Macroscopic optical response and photonic bands

TL;DR

This paper presents a comprehensive formalism for calculating the macroscopic dielectric response and photonic band structure of complex, periodic composite systems with arbitrary dielectric properties, including retardation effects.

Contribution

It introduces a general formalism that goes beyond the long-wavelength approximation to accurately compute electromagnetic responses in arbitrary photonic crystal geometries.

Findings

The formalism accurately predicts photonic band structures.

It reveals regions of left-handedness in the dispersion relation.

The approach accounts for spatial dispersion via an effective magnetic permeability.

Abstract

We develop a formalism for the calculation of the macroscopic dielectric response of composite systems made of particles of one material embedded periodically within a matrix of another material, each of which is characterized by a well defined dielectric function. The nature of these dielectric functions is arbitrary, and could correspond to dielectric or conducting, transparent or opaque, absorptive and dispersive materials. The geometry of the particles and the Bravais lattice of the composite are also arbitrary. Our formalism goes beyond the longwavelenght approximation as it fully incorporates retardation effects. We test our formalism through the study the propagation of electromagnetic waves in 2D photonic crystals made of periodic arrays of cylindrical holes in a dispersionless dielectric host. Our macroscopic theory yields a spatially dispersive macroscopic response which allows the calculation of the full photonic band structure of the system, as well as the characterization of its normal modes, upon substitution into the macroscopic field equations. We account approximately for the spatial dispersion through a local magnetic permeability and we analyze the resulting dispersion relation, obtaining a region of left-handedness.