# Wave Diffraction by Metamaterial-Coated Wedges: The UAPO Solution for Skew Incidence
## 1 Introduction  
The electromagnetic scattering involving a perfect electric conducting (PEC) wedge with arbitrary apex angle and uniform coating by a double-negative metamaterial (DNG MTM) with losses has been tackled in [1, 2] by considering an incident plane wave that is perpendicular to the edge. This two-dimensional problem has been studied at high frequencies by applying the Uniform Asymptotic Physical Optics (UAPO) method in the framework of the Uniform Geometrical Theory of Diffraction (UTD) [3], which gives a very attractive model that combines the diffraction contribution and the Geometrical Optics (GO) ones for the scattering computation. In the UAPO method, the electric and magnetic surface currents have been used as equivalent sources on the external surfaces of the composite structure and incorporated in the scattering integral. Closed-form expressions have been obtained for parallel and perpendicular polarizations with respect to the edge of the structure by adopting useful approximations and integral representations, and performing asymptotic evaluations of the resulting integrals. The capability of the UAPO diffracted field to balance the GO field at the shadow boundaries has been demonstrated by means of numerical tests [1, 2]. The solution for the two-dimensional plane wave diffraction in [1, 2], as well as that which was proposed in [4], has an application that is limited to schemes concerning normal incidences. Accordingly, the UAPO solution for the three-dimensional scenario of plane waves at a skew incidence is needed. This article addresses the above need by accounting for the relevance of scattering problems involving coated metallic wedges, the metallic part of which can be replaced by a PEC structure with the same geometry at high frequencies, where the UAPO approach works. As it is well known, the electromagnetic scattering of dielectric and metallic structures can be modified by adding layers on their surfaces, and therefore such a practice results very attractive for civil and military applications, i.e., airplanes and ships can necessitate tolower their radar cross section by covering the metallic surfaces. Moreover, metamaterials have been recently proposed in Intelligent Reflecting Surfaces (IRS). This article refers to a particular structure that can be allocated in a smart wireless propagation scenario. The unusual electromagnetic peculiarities of DNG MTMs are related to the negative real parts of their permittivity and permeability, and offer unconventional solutions with respect to the standard materials, so opening new frontiers for research activities and applications. The reader that is fascinated by such materials can find interesting information and applications, as well as available manufacturing processes, in several books and articles (see [5; 6; 7; 8; 9; 10; 11; 12] also for help in finding other references). The UAPO approach has been already applied to solve three-dimensional scattering problems. A planar junction between a double-positive material and a DNG MTM with a PEC backing has been considered in [13], whereas the scattering by a structure that is formed by a truncated DNG MTM sheet accommodated on a PEC support has been studied in [14]. Some references [15; 16; 17; 18; 19; 20] contain solutions for the plane wave diffraction problems involving dielectric wedges. In particular, the time domain counterpart of the UAPO solution in [19] has been obtained in [20] with reference to a penetrable wedge, which has an acute apex angle. The corresponding UAPO diffracted field has been always formulated in explicit and easy-to-use forms, and the effectiveness of the solutions has been demonstrated by adopting well-assessed numerical tools. As pointed out in a previous statement, the UAPO method is applied in this article for solving the three-dimensional scattering problem of obliquely incident plane waves, which illuminate a PEC wedge with a DNG MTM planar coating.