Mathematical analysis of electromagnetic plasmonic metasurfaces

We study the anomalous electromagnetic scattering in the homogenization regime, by a subwavelength thin layer consisting of periodically distributed plasmonic nanoparticles on a perfectly conducting plane. By using quasi-periodic layer potential techniques, we derive the asymptotic expansion of the electromagnetic field away from the thin layer and quantitatively analyze the field enhancement induced by the excitation of the mixed collective plasmonic resonances, which can be characterized by the spectra of two types of periodic Neumann-Poincare operators. Based on the asymptotic behavior of the scattered field in the macroscopic scale, characterize the reflection scattering matrix for the thin layer and demonstrate that the optical effect of this metasurface can be effectively approximated by a Leontovich impedance boundary condition, which is uniformly valid no matter whether the incident frequency is near the resonant range. The quantitative approximation clearly shows the blow-up of the field energy and the conversion of the field polarization when the resonance occurs, resulting in a significant change of the reflection property of the conducting plane. These results confirm essential physical changes of electromagnetic metasurface at resonances mathematically, whose occurrence was verified earlier for the acoustic case and the transverse magnetic case. © 2020 Society for Industrial and Applied Mathematics.