132 research outputs found
Wave mechanics in media pinned at Bravais lattice points
The propagation of waves through microstructured media with periodically
arranged inclusions has applications in many areas of physics and engineering,
stretching from photonic crystals through to seismic metamaterials. In the
high-frequency regime, modelling such behaviour is complicated by multiple
scattering of the resulting short waves between the inclusions. Our aim is to
develop an asymptotic theory for modelling systems with arbitrarily-shaped
inclusions located on general Bravais lattices. We then consider the limit of
point-like inclusions, the advantage being that exact solutions can be obtained
using Fourier methods, and go on to derive effective medium equations using
asymptotic analysis. This approach allows us to explore the underlying reasons
for dynamic anisotropy, localisation of waves, and other properties typical of
such systems, and in particular their dependence upon geometry. Solutions of
the effective medium equations are compared with the exact solutions, shedding
further light on the underlying physics. We focus on examples that exhibit
dynamic anisotropy as these demonstrate the capability of the asymptotic theory
to pick up detailed qualitative and quantitative features
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
Pinning and Tribology of Tethered Monolayers on Disordered Substrates
We study the statistical mechanics and dynamics of crystalline films with a
fixed internal connectivity on a random substrate. Defect free triangular
lattices exhibit a sharp transition to a low temperature glassy phase with
anomalous phonon fluctuations and a nonlinear force-displacement law with a
continuously variable exponent, similar to the vortex glass phase of directed
lines in 1+1 dimensions. The periodicity of the tethered monolayer acts like a
filter which amplifies particular Fourier components of the disorder. However,
the absence of annealed topological defects like dislocations is crucial: the
transition is destroyed when the constraint of fixed connectivity is relaxed
and dislocations are allowed to proliferate.Comment: revtex, preprint style, 27 pages. This submission is a revision of
cond-mat/9607184. The revisions affect only Appendix B, Appendix C, and Eqs.
2.27, 2.28, 2.3
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
High frequency homogenization for travelling waves in periodic media
We consider high frequency homogenization in periodic media for travelling
waves of several different equations: the wave equation for scalar-valued waves
such as acoustics; the wave equation for vector-valued waves such as
electromagnetism and elasticity; and a system that encompasses the
Schr{\"o}dinger equation. This homogenization applies when the wavelength is of
the order of the size of the medium periodicity cell. The travelling wave is
assumed to be the sum of two waves: a modulated Bloch carrier wave having
crystal wave vector \Bk and frequency plus a modulated Bloch
carrier wave having crystal wave vector \Bm and frequency . We
derive effective equations for the modulating functions, and then prove that
there is no coupling in the effective equations between the two different waves
both in the scalar and the system cases. To be precise, we prove that there is
no coupling unless and (\Bk-\Bm)\odot\Lambda \in 2\pi
\mathbb Z^d, where is the
periodicity cell of the medium and for any two vectors the product is defined to be
the vector This last condition forces the
carrier waves to be equivalent Bloch waves meaning that the coupling constants
in the system of effective equations vanish.
We use two-scale analysis and some new weak-convergence type lemmas. The
analysis is not at the same level of rigor as that of Allaire and coworkers who
use two-scale convergence theory to treat the problem, but has the advantage of
simplicity which will allow it to be easily extended to the case where there is
degeneracy of the Bloch eigenvalue.Comment: 30 pages, Proceedings of the Royal Society A, 201
Trapped Modes and Steered Dirac Cones in Platonic Crystals
This paper discusses the properties of flexural waves obeying the biharmonic
equation, propagating in a thin plate pinned at doubly-periodic sets of points.
The emphases are on the properties of dispersion surfaces having the Dirac cone
topology, and on the related topic of trapped modes in plates with a finite set
(cluster) of pinned points. The Dirac cone topologies we exhibit have at least
two cones touching at a point in the reciprocal lattice, augmented by another
band passing through the point. We show that the Dirac cones can be steered
along symmetry lines in the Brillouin zone by varying the aspect ratio of
rectangular lattices of pins, and that, as the cones are moved, the involved
band surfaces tilt. We link Dirac points with a parabolic profile in their
neighbourhood, and the characteristic of this parabolic profile decides the
direction of propagation of the trapped mode in finite clusters.Comment: 21 pages, 12 figure
Photonic crystal design and fabrication assisted by tunable femtosecond oscillator laser
Photonic Crystals have the potential for engineered light interaction as directed by the photonic bandgap, a property that details the prohibited propagation region and ultimately the ability to guide light. A so called, photonics-on-chip, would incorporate arbitrary light guiding with functional elements in one package. The realization of such a device is subject to the fabrication paradigms implemented and this thesis is concerned with holographic lithography as a means for creating polymer photonics crystal templates and two-photon polymerization for incorporating light guiding pathways.
In this thesis, the design of the to-be-fabricated structures has been co-opted by computational exploration of the photonic crystals possible under the fabrication paradigms with the photonic bandgap considered as a target for design optimization, and tolerancing. Introduction of defects into photonic crystal templates has been successfully realized with the assistance of a tunable femtosecond oscillator laser after the processing conditions have been investigated and demonstrated
Mechanics study and application of micro-engineered smart surface
Naturally existing functional surfaces with micro-structure arose competing interests due to their potential application in engineering filed such as wetting control, optical control, micro-fluidic, tissue scaffolds, marine engineering, oil field, etc al. A patterned surface with stimuli responsive properties attracts considerable interest for its importance in advanced engineering, partly due to its reversibility, easy design and control, good compatibility and responsive behaviour to external stimuli. In this work, we have investigated various surface instabilities that enable a convenient strategy of micro-engineered structure impart reversible patterned feature to an elastic surface. We focus on the classic bi-layer system contains a stiff layer on a soft substrate that produces parallel harmonic wrinkles at uniaxial compression and ultimately develop into deep creases and fold. By introducing the microscale planar Bravais lattice holes, we guided these instabilities into various patterns to achieve an anisotropic manipulation of single liquid droplet by initialize localized surface morphologies. The Finite Element Analysis provided the fundamental theory on the surface instabilities evolution and development. The finding demonstrates considerable control over the threshold of a surface elastic instability and bi-axial switching of droplet shape that relevant to many novel applications including wearable electronic devices, bio-medical systems, micro-fluidics and optical devices
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