The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms

TL;DR

This paper develops an asymptotic theory to model electromagnetic wave behavior in complex periodic photonic structures, capturing short-scale details and applying it to photonic crystal fibres and metafilms with highly accurate results.

Contribution

The paper introduces a novel asymptotic approach that models electromagnetic waves in periodic structures without requiring long wavelengths, effectively capturing complex local effects.

Findings

Accurately models wave behavior in photonic crystal fibres.

Captures directional anisotropy in structured metafilms.

Provides insight into elliptic to hyperbolic transition in wave behavior.

Abstract

An asymptotic theory is developed to generate equations that model the global behaviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. The theory we develop is then applied to two topical examples, the first being the case of aligned dielectric cylinders, which has great importance in the modelling of photonic crystal fibres. We then consider the propagation of waves in a structured metafilm, here chosen to be a planar array of dielectric spheres. At certain frequencies strongly directional dynamic anisotropy is observed, and the asymptotic theory is shown to capture the effect, giving highly accurate qualitative and quantitative results as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour.