Dynamic homogenisation of Maxwell’s equations with applications to photonic crystals and localised waveforms on gratings
File(s)
Published version
Author(s)
Maling, B
Colquitt, D
Craster, RV
Type
Journal Article
Abstract
A two-scale asymptotic theory is developed to generate continuum equations that model the macroscopic be-
haviour 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 resulting equations include tensors that represent effective refractive indices near band
edge frequencies along all principal axes directions, and these govern scalar functions providing long-scale mod-
ulation of short-scale Bloch eigenstates, which can be used to predict the propagation of waves at frequencies
outside of the long wavelength regime; these results are outside of the remit of typical homogenisation schemes.
The theory we develop is applied to two topical examples, the first being the case of aligned dielectric cylin-
ders, which has great importance in modelling photonic crystal fibres. Results of the asymptotic theory are veri-
fied against numerical simulations by comparing photonic band diagrams and evanescent decay rates for guided
modes. The second example is the propagation of electromagnetic waves localised within a planar array of di-
electric spheres; at certain frequencies strongly directional propagation is observed, commonly described as dy-
namic anisotropy. Computationally this is a challenging three-dimensional calculation, which we perform, and
then demonstrate that the asymptotic theory captures the effect, giving highly accurate qualitative and quantitative
comparisons as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour.
haviour 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 resulting equations include tensors that represent effective refractive indices near band
edge frequencies along all principal axes directions, and these govern scalar functions providing long-scale mod-
ulation of short-scale Bloch eigenstates, which can be used to predict the propagation of waves at frequencies
outside of the long wavelength regime; these results are outside of the remit of typical homogenisation schemes.
The theory we develop is applied to two topical examples, the first being the case of aligned dielectric cylin-
ders, which has great importance in modelling photonic crystal fibres. Results of the asymptotic theory are veri-
fied against numerical simulations by comparing photonic band diagrams and evanescent decay rates for guided
modes. The second example is the propagation of electromagnetic waves localised within a planar array of di-
electric spheres; at certain frequencies strongly directional propagation is observed, commonly described as dy-
namic anisotropy. Computationally this is a challenging three-dimensional calculation, which we perform, and
then demonstrate that the asymptotic theory captures the effect, giving highly accurate qualitative and quantitative
comparisons as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour.
Date Issued
2016-11-10
Date Acceptance
2016-11-01
Citation
Wave Motion, 2016, 69, pp.35-49
ISSN
0165-2125
Publisher
Elsevier
Start Page
35
End Page
49
Journal / Book Title
Wave Motion
Volume
69
Copyright Statement
© 2016 The Authors. Published by Elsevier B.V. This manuscript is made available under the CC-BY licence 4.0 (https://creativecommons.org/licenses/by/4.0/)
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/J009636/1
EP/DCT/DC/P49839
EP/L024926/1
Subjects
Applied Mathematics
0102 Applied Mathematics
0203 Classical Physics
Publication Status
Published