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Effective Light Dynamics in Perturbed Photonic Crystals

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Abstract

In this work, we rigorously derive effective dynamics for light from within a limited frequency range propagating in a photonic crystal that is modulated on the macroscopic level; the perturbation parameter \({\lambda \ll 1}\) quantifies the separation of spatial scales.We do that by rewriting the dynamical Maxwell equations as a Schrödinger-type equation and adapting space-adiabatic perturbation theory. Just like in the case of the Bloch electron, we obtain a simpler, effective Maxwell operator for states from within a relevant almost invariant subspace. A correct physical interpretation for the effective dynamics requires establishing two additional facts about the almost invariant subspace: (1) The source-free condition has to be verified and (2) it has to support real states. The second point also forces one to consider a multiband problem even in the simplest possible setting; This turns out to be a major difficulty for the extension of semiclassical methods to the domain of photonic crystals.

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Authors and Affiliations

  1. Department Mathematik, Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058, Erlangen, Germany

    Giuseppe De Nittis

  2. Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George Street, Toronto, ON, M5S 2E4, Canada

    Max Lein

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  1. Giuseppe De Nittis
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  2. Max Lein
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Correspondence to Max Lein.

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Communicated by H. Spohn

Dedicated to Herbert Spohn on the occasion of his (66 – ε)th birthday

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De Nittis, G., Lein, M. Effective Light Dynamics in Perturbed Photonic Crystals. Commun. Math. Phys. 332, 221–260 (2014). https://doi.org/10.1007/s00220-014-2083-0

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  • DOI: https://doi.org/10.1007/s00220-014-2083-0

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