# Wave Diffraction by Metamaterial-Coated Wedges: The UAPO Solution for Skew Incidence
## 4 Numerical Examples  
Two sets of figures are shown in Section 4 to prove the effectiveness of the UAPO-based method for evaluating the electromagnetic scattering from an arbitrary-angled PEC wedge that is covered by a DNG MTM uniform layer. One should note that no limitations exist for the values of permittivity and permeability. The first set (Figures 2-5) is devoted to the ability of the UAPO diffracted field to compensate for the jumps of the GO field at the shadow boundaries, whereas the second set (Figures 6-10) includes comparisons with the Comsol Multiphysics(r) data. Both sets refer to a composite wedge that is characterized by \(\alpha=135^{\circ}\) (\(n=1.25\)), \(\varepsilon_{r}=-(3.2+j0.08)\), \(\mu_{r}=-(8+j0.09)\), and \(d=0.1\lambda_{0}\) if \(\lambda_{0}\) is the free-space wavelength. Moreover, \(P\) moves on a circular arc with \(\rho=6\lambda_{0}\) and, according to the \(\alpha\) value, \(\phi\) varies from \(0\) to \(225^{\circ}\). The magnitudes of the \(\beta-\) and \(\phi-\) components of the GO field, the UAPO diffracted field, and the total field are reported in Figures 2 and 3, respectively, where \(\beta^{\prime}=70^{\circ}\), \(\phi^{\prime}=80^{\circ}\). Both of the wedge surfaces are lit by the incident wave (\(U_{0,n\pi}=1\)) so that two reflection boundaries exist at \(\phi=100^{\circ},190^{\circ}\), where the GO field fails and exhibits inacceptable jumps. As expected, the UAPO diffracted field assumes significant magnitude values in the angular regions around such directions, and it is able to balance the GO jumps, so offering a continuous total field from \(0\) to \(225^{\circ}\). This ability is due to the behavior of the diffraction contribution at the shadow boundaries: the value on the left side is different from the value on the right side, so obtaining the needed discontinuity for the compensation. Figures 4 and 5, which refer to \(\beta^{\prime}=70^{\circ},\phi^{\prime}=40^{\circ}\) also confirm such an ability when one reflected field boundary (\(\phi=140^{\circ}\)) and one incident field boundary (\(\phi=220^{\circ}\)) exist since the incident wave impacts only one surface (\(U_{0}=1\), \(U_{n\pi}=0\)). Figure 3: Modulus of the \(\varphi\)-components with \(\beta^{\prime}=70^{\circ},\varphi^{\prime}=80^{\circ}\), \(E^{i}_{\beta^{\prime}}=0,E^{i}_{\phi^{\prime}}=1\). Figure 4: Modulus of the \(\beta\)-components with \(\beta^{\prime}=70^{\circ},\phi^{\prime}=40^{\circ}\), \(E^{i}_{\beta^{\prime}}=1,E^{i}_{\phi^{\prime}}=0\). Figure 6: Test by using Comsol Multiphysics® output with \(\beta^{\prime}=90^{\circ},\phi^{\prime}=80^{\circ}\), \(E^{i}_{\beta^{\prime}}=1,E^{i}_{\phi^{\prime}}=0\). (**a**) Modulus of the \(\beta\)-component. (**b**) Phase of the \(\beta\)-component. Figure 7: Test by using Comsol Multiphysics® output with \(\beta^{\prime}=90^{\circ},\phi^{\prime}=80^{\circ}\), \(E^{i}_{\beta^{\prime}}=0,E^{i}_{\phi^{\prime}}=1\). (a) Modulus of the \(\varphi\)-component. (b) Phase of the \(\varphi\)-component. Figure 8: Test by using Comsol Multiphysics® output with \(\beta^{\prime}=90^{\circ}\), \(\phi^{\prime}=80^{\circ}\). (**a**) Modulus of the \(\beta\)-component with \(E^{i}_{\beta^{\prime}}=1,E^{i}_{\phi^{\prime}}=0\). (**b**) Modulus of the \(\varphi\)-component with \(E^{i}_{\beta^{\prime}}=0,E^{i}_{\phi^{\prime}}=1\). Figure 9: Test by using Comsol Multiphysics® output with \(\beta^{\prime}=90^{\circ},\phi^{\prime}=40^{\circ}\), \(E^{i}_{\psi^{\prime}}=1,E^{i}_{\psi^{\prime}}=0\). (a) Modulus of the \(\beta\)-component. (b) Phase of the \(\beta\)-component. With reference to the figures of the second set, the wave propagation is considered to be orthogonal to the edge (\(\beta^{\prime}=90^{\circ}\)) to reduce the needed computation effort when one is running the RF unit of Comsol Multiphysics(r). The apex of the wedge is centered in a square propagation scenario with the side length of \(33.3\lambda_{0}\) and the thickness of the perfectly matched layer at the box boundary is \(2\lambda_{0}\).