Antenna elements and systems and other radio and microwave frequency devices are constructed with a high index of refraction medium having high matched values of relative permeability and relative permittivity, and a low loss tangent. By making the permeability of the transmission medium substantially equal to its relative permittivity, the impedance of the material is matched to that of the surrounding free space or air. By immersing a radiating element in such a material, and/or by using such a material between adjacent radiating elements or between a radiating element and a reflective ground plane, the physical size and/or the spacing of the elements may be substantially reduced without appreciable performance loss, thereby resulting in a more compact device that is particularly desirable for mobile applications. At least one exemplary such material is formed in layers and has electrical properties which are anisotropic and homogeneous and which vary as a function of frequency; the layers of such a material are preferably oriented such that the particular frequencies of radiation propagating through each layer are presented with high matched values of relative permittivity and relative permeability, and low values of dielectric and magnetic loss tangents.