Propagation in and scattering from a matched metamaterial having a zero index of refraction

Planar metamaterials that exhibit a zero index of refraction have been realized experimentally by several research groups. Their existence stimulated the present investigation, which details the properties of a passive, dispersive metamaterial that is matched to free space and has an index of refraction equal to zero. Thus, unlike previous zero-index investigations, both the permittivity and permeability are zero here at a specified frequency. One-, two-, and three-dimensional source problems are treated analytically. The one- and two-dimensional source problem results are confirmed numerically with finite difference time domain (FDTD) simulations. The FDTD simulator is also used to treat the corresponding one- and two-dimensional scattering problems. It is shown that in both the source and scattering configurations the electromagnetic fields in a matched zero-index medium take on a static character in space, yet remain dynamic in time, in such a manner that the underlying physics remains associated with propagating fields. Zero phase variation at various points in the zero-index medium is demonstrated once steady-state conditions are obtained. These behaviors are used to illustrate why a zero-index metamaterial, such as a zero-index electromagnetic band-gap structured medium, significantly narrows the far-field pattern associated with an antenna located within it. They are also used to show how a matched zero-index slab could be used to transform curved wave fronts into planar ones.