Mathematical formulation for zero reflection from multilayer metamaterial structures and their notable applications

The behaviour of bilayer structures composed of common materials and metamaterials (MTMs) under oblique incidence of plane waves is investigated by exact analytical methods. The TE, TM and elliptical polarisations are analysed. There are several combinations of double positive (DPS), double negative (DNG), epsilon negative (ENG) and mu negative (MNG) media for the bilayer structures, but only DPS?DPS, DPS?DNG and ENG?MNG bilayers with TE, TM and circular polarisations are analysed in detail. For homogeneous and isotropic MTM media, exact mathematical relations are derived for the design of reflectionless bilayer structures as a function of their geometry (thickness) and electric and magnetic parameters. Frequency dispersion is included in the formulations. It is shown that bilayers composed of common materials are not effective for the construction of zero reflection bilayer surfaces, whereas the application of MTMs is required to realise reflectionless phenomena. For the design of zero reflection bilayer structures, their thicknesses and values of ε and μ are determined. Finally, the performance of forward and backward notch filters observed by MTM bilayer structures are studied in detail and their designs and applications are investigated. The bandwidth of lossy MTMs increases considerably.