Electromagnetic scattering by a semielliptical trench filled with DNG metamaterial

A semielliptical trench flush-mounted under an infinite perfect electric conductor plane is considered. The trench is slotted along the interfocal distance of its cross-section, and is filled with a double-negative (DNG) metamaterial that is isoimpedance to the material filling the half-space above the trench. The primary field is a plane wave that is obliquely incident upon the trench from the half-space above it, in a direction perpendicular to the trench axis and polarized either parallel or perpendicular to the axis. This two-dimensional boundary-value problem is solved exactly by separation of variables. The field components are written as infinite series of products of radial and angular Mathieu functions. Since the indexes of refraction inside and outside the trench are real and of opposite sign, the properties of the angular Mathieu functions permit the explicit, analytical determination of the modal expansion coefficients, thus yielding a canonical solution to the problem. Numerical results are exhibited and discussed.