On band gaps in photonic crystal fibers

We consider the Maxwell's system for a periodic array of dielectric `fibers' embedded into a `matrix', with respective electric permittivities $\epsilon_0$ and $\epsilon_1$, which serves as a model for cladding in photonic crystal fibers (PCF). The interest is in describing admissible and forbidden (gap) pairs $(\omega,k)$ of frequencies $\omega$ and propagation constants $k$ along the fibers, for a Bloch wave solution on the cross-section. We show that, for "pre-critical" values of $k(\omega)$ i.e. those just below $\omega (\min\{\epsilon_0,\epsilon_1\}\mu)^{1/2}$ (where $\mu$ is the magnetic permeability assumed constant for simplicity), the coupling specific to the Maxwell's systems leads to a particular partially degenerating PDE system for the axial components of the electromagnetic field. Its asymptotic analysis allows to derive the limit spectral problem where the fields are constrained in one of the phases by Cauchy-Riemann type relations. We prove related spectral convergence. We finally give some examples, in particular of small size "arrow" fibers ($\epsilon_0>\epsilon_1$) where the existence of the gaps near appropriate "micro-resonances" is demonstrated by a further asymptotic analysis.Comment: 36 pages, 1 figur

Fibre Homogenisation

In this article we present a novel method for studying the asymptotic behaviour, with order-sharp error estimates, of the resolvents of parameter-dependent operator families. The method is applied to the study of differential equations with rapidly oscillating coefficients in the context of second-order PDE systems and the Maxwell system. This produces a non-standard homogenisation result that is characterised by `fibre-wise' homogenisation of the related Floquet-Bloch PDEs. These fibre-homogenised resolvents are shown to be asymptotically equivalent to a whole class of operator families, including those obtained by standard homogenisation methods

Two-Scale Homogenisation of Partially Degenerating PDEs with Applications to Photonic Crystals and Elasticity

In this thesis we study elliptic PDEs and PDE systems with e-pcriodic coeffi- cients, for small E, using the theory of two-scale homogenisation. We study a class of PDEs of partially degenerating type: PDEs with coefficients that are not uniformly elliptic with respect to E, and become degenerate in the limit E -t O. We review a recently developed theory of homogenisation for a general class of partially degenerating PDEs via the theory of two-scale convergence, and study two such problems from physics. The first problem arises from the study of a linear elastic composite with periodically dispersed inclusions that are isotropic and (soft' in shear: the shear modulus is of order E2. By passing to the two- scale limit as E -t 0 we find the homogenised limit equations to be a genuinely two-scale system in terms of both the macroscopic variable x and the micro- scopic variable y. We discover that the corresponding two-scale limit solutions must satisfy the incompressibility condition in y and therefore the composite only undergoes microscopic deformations when a (microscopically rotational' force is applied. We analyse the corresponding limit spectral problem and find that, due to the y-incompressibility, the spectral problem is an uncoupled two-scale prob- lem in terms of x and y. This gives a simple representation of the two-scale limit spectrum. We prove the spectral compactness result that states: the spectrum of the original operator converges to the spectrum of the limit operator in the sense of Hausdorff. The second problem we study is the propagation of electro- magnetic waves down a photonic fibre with a periodic cross section. We seek solutions to Maxwell's equations, propagating down the waveguide with wave number k E2-close to some (critical' value. In this setting, Maxwell's equations are reformulated as a partially degenerating PDE system with z-periodic coeffi- cients. Using the theory of homogenisation we pass to the limit as E -t 0 to find a non-standard two-scale homogenised limit and prove that the spectral compact- ness result holds. We finally prove that there exist gaps in the limit spectrum for two particular examples: a one-dimensionally periodic 'multilayer ' photonic crystal and a two-dimensionally periodic two-phase photonic crystal with the in- clusion phase consisting of arbitrarily small circles. Therefore, we prove that these photonic fibres have photonic band gaps for certain k.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Uniform asymptotics for a family of degenerating variational problems and error estimates in homogenisation theory

We study an abstract family of asymptotically degenerating variational problems. Those are natural generalisations of families of problems emerging upon application of a rescaled Floquet-Bloch-Gelfand transform to resolvent problems for highly oscillatory high-contrast elliptic PDEs. An asymptotic analysis of these problems leads us to a hierarchy of approximation results with uniform operator-type error estimates under various assumptions, satisfied by specific examples. Associated spectral problems are considered, and we provide approximations of the spectrum in terms of the spectrum of a certain `bivariate' operator which appears an abstract generalisation of the two-scale limit operators for highly oscillatory high-contrast PDEs. An explicit description of the limit spectrum in the abstract setting is provided, and tight error estimates on the distance between the original and limit spectra are established. Our generic approach allows us to readily consider a wide class of asymptotically degenerating problems including but also going beyond high-contrast highly oscillatory PDEs. The obtained results are illustrated by various examples

Flip, feedback and fly: Using LOOP to Enhance the Professional Experience of Initial Teacher Education

The Australian Professional Teaching Standards require pre-service teachers to complete a minimum number of days of professional experience in order to graduate. Problems can arise, however, when the evaluation of their professional experience against the Standards shifts from the providers of teacher education programmes to school-based supervising teachers. The Lesson Observation On-line Platform (LOOP) begins to address these problems by utilising a secure, shared digital platform to facilitate evidence-based evaluation of the performance of pre-service teachers. In this research, we evaluated the potential of LOOP to assess pre-service teachers against the Standards as well as to enhance the professional development of both pre-service teachers and their supervising teachers. The responses from two pre-service teachers and their supervising teachers demonstrate that the methodological matters can be easily overcome. Nevertheless our findings indicate that there are several practical issues that need to be overcome if LOOP were to be fully successful

Homogenization Techniques for Periodic Structures

International audienceWe describe a selection of mathematical techniques and results that suggest interesting links between the theory of gratings and the theory of homogenization, including a brief introduction to the latter. We contrast the "classical" homogenization, which is well suited for the description of composites as we have known them since their advent until about a decade ago, and the "non-standard" approaches, high-frequency homogenization and high-contrast homogenization, developing in close relation to the study of photonic crystals and metamaterials, which exhibit properties unseen in conventional composite media, such as negative refraction allowing for super-lensing through a flat heterogeneous lens, and cloaking, which considerably reduces the scattering by finite size objects (invisibility) in certain frequency range. These novel electromagnetic paradigms have renewed the interest of physicists and applied mathematicians alike in the theory of gratings