Duality relation for the Maxwell system.
This paper is intended to establish a link between the vector Maxwell system for three-dimensional (3D) and 2D finite photonic crystals in the low-frequency limit. For this, we generalize the classical results of Keller and Dykhne (chessboard problem) to periodic media described by piecewise continuous permittivity profiles: our theorem enlights the result of Mendelson (polycrystalline and multiphase media) in the framework of homogenization theory of elliptic operators. In fine, we give illustrative examples by using both integral equation and variational approaches via the so-called method of fictitious charges and finite-element method.
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