Effective Light Dynamics in Perturbed Photonic Crystals

TL;DR

This paper develops a rigorous mathematical framework to derive effective light dynamics in modulated photonic crystals using space-adiabatic perturbation theory, addressing multiband complexities and physical interpretation issues.

Contribution

It introduces a novel application of space-adiabatic perturbation theory to photonic crystals, deriving effective Maxwell operators and clarifying physical conditions for invariant subspaces.

Findings

Derived effective Maxwell operator for photonic crystals

Identified conditions for physical interpretation of effective dynamics

Highlighted challenges in multiband semiclassical methods

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 Schr\"odinger-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 to establish 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.