A few aspects of topological waves
Résumé
This manuscript adresses works in theoretical physics that deal with topological aspects of propagating waves. The investigation of the topological properties of waves have mainly consisted in the study of waves singularities in their intensity, phase or polarization. Inspired by the discovery of
topological phases of matter, whose quantum Hall effect discovered in 1980 and topological insulators discovered in 2005 are some manifestations, new topological properties of waves have been identified these last years. A few topological tools used and developed to describe these quantum materials revealed the key cornerstone concept of bulk-edge correspondence, that relates the topological characterization of electronic wavefunctions in a quantum material to the existence of confined states at its boundaries, that make for instance possible the transport of information without dissipation. This correspondence being applicable to classical waves, it turned out to be a powerful tool to describe and predict the existence of guided waves, from specifically designed metamaterials to natural systems.
This thesis, comprised of two parts, first presents this evolution by illustrating it with recent results dealing with the existence of unidirectional confined waves (so-called ”chiral waves”) in driven systems, random graphs, and continuous media such as gyrotropic optical systems and geophysical fluids. The second part is an introduction of mathematical tools in topology that are necessary to capture more technical aspects that underlie this common phenomenology shared by plentiful waves physical systems.
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