3-D Implementation of Transformation Optics Using a Tetrahedron-Based Meshing Technique and Homogeneous Materials

Abstract

A tetrahedron meshing and transforming method for 3-D implementation of transformation optics is proposed and verified for a few waveguide components. The method is applicable to a wide range of electromagnetic problems. An important feature of the proposed method is that the transformed medium will be piecewise homogeneous and free of any singularities. In this method, the region under transformation is divided into a few tetrahedrons, both in the physical and virtual domains. Each tetrahedron of the physical domain transforms to a tetrahedron in the virtual domain. The material properties of each tetrahedron are then calculated using the Jacobean matrix of the transformation. Since the elements of the Jacobean matrix are constant, the $\varepsilon$ and $\mu$ tensors are finite and homogeneous in each tetrahedron. The method is verified for several rectangular waveguide coupler examples. Features of the $\varepsilon$ and $\mu$ tensors of the discussed examples are described. For instance, it is shown that the material properties may reduce to 2-D material types, or material anisotropy may reduce, under certain conditions.