Reduction of FDTD Stair-Casing Error regarding to Metamaterials by Using High-Order Polynomial Transformation Functions

The material parameters of a metamaterial (MTM) are determined by the transformation function used in the optical transformation. Some previously reported MTMs, such as the invisibility cloak, the field rotator, and the field concentrator, were designed by a linear transformation. Their impedance was matched to the background so that no reflection was found; however, the material parameters were mismatched to the background due to the linear transformation function. In the present work, the parameters were matched by using high-order polynomial functions as the transformation function. Since similar materials are filled in boundary cells of the finite-difference time-domain (FDTD) algorithm, the stair-casing error was reduced and the tolerance against boundary abrasion was increased. The frequency response of the proposed method was analyzed. The proposed method is applicable to MTM structures that have complex boundary shapes. In this work, circular and elliptic boundary shapes were considered as examples.