Wave propagation in periodic structures and negative refraction mechanisms

• Applicants: Professor Dr. Mario Ohlberger; Professor Dr. Ben Schweizer
• Subject Area: Mathematics
• Term: from 2014 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 258039653

• Final Report Year: 2018

Final Report Abstract

This research project was very successful in different fields. With the common interest in wave propagation in heterogeneous media, two research teams could interact and collaborate, they could share useful methods and they succeeded to deepen today’s knowledge on wave propagation in periodic structures and negative refraction. Some specific questions turned out to be of particular interest and much of the research efforts have been directed to these fields: 1.) Homogenization of periodic structures, 2.) Bloch wave analysis, 3.) Multi-scale methods. In the first field, the first mathematically rigorous derivation of a meta-material with negative refractive index was obtained. Moreover, the influence of the microstructure’s geometry on the homogenization results was investigated. The second field considers the expansion of solutions in Bloch waves to obtain appropriate radiation conditions in photonic crystals, which is also implemented in a numerical scheme. The third field embraces the design and numerical analysis of multi-scale methods, in particular the convergence of the schemes. Approaches for general heterogeneous media based on the Localized Orthogonal Decomposition have also been examined. The scientific output of this project is large (and, as a consequence, the number of publications is considerable). We emphasize that most publications have been made possible by some more informal exchange between the partners in this research project.

Publications

• A negative index meta-material for Maxwell’s equations. SIAM J. Math. Anal., 48(6):4155–4174, 2016
  A. Lamacz and B. Schweizer
  (See online at https://doi.org/10.1137/16M1064246)
• A new Heterogeneous Multiscale Method for time-harmonic Maxwell’s equations. SIAM J. Numer. Anal., 54(6):3493–3522, 2016
  P. Henning, M. Ohlberger, and B. Verfürth
  (See online at https://doi.org/10.1137/15M1039225)
• Effective Maxwell’s equations in general periodic microstructures. Applicable Analysis, 97(13):2210–2230, 2017
  B. Schweizer and M. Urban
  (See online at https://doi.org/10.1080/00036811.2017.1359563)
• Localized Orthogonal Decomposition for two-scale Helmholtz-type problems. AIMS Mathematics, 2(3):458–478, 2017
  M. Ohlberger and B. Verfürth
  (See online at https://doi.org/10.3934/Math.2017.2.458)
• A Bloch wave numerical scheme for scattering problems in periodic waveguides. SIAM J. Numer. Anal., 56(3):1848–1870, 2018
  T. Dohnal and B. Schweizer
  (See online at https://doi.org/10.1137/17M1141643)
• A new Heterogeneous Multiscale Method for the helmholtz equation with high contrast. Mulitscale Model. Simul., 16(1):385–411, 2018
  M. Ohlberger and B. Verfürth
  (See online at https://doi.org/10.1137/16M1108820)
• Effective Maxwell’s equations for perfectly conducting split ring resonators. Arch. Ration. Mech. Anal., 229(3):1197–1221, 2018
  R. Lipton and B. Schweizer
  (See online at https://doi.org/10.1007/s00205-018-1237-1)
• Numerical homogenization of H(curl)-problems. SIAM J. Numer. Anal., 56:1570–1596, 2018
  D. Gallistl, P. Henning, and B. Verfürth
  (See online at https://doi.org/10.1137/17M1133932)
• Outgoing wave conditions in photonic crystals and transmission properties at interfaces. ESAIM: Math. Model. Numer. Anal., 2018
  A. Lamacz and B. Schweizer
  (See online at https://doi.org/10.1051/m2an/2018026)