Superdimensional Metamaterial Resonators From Sub-Riemannian Geometry

We introduce a fundamentally new method for the design of metamaterial arrays. These behave superdimensionally, exhibiting a higher local density of resonant frequencies, giant focusing of rays, and stronger concentration of waves than expected from the physical dimension. This sub-Riemannian optics allows planar designs to function effectively as 3- or higher-dimensional media, and bulk material as dimension 4 or higher. Valid for any waves modeled by the Helmholtz equation, including scalar optics and acoustics, and with properties derived from the behavior of waves in sub-Riemannian geometry, these arrays can be assembled from nonresonant metamaterial cells and are potentially broadband. Possible applications include antenna design and energy harvesting.