A spatial transformation technique is applied to an important class of two-dimensional electromagnetic cloaks that do not possess rotational symmetry around the longitudinal axis. These cloaks are based on an eccentric elliptic annular geometry, which represents a generalization of previously published cloaking configurations. The required material properties of the cloak are derived in terms of the geometrical parameters and the coordinates in the transformed system. The validity of the cloak parameters is verified by a full-wave finite-element simulation of plane wave scattering from uncloaked and cloaked perfectly conducting cylinders with elliptic cross section.

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