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Mathematical analysis of electromagnetic plasmonic metasurfaces

by H. Ammari and B. Li and J. Zou

(Report number 2019-38)

Abstract
We study the anomalous electromagnetic scattering in the homogenization regime, by a subwavelength thin layer of periodically distributed plasmonic nanoparticles on a perfect conducting plane. By using layer potential techniques, we derive the asymptotic expansion of the electromagnetic field away from the thin layer and quantitatively analyze the field enhancement due to the mixed collective plasmonic resonances, which can be characterized by the spectra of periodic Neumann-Poincar\'{e} type operators. Based on the asymptotic behavior of the scattered field in the macroscopic scale, we further demonstrate that the optical effect of this thin layer can be effectively approximated by a Leontovich boundary condition, which is uniformly valid no matter whether the incident frequency is near the resonant range but varies with the magnetic property of the plasmonic nanoparticles. The quantitative approximation clearly shows the blow-up of the field energy and the conversion of polarization when 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 by Ammari et al..

Keywords: metasuraces, plasmonic particles, Maxwell equations, approximate boundary conditions.

@Techreport{ALZ19_842,
  author = {H. Ammari and B. Li and J. Zou},
  title = {Mathematical analysis of electromagnetic plasmonic metasurfaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-38},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-38.pdf },
  year = {2019}
}

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