In many contexts of physics, engineering, and materials science, one is faced with the task of understanding and quantifying the behaviour of a mixture ("composite") of a large number of individual material components, which could have, for example, acoustic or electromagnetic properties. To address this task, many analytical and computational approaches have been developed, working well under some conditions on the geometric and material properties of the components of the composite but providing poor approximations when these conditions are relaxed. The challenge to widen the range of media amenable to analytical approaches has driven the wider subject area of mathematical homogenisation, where new methods have to be developed for dealing with problems that do not fall into the existing frameworks.
One of the contexts where new approaches are called for are composite media with components that "resonate" with a wave motion taking place in the medium, that is those situations when the wavelength (wave speed times the temporal period) in some of the constituent elements of the medium is comparable to the actual size of these elements. The existing approximations, based on a standard "homogenisation" rationale, are unable to capture the essential features of interactions between the medium and the wave motion in such cases, as the related averaging techniques work under the assumption of smallness between the size of heterogeneity and associated wavelength.
The project will provide a quantitative framework for electromagnetic composites where some of the components exhibits resonant properties, this leading to non-classical behaviour on the macro-scale. We will study the case of conducting (for example, metallic) inclusions in a dielectric medium, which is important for obtaining an insight into the behaviour of composite materials with unusual, counter-intuitive properties (such as "negative" refraction), the key ingredient of many active devices and functional materials. Such materials have recently been the focus of ground-breaking physics experiments (for example, those involving "split-ring" resonators), although little is known about their dispersive properties (i.e. the dependence of the wave-speed on the frequency). Although qualitative progress has been made in recent years, quantitative mathematical formulations for composites with conducting components are beyond the reach of the existing methods of mathematical theory of homogenisation (used to analyse the overall properties of composites), due to the intrinsic dispersive nature of conductors.
Inspired by recent advances in applications of operator theory to the asymptotic analysis of boundary-value problems of mathematical physics and the subsequent quantitative results for composites with arbitrary micro-geometries, we will develop a mathematical framework for inhomogeneous periodic media with frequency-dependent interface conditions and will use it for an explicit derivation of effective properties of composites with conducting inclusions, with a sharp control of the approximation error. We will then provide quantitatively sharp formulae for frequency dispersion in photonic crystals with conducting inclusions, opening up new avenues for on-demand metallic photonic fibre design with explicit control of bandgap propagation for a wide range of frequencies.