Exact Transmission through a Metamaterial Parabolic-Cylinder Radome

A radome whose interfaces with the surrounding medium are confocal parabolic cylinders with the same symmetry plane is considered. The medium on both convex and concave sides of the radome is linear, homogeneous, lossless and isotropic; it is characterized by a real positive electric permittivity $\varepsilon_{1}$ and a real positive magnetic permeability $\mu_{1}$ or, equivalently, by a refractive index $n=\sqrt{ }\left(\varepsilon_{1} \mu_{1}\right)$ and an intrinsic impedance $Z_{1}=\sqrt{ }\left(\mu_{1} / \varepsilon_{1}\right)$. The radome is made of a linear, homogeneous, lossless and isotropic metamaterial characterized by a real negative permittivity $-\varepsilon_{2}$ and a real negative permeability $-\mu_{2}$ such that $\varepsilon_{1} \mu_{1}=\varepsilon_{2} \mu_{2}$. Because of causality [1], this entails that the media outside and inside the radome have real and opposite refractive indexes, and positive intrinsic impedances: $Z_{2}=\sqrt{ }\left(\mu_{2} / \varepsilon_{2}\right)$ is real positive and possibly different from $Z_{1}$. The primary field is a plane wave symmetrically incident on the concave side of the radome, and the analysis is conducted in the phasor domain, with a time-dependent factor $\exp (+\mathrm{j} \omega \mathrm{t})$ omitted throughout.