Radiation from structured-ring resonators

We investigate the scalar-wave resonances of systems composed of identical Neumann-type inclusions arranged periodically around a circular ring. Drawing on natural similarities with the undamped Rayleigh-Bloch waves supported by infinite linear arrays, we deduce asymptotically the exponentially small radiative damping in the limit where the ring radius is large relative to the periodicity. In our asymptotic approach, locally linear Rayleigh-Bloch waves that attenuate exponentially away from the ring are matched to a ring-scale WKB-type wave field. The latter provides a descriptive physical picture of how the mode energy is transferred via tunnelling to a circular evanescent-to-propagating transition region a finite distance away from the ring, from where radiative grazing rays emanate to the far field. Excluding the zeroth-order standing-wave modes, the position of the transition circle bifurcates with respect to clockwise and anti-clockwise contributions, resulting in striking spiral wavefronts

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

Spectral algorithms for reaction-diffusion equations

A collection of codes (in MATLAB & Fortran 77), and examples, for solving reaction-diffusion equations in one and two space dimensions is presented. In areas of the mathematical community spectral methods are used to remove the stiffness associated with the diffusive terms in a reaction-diffusion model allowing explicit high order timestepping to be used. This is particularly valuable for two (and higher) space dimension problems. Our aim here is to provide codes, together with examples, to allow practioners to easily utilize, understand and implement these ideas; we incorporate recent theoretical advances such as exponential time differencing methods and provide timings and error comparisons with other more standard approaches. The examples are chosen from the literature to illustrate points and queries that naturally arise

Transformation seismology: composite soil lenses for steering surface elastic Rayleigh waves.

Metamaterials are artificially structured media that exibit properties beyond those usually encountered in nature. Typically they are developed for electromagnetic waves at millimetric down to nanometric scales, or for acoustics, at centimeter scales. By applying ideas from transformation optics we can steer Rayleigh-surface waves that are solutions of the vector Navier equations of elastodynamics. As a paradigm of the conformal geophysics that we are creating, we design a square arrangement of Luneburg lenses to reroute Rayleigh waves around a building with the dual aim of protection and minimizing the effect on the wavefront (cloaking). To show that this is practically realisable we deliberately choose to use material parameters readily available and this metalens consists of a composite soil structured with buried pillars made of softer material. The regular lattice of inclusions is homogenized to give an effective material with a radially varying velocity profile and hence varying the refractive index of the lens. We develop the theory and then use full 3D numerical simulations to conclusively demonstrate, at frequencies of seismological relevance 3–10 Hz, and for low-speed sedimentary soil (v(s): 300–500 m/s), that the vibration of a structure is reduced by up to 6 dB at its resonance frequency

Quantum Plasmonics

Quantum plasmonics is an exciting subbranch of nanoplasmonics where the laws of quantum theory are used to describe light–matter interactions on the nanoscale. Plasmonic materials allow extreme subdiffraction confinement of (quantum or classical) light to regions so small that the quantization of both light and matter may be necessary for an accurate description. State-of-the-art experiments now allow us to probe these regimes and push existing theories to the limits which opens up the possibilities of exploring the nature of many-body collective oscillations as well as developing new plasmonic devices, which use the particle quality of light and the wave quality of matter, and have a wealth of potential applications in sensing, lasing, and quantum computing. This merging of fundamental condensed matter theory with application-rich electromagnetism (and a splash of quantum optics thrown in) gives rise to a fascinating area of modern physics that is still very much in its infancy. In this review, we discuss and compare the key models and experiments used to explore how the quantum nature of electrons impacts plasmonics in the context of quantum size corrections of localized plasmons and quantum tunneling between nanoparticle dimers. We also look at some of the remarkable experiments that are revealing the quantum nature of surface plasmon polaritons

High frequency homogenisation for elastic lattices

A complete methodology, based on a two-scale asymptotic approach, that enables the homogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distinguished from scalar lattices in that two or more types of coupled waves exist, even at low frequencies. Such a theory enables the determination of effective material properties at both low and high frequencies. The theoretical framework is developed for the propagation of waves through lattices of arbitrary geometry and dimension. The asymptotic approach provides a method through which the dispersive properties of lattices at frequencies near standing waves can be described; the theory accurately describes both the dispersion curves and the response of the lattice. The leading order solution is expressed as a product between the standing wave solution and the long-scale envelope functions that are eigensolutions of the homogenised partial differential equation. The general theory is supplemented by a pair of illustrative examples for two archetypal classes of two-dimensional elastic lattices. The efficiency of the asymptotic approach in accurately describing several interesting phenomena is demonstrated, including dynamic anisotropy and Dirac cones. A significant advantage of the method exposited herein is that it allows one to obtain analytical expressions for the leading order asymptotic solutions and effective material properties, even for complex systems; the methodology allows for greater physical understanding of the behaviour of the system than is usually the case for purely numerical homogenisation schemes

Infinite product expansion of the Fokker-Planck equation with steady-state solution

We present an analytical technique for solving Fokker–Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum. This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour. The efficacy of the technique is demonstrated via comparisons with computations of typical examples

Building on Oldroyd’s viscoplastic legacy: Perspectives and new developments

The decade following the second world war heralded the publication of a collection of important papers on non-Newtonian fluid mechanics; Oldroyd’s work featured heavily in this collection. Not only did these articles establish important results, but Oldroyd’s style and methods set the scene for subsequent work in the area, exploiting mathematical analysis to formulate problems, establish results and guide further research. While Oldroyd’s name will forever be linked with the study of elastic fluids, the purpose of the present paper is to offer a modern perspective on a number of Oldroyd’s papers on viscoplastic fluids from 1947–1951 [1], [2], [3], [4], [5], [6], [7], [8]. Along the way, we sprinkle in a brief review of some of the subsequent developments stemming from Oldroyd’s advances, together with a few new results guided by his work. Following the approach of most of Oldroyd’s original papers, we focus on unidirectional flow down conduits. In an Appendix, we complement this discussion with a lubrication analysis, extending, clarifying and correcting the important original analysis of Walton and Bittleston (1991) [9]; although lubrication theory was not directly utilized by Oldroyd, the methodology aligns with his philosophy of using asymptotic and analytical approaches

Dynamics of spreading thixotropic droplets

The e ect of thixotropy on the two-dimensional spreading of a sessile drop is modelled using lubrication theory. Thixotropy is incorporated by the inclusion of a structure parameter, , measuring structure build-up governed by an evolution equation linked to the droplet micromechanics. A number of models are derived for coupled to the interface dynamics; these range from models that account for the cross-stream dependence of to simpler ones in which this dependence is prescribed through appropriate closures. Numerical solution of the governing equations show that thixotropy has a profound e ect on the spreading characteristics; the long-time spreading dynamics, however, are shown to be independent of the initial structural state of the droplet. We also compare the predictions of the various models and determine the range of system parameters over which the simple models provide su ciently good approximations of the full, two-dimensional spreading dynamics