### 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 λ ≪ 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.

Original language | English |
---|---|

Pages (from-to) | 221-260 |

Number of pages | 40 |

Journal | Communications in Mathematical Physics |

Volume | 332 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 Jan 1 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

## Fingerprint Dive into the research topics of 'Effective light dynamics in perturbed photonic crystals'. Together they form a unique fingerprint.

## Cite this

*Communications in Mathematical Physics*,

*332*(1), 221-260. https://doi.org/10.1007/s00220-014-2083-0