## 1 Introduction  
Artificial engineered materials propose new solutions in designing and manufacturing structures with given peculiarities at working frequencies and allow one to overcome the intrinsic limitations of standard materials. Negative real parts of permittivity and permeability, as well as the negative refraction index, define very important artificial materials in this context. They are known as double-negative metamaterials (DNG MTMs), negative index materials (NIMs), left-handed materials (LHMs), or backward (BW) media (information about characteristics and applications, as well as additional references, can be found in [1; 2; 3; 4; 5; 6]). The above acronyms are associated with the unconventional characteristics, e.g., the mutual position of the electric field, magnetic field and wave vector of a propagating plane wave describe LHMs, and BW is related to the backward propagation with the wave vector antiparallel to the Poynting one. The acronym DNG MTM will be used from this point on to denote this class of artificial materials. From the viewpoint of their fabrication, they can be manufactured by implanting small inclusions in host structures or by adding inhomogeneity to host surfaces [1; 2; 3; 4; 5; 6]. Studies, as well as communication and space applications, can take benefit from techniques for the evaluation of the electromagnetic scattering from composite structure hosting DNG MTMs. The uniform asymptotic physical optics (UAPO) approach suggests alternative analytical solutions to study scattering problems [7; 8; 9; 10; 11; 12; 13; 14]. It has been already applied to DNG MTM structures [7; 8; 9], and its usefulness has been proved by comparisons with data resulting from well-assessed numerical tools. The approach implements convenient approximations and asymptotic techniques to extract the high-frequency diffraction term from the radiation integral, including electric and magnetic currents as equivalent surface sources. The UAPO diffracted field is always expressed in closed form without requiring the solution of differential/integral equations or the computation of special functions, save for the transition function of the uniform geometrical theory of diffraction (UTD) [15]. The electromagnetic field in the region surrounding the scattering structure is then determined by adding the UAPO diffraction contribution to the geometrical optics (GO) one. This article refers to the exploitation of the UAPO method to evaluate the diffraction of plane waves from the rectilinear discontinuity of a DNG MTM layer hosted by a planar perfect electric conductor (PEC) support (see Figure 1). As is well known, a PEC is an ideal medium typically adopted to approximate the electromagnetic characteristics of a metal at high frequencies, where the UAPO approach works. Accounting for this, such a particular composite structure can be useful, for example, to modify the scattering response of a metallic body as requested by a smart radio environment. Therefore, the availability of an efficient analytical approach to be used in the design process can be of interest for electromagnetic and communication engineers from the application point of view as well as for researchers and scientists from the theoretical point of view. Analytical and numerical preliminary results were presented in [16], where the ability of the UAPO diffracted field to compensate for the jump of the GO field at the reflection shadow boundary in the upper half-space was demonstrated. The equivalent transmission line (ETL) models were applied to the evaluation of the reflection coefficients associated to the DNG MTM external surface. This step is important not only for determining the GO response of the composite structure, but also for formulating the electric and magnetic PO equivalent surface currents in terms of the incident electric field. Note that the PEC support is not present in the DNG MTM structures considered in [7,8] and, therefore, the evaluation of reflection and transmission coefficients needs a different method.