Discrete coordinate transformation for designing all-dielectric flat antennas

Transformation electromagnetics provides a practical approach to control electromagnetic fields at will. Based on this principle, novel devices such as the invisible cloak have been proposed. Here we examine the extension of this technique as applied to the design of flat devices in antenna systems. A method using discrete coordinate transformation is proposed, which allows the conversion of conventional devices with curved shapes into flat systems, while preserving their non-dispersive, isotropic, broadband, and lossless properties. Two specific design examples, a flat reflector and a flat lens embedded in free space, are presented. To avoid the loss and narrow bandwidth issues typically present in metamaterials, appropriate approximations and simplifications are introduced to make the all-dielectric devices, which are more practical to build. It is also shown that the discrete coordinate transformation is valid for both the E and H polarizations, as long as the local coordinates of the system remain near-orthogonal. Finite-Difference Time-Domain simulations are used to verify the performances of these designs, and show that the all-dielectric devices have similar broadband performances compared to the conventional ones, while possessing the advantages of flat profiles and small volumes.