perp_{0,n\pi}} (\hat{n}_{0,n\pi}\times\hat{n}_{\perp_{0,n\pi}})\right]\exp(-jk^{i}\cdot r^{ \prime}_{-0,n\pi}).\end{split} \tag{1}\]  
In the above expressions, \(\zeta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}\), the symbols \(\parallel\) and \(\perp\) refer to parallel and perpendicular polarizations, respectively, \(\hat{n}_{0,n\pi}\) is the unit vector normal to the involved surface, \(\hat{n}_{\perp_{0,n\pi}}=\left(k^{i}\times\hat{n}_{0,n\pi}\right)/\left|k^{i} \times\hat{n}_{0,n\pi}\right|\), and \(\cos\theta^{i}_{0,n\pi}=-k^{i}\cdot\hat{n}_{0,n\pi}\). The ETL model is applied to calculate the reflection coefficients \(R\):  
\[R_{\parallel,\perp_{0,n\pi}}=\frac{Z^{in}_{\parallel,\perp_{0,n\pi}}-Z^{0}_{ \parallel,\perp_{0,n\pi}}}{Z^{in}_{\parallel,\perp_{0,n\pi}}+Z^{0}_{ \parallel,\perp_{0,n\pi}}}, \tag{3}\]  
in which \(Z^{0}_{\parallel_{0,n\pi}}=\zeta_{0}\cos\theta^{i}_{0,n\pi}\) and \(Z^{0}_{\perp_{0,n\pi}}=\zeta_{0}/\cos\theta^{i}_{0,n\pi}\) are the free-space ETL characteristic impedances, and the ETL input impedances \(Z^{in}\) for a DNG MTM layer covering a PEC support can be evaluated according to [1; 2; 13; 20].  
Formulas (1) and (2) provide zero surface currents if the incident wave does not illuminate the corresponding surface.  
Figure 1: The diffraction problem.