## 1. Introduction

[2] We consider the two-dimensional geometry of Figure 1 where a partially covered cavity is located at the corner of two metallic walls perpendicular to each other. The cavity has a cross section that is a quarter ellipse, and is slotted from the focus to the center of the ellipse. The cavity is partially covered by a thin metallic strip that extends from the focal line away from the central line of the ellipse, as part of the metallic wall under which the cavity is flush mounted. The cavity is filled with a double-negative lossless metamaterial whose electric permittivity and magnetic permeability are real and opposite to the corresponding parameters of the quarter-space above the cavity. Causality requires that the index of refraction of the DNG material be negative and its intrinsic impedance positive.

[3] Two types of sources are considered. One is a plane wave with arbitrary direction of incidence in the quarter space (*x* > 0, *y* > 0) and polarized with the electric or the magnetic field parallel to the *z* axis. The other one is an electric or magnetic line source parallel to the *z* axis. This two-dimensional boundary value problem is solved exactly, in the frequency domain. In elliptic cylindrical coordinates, the primary and secondary field components are expanded in infinite series of eigenfunctions that are products of radial and angular Mathieu functions, where the Stratton-Chu normalization is adopted [see, e.g., *Stratton*, 1941; *Staff of the Computation Laboratory*, 1967; *Bowman et al.*, 1987]. Since the angular Mathieu functions are the same for positive and negative refractive index, it is possible to determine analytically the modal expansion coefficients of the secondary fields, by imposing the boundary conditions.

[4] The only two-dimensional problem involving radiation and scattering by a cavity flush mounted under a metallic plane for which an exact analytical solution exists is that of a slotted semielliptical channel [*Uslenghi*, 1992, 2004a]. A related geometry is the cavity-backed gap in a corner [*Uslenghi*, 1999; *Erricolo and Uslenghi*, 2005]. These geometries involve materials inside and outside the channel that are isorefractive to each other. Recently, the analysis performed by *Uslenghi* [1992, 2004a] was extended to the case of a trench filled with DNG metamaterial [*Akgol et al.*, 2009a, 2009b]. The present work is an extension of the geometry analyzed by *Uslenghi* [1999] and *Erricolo and Uslenghi* [2005] to the case of a corner cavity filled with DNG metamaterial.

[5] Numerical results are shown for fields both inside and outside the cavity, for several cavity configurations and different primary sources.