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Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem

Anne-Sophie Bonnet-Ben Dhia 1 Lucas Chesnel 2 Vincent Pagneux 3
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
2 DeFI - Shape reconstruction and identification
Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
Abstract : We consider the reflection-transmission problem in a waveguide with obstacle. At certain frequencies, for some incident waves, intensity is perfectly transmitted and the reflected field decays exponentially at infinity. In this work, we show that such reflectionless modes can be characterized as eigenfunctions of an original non-selfadjoint spectral problem. In order to select ingoing waves on one side of the obstacle and outgoing waves on the other side, we use complex scalings (or Perfectly Matched Layers) with imaginary parts of different signs. We prove that the real eigenvalues of the obtained spectrum correspond either to trapped modes (or bound states in the continuum) or to reflectionless modes. Interestingly, complex eigenvalues also contain useful information on weak reflection cases. When the geometry has certain symmetries, the new spectral problem enters the class of PT-symmetric problems.
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Contributeur : Lucas Chesnel <>
Soumis le : mercredi 21 novembre 2018 - 22:07:35
Dernière modification le : vendredi 22 janvier 2021 - 11:54:03
Archivage à long terme le : : vendredi 22 février 2019 - 16:34:11


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  • HAL Id : hal-01692297, version 2


Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, Vincent Pagneux. Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem. Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, Royal Society, The, 2018. ⟨hal-01692297v2⟩



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