Radiation of a line source by a slotted semielliptical trench filled with DNG metamaterial

A semielliptical trench flush-mounted under a metal plane and filled with double-negative (DNG) metamaterial is considered. The primary field is an electric line source parallel to the trench and located inside the trench or in the half-space above it. The two-dimensional boundary-value problem is solved exactly in terms of infinite series of elliptic-cylinder wavefunctions. In particular, the far field is explicitly determined. Numerical results are provided and discussed.