@article{discovery1462504,
          volume = {78},
            note = {This is the published version of record. For information on re-use, please refer to the publisher's terms and conditions.},
           pages = {437--456},
           title = {Superdimensional Metamaterial Resonators From Sub-Riemannian Geometry},
            year = {2018},
          number = {1},
         journal = {SIAM Journal on Applied Mathematics},
        keywords = {transformation optics, metamaterials, sub-Riemannian geometry},
             url = {https://doi.org/10.1137/17M1130964},
          author = {Greenleaf, A and Kettunen, H and Kurylev, Y and Lassas, M and Uhlmann, G},
        abstract = {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.},
            issn = {0036-1399}
}