Phase response at resonance frequency for metamaterial-insert mediums

The phase propagation across the metamaterial - insert is essential to characterize the medium as epsilon-negative (ENG), mu-negative (MNG) or double negative (DNG). In this paper, we study the phase behavior on propagating through metamaterial-insert. For ENG or MNG media, we find that the phase propagation through the metamaterial insert has zero slope at the resonant frequency and positive slope at other frequencies. In other words, the medium has a band-gap at resonance, which results in constant phase across the medium. For the DNG media, the phase propagation through metamaterial-insert has negative slope at the resonance frequency and positive slope at other frequencies. In other words, the propagation of the wave is backward at the resonance which results in negative phase slope across the medium.