Spatial frequency maps of power flow in metamaterials and photonic crystals: Investigating backward-wave modes across the electromagnetic spectrum

We present spatial frequency maps of power flow in metamaterials and photonic crystals in order to provide insights into their electromagnetic responses and further our understanding of backward power in periodic structures. Since 2001, many different structures across the electromagnetic spectrum have been presented in the literature as exhibiting an isotropic negative effective index. Although these structures all exhibit circular or spherical equifrequency contours that resemble those of left-handed media, here we show through $k$-space diagrams that the distribution of power in the spatial frequency domain can vary considerably across these structures. In particular, we show that backward power arises from high-order right-handed harmonics in photonic crystals, magnetodielectric crystals, and across the layers of coupled-plasmonic-waveguide metamaterials, while arising from left-handed harmonic pairs in split-ring resonator and wire composites, plasmonic crystals, and along the layers of coupled-plasmonic-waveguide metamaterials. We also show that the fishnet structure exhibits the same left-handed harmonic pairs as the latter group. These observations allow us to categorize different metamaterials according to their spatial spectral source of backward power and identify the mechanism behind negative refraction at a given interface. Finally, we discuss how $k$-space maps of power flow can be used to explain the high or low transmittance of power into different metamaterial or photonic crystal structures.