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1 IECN - Institut Élie Cartan de Nancy 2 CORIDA - Robust control of infinite dimensional systems and applications IECN - Institut Élie Cartan de Nancy, LMAM - Laboratoire de Mathématiques et Applications de Metz, Inria Nancy - Grand Est 3 Department of Mathematics

Abstract : For the scalar wave equation in periodic structures, we establish rigorously the emergence of a spectral bandgap as a frequency interval of exponentially small transmission of energy through a slab of the crystal when the thickness of the slab tends to infinity. We extend the result to slabs that contain a planar defect for frequencies that do not coincide with guided mode frequencies for the defect. For one-dimensional crystals, we prove that the transmission approaches a nonzero value at guided mode frequencies and give an explicit formula for the transmission. The main analytical tool in the multidimensional case is the Calderón boundary integral projectors for the Helmholtz equation.

Keywords : Calder on projector potential theory Photonic crystals periodic structures Helmholtz equation Green functions

Author: Karim Ramdani - Stephen Shipman -



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