Three cases of discontinuous refractive index in metamaterial study

We investigate three cases of metamaterials presented in the literature displaying refractive index with one or more discontinuities along the frequency spectrum. We reproduce the numerical simulations of these metamaterials and compare our simulations to each reported case. For each case, we perform a geometrical investigation of each metamaterial's refractive index by mean of a numerical simulation of a prism made of the reported metamaterials upon which is incident a plane electromagnetic wave. Such investigation allows us to infirm or confirm negative refraction at resonance frequency. Finally, we carry a numerical and theoretical investigation of this discontinuity and show that, as the refractive index crosses a discontinuity, while the topology of the effective wave has changed within the metamaterial, the dynamics of the phases remain unchanged at any time at the metamaterial's boundaries.