Abstract

An asymptotic procedure based upon the method of multiple scales has been developed by Elena Cherkaev ( Department of Mathematics, University of Utah, USA ), Niklas Wellander (Swedish Defence Research Agency, Linkoping, Sweden), Frederic Zolla (Institut Fresnel, Aix-Marseille University, CNRS , Marseille, France) and the speaker for homogenization of wave equations in quasiperiodic inhomogeneous media deduced from a cut-and-projection [1]. Partial differential operators (gradient, divergence and curl) acting on periodic functions with m variables in a higher-dimensional space are projected onto operators acting on quasiperiodic functions with n variables in the physical space (m>n). When the wave wavelength is much larger than the periodic cell in the m-dimensional space, heterogeneous photonic quasicrystals can be approximated by homogeneous media described by anisotropic tensors of permittivity and permeability, deduced from the resolution of annex problems of electrostatic type on the periodic cell in m-dimensional space. We point out that this cut-and-project multiple scale method can be applied to other wave problems described by systems of linear partial differential equations in quasiperiodic structures deduced from a cut-and-projection.   [1] S. Guenneau, F. Zolla, E. Cherkaev and N. Wellander, Multiple scale method applied to homogenization of irrational metamaterials, IEEE Proceedings of the Fourteenth International Congress on Artificial Materials for Novel Wave Phenomena, 162-164, 2020.