# Wave Diffraction by Metamaterial-Coated Wedges: The UAPO Solution for Skew Incidence
## 3 UAPO Diffracted Field  
The equivalent sources (1) and (2) are then assumed to radiate in the free space around the wedge, and the following integral formulation can be used to evaluate the scattered field \(E_{-}^{s}\) by accounting for the linearity of the integral operator:  
\[ \begin{array}{rl}E_{-}^{s}=&-jk_{0}U_{0}\underset{S_{0}}{\iint}\left[\left( \underset{=}{I}-\hat{u}_{0}\hat{u}_{0}\right)\zeta_{0}\underset{-s_{0}}{J}+ \underset{-s_{0}}{M}\times\hat{u}_{0}\right]\!G\!\left(\underset{-}{r}, \underset{-}{r}^{\prime}\right)\!\text{d}S+\\ &-jk_{0}U_{n\pi}\underset{S_{n\pi}}{\iint}\left[\left(\underset{=}{I}-\hat{u} _{n\pi}\hat{u}_{n\pi}\right)\zeta_{0}\underset{-s_{n\pi}}{J}+\underset{-s_{n \pi}}{M}\times\hat{u}_{n\pi}\right]\!G\!\left(\underset{-}{r},\underset{-}{r} ^{\prime}\right)\!\text{d}S,\end{array}  \tag{4}\]  
where \(\zeta\!\left(\underset{-}{r},\underset{-}{r}^{\prime}\right)=\exp\!\left(- \!\beta\!\left|\underset{-}{r}-\underset{-0\pi}{r}\right|\right)\!/\!\left( \underset{-}{4\pi}\right|\!\left|\underset{-}{r}-\underset{-0\pi}{r}\right| \right)\) and \(\hat{u}_{0,n\pi}=\left(\underset{-}{r}-\underset{-0,n\pi}{r}\right)/\!\left| \underset{-}{r}-\underset{-0,n\pi}{r}\right|\) account for the positions of \(P\) and the source points at \(\underset{-0,n\pi}{r}\). The term \(\underset{=}{I}\) is the \(3\times 3\) identity matrix and \(U_{0,n\pi}\) is equal to \(1\) if the related surface is lit by the incident plane wave or its value is \(0\). The incidence direction \(\hat{k}^{i}=-\cos\phi^{\prime}\sin\beta^{\prime}\hat{x}-\sin\phi^{\prime}\sin \beta^{\prime}\hat{y}+\cos\beta^{\prime}\hat{z}\) is assumed (see Figure 1). According to [13; 14], the UAPO approach is able to extract the diffraction contributions from (4) and to provide the diffracted field \(E_{-}^{d}\) at \(P(s,\beta,\phi)\) with \(\beta=\beta^{\prime}\) (s symbolizes the distance from the diffraction point to \(P\) and \(\phi\) varies from \(0\) to \(n\pi\) on the arc of circumference with radius \(\rho=s\sin\beta\) and center on the external edge):  
\[ \begin{pmatrix}E_{\beta}^{d}\\ E_{\phi}^{d}\end{pmatrix} =\frac{\exp(-jk_{0}s)}{\sqrt{s}}\!\left[U_{0}F_{0}^{d} \underset{=}{N}+U_{n\pi}F_{n\pi}^{d}\underset{=}{N}\right]\!\left( \begin{array} []{c}E_{\beta^{\prime}}^{i}\\ E_{\phi^{\prime}}^{i}\end{array} \right)=\frac{\exp(-jk_{0}s)}{\sqrt{s}}\!\! \underset{=}{D}\!\left( \begin{array}{c}E_{\beta^{\prime}}^{i}\\ E_{\phi^{\prime}}^{i}\end{array} \right). \tag{5}\]  
Accordingly, the matrix \(\underset{=}{D}\) of the diffraction coefficients is given by the superposition of the surface contributions and uses:  
\[F_{0}^{d}=\frac{\exp(-j\pi/4)}{2\sqrt{2\pi k_{0}}}\frac{F\!\left(2k_{0}s\sin^{ 2}\beta^{\prime}\cos^{2}\!\left(\frac{\phi^{\prime}+\phi}{2}\right)\right)}{ \left(\cos\phi^{\prime}+\cos\phi\right)\sin^{2}\beta^{\prime}} \tag{6}\]  
\[F_{n\pi}^{d}=\frac{\exp(-j\pi/4)}{2\sqrt{2\pi k_{0}}}\frac{F\!\left(  
\[ \begin{split}\underset{=}{N}_{A_{5}}&=\begin{pmatrix}0&(1-R_{\perp _{0}})\sin\phi^{\prime}\sin\beta^{\prime}\\ 1+R_{\parallel_{0}}&0\end{pmatrix}\end{split}  \tag{14}\]  
\[\underset{=}{N}_{A_{6}}= \begin{pmatrix}(1-R_{\parallel_{0}})\sin\phi^{\prime} \sin\beta^{\prime}&0\\ 0&-(1+R_{\perp_{0}})\end{pmatrix}  \tag{15}\]  
\[\underset{=}{N}_{A_{7}}=\frac{1}{\sqrt{1-(\sin\phi^{\prime}\sin\beta^{\prime} )^{2}}} \begin{pmatrix}\sin\phi^{\prime}\cos\beta^{\prime}&\cos\phi^{\prime}\\ -\cos\phi^{\prime}&\sin\phi^{\prime}\cos\beta^{\prime}\end{pmatrix}  \tag{16}\]  
\[\underset{=}{N}_{B_{1}}=\underset{=}{N}_{A_{1}} \tag{17}\]  
\[\underset{=}{N}_{B_{2}}= \begin{pmatrix}1-(\cos\phi\sin\beta^{\prime})^{2}&- \sin\phi\cos\phi\sin^{2}\beta^{\prime}&-\cos\phi\sin\beta^{\prime}\cos\beta^{ \prime}\\ -\sin\phi\cos\phi\sin^{2}\beta^{\prime}&1-(\sin\phi\sin\beta^{\prime})^{2}&- \sin\phi\sin\beta^{\prime}\cos\beta^{\prime}\\ -\cos\phi\sin\beta^{\prime}\cos\beta^{\prime}&-\sin\phi\sin\beta^{\prime}\cos \beta^{\prime}&\sin^{2}\beta^{\prime}\end{pmatrix}  \tag{18}\]  
\[\underset{=}{N}_{B_{3}}=