Communications in Mathematical Physics

, Volume 332, Issue 1, pp 221–260 | Cite as

Effective Light Dynamics in Perturbed Photonic Crystals



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.


Vector Bundle Photonic Crystal Pseudodifferential Operator Topological Insulator Selfadjoint Operator 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany
  2. 2.Department of MathematicsUniversity of Toronto, Bahen CentreTorontoCanada

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