847 research outputs found
The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms
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
Dynamic homogenisation of Maxwell’s equations with applications to photonic crystals and localised waveforms on gratings
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
Mesoscopic magnetism in dielectric photonic crystal meta materials: topology and inhomogeneous broadening
We consider meta materials made from a two-dimensional dielectric rod-type
photonic crystal. The magnetic response is studied within the recently
developed homogenization theory and we in particular study the effects of
topology and inhomogeneous broadening. While topology itself mainly affects the
Mie resonance frequency we find that the dispersion in the topological radius R
of the dielectric rods may lead to significant inhomogeneous broadening and
suppression of the negative-mu phenomena for dR/R0 >> epsilon''/epsilon', with
epsilon=epsilon'+i*epsilon'' being the relative dielectric function of the
rods.Comment: 13 pages including 1 table and 5 figure
Phase diagram for the transition from photonic crystals to dielectric metamaterials
Photonic crystals and metamaterials represent two seemingly different classes
of artificial electromagnetic media but often they are composed of similar
structural elements arranged in periodic lattices. The important question is
how to distinguish these two types of periodic photonic structures when their
parameters, such as dielectric permittivity and lattice spacing, vary
continuously. Here, we discuss transitions between photonic crystals and
all-dielectric metamaterials and introduce the concept of a phase diagram and
an order parameter for such structured materials, based on the physics of Mie
and Bragg resonances. We show that a periodic photonic structure transforms
into a metamaterial when the Mie gap opens up below the lowest Bragg bandgap
where the homogenization approach can be justified and the effective
permeability becomes negative. Our theoretical approach is confirmed by
detailed microwave experiments for a metacrystal composed of a square lattice
of glass tubes filled with heated water. This analysis yields deep insight into
the properties of periodic photonic structures, and it also provides a useful
tool for designing different classes of electromagnetic materials in a broad
range of parameters.Comment: 7 pages, 6 figure
Transformation Optics with Photonic Band Gap Media
We introduce a class of optical media based on adiabatically modulated,
dielectric-only, and potentially extremely low-loss, photonic crystals. The
media we describe represent a generalization of the eikonal limit of
transformation optics (TO). The foundation of the concept is the possibility to
fit frequency isosurfaces in the k-space of photonic crystals with elliptic
surfaces, allowing them to mimic the dispersion relation of light in
anisotropic effective media. Photonic crystal cloaks and other TO devices
operating at visible wavelengths can be constructed from optically transparent
substances like glasses, whose attenuation coefficient can be as small as 10
dB/km, suggesting the TO design methodology can be applied to the development
of optical devices not limited by the losses inherent to metal-based, passive
metamaterials.Comment: 4 pages, 4 figure
Sub-Wavelength Plasmonic Crystals: Dispersion Relations and Effective Properties
We obtain a convergent power series expansion for the first branch of the
dispersion relation for subwavelength plasmonic crystals consisting of
plasmonic rods with frequency-dependent dielectric permittivity embedded in a
host medium with unit permittivity. The expansion parameter is , where is the norm of a fixed wavevector, is the period of
the crystal and is the wavelength, and the plasma frequency scales
inversely to , making the dielectric permittivity in the rods large and
negative. The expressions for the series coefficients (a.k.a., dynamic
correctors) and the radius of convergence in are explicitly related to
the solutions of higher-order cell problems and the geometry of the rods.
Within the radius of convergence, we are able to compute the dispersion
relation and the fields and define dynamic effective properties in a
mathematically rigorous manner. Explicit error estimates show that a good
approximation to the true dispersion relation is obtained using only a few
terms of the expansion. The convergence proof requires the use of properties of
the Catalan numbers to show that the series coefficients are exponentially
bounded in the Sobolev norm
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