Quantitative tools for upscaling the micro-geometry of resonant media

在物理学、工程学和材料科学的许多背景下，人们面临着理解和量化大量单个材料组分的混合物（“复合物”）的行为的任务，这些材料组分可能具有例如声学或电磁特性。为了解决这个问题，许多分析和计算方法已经开发出来，在某些条件下工作良好的复合材料的组件的几何和材料特性，但提供穷人的近似，当这些条件放宽。扩大适合分析方法的媒体范围的挑战推动了更广泛的数学均质化主题领域，其中必须开发新的方法来处理不属于现有框架的问题。 需要新方法的背景之一是具有与介质中发生的波动“共振”的成分的复合介质，也就是当介质的某些组成元素中的波长（波速乘以时间周期）与这些元素的实际大小相当时的情况。现有的近似，基于一个标准的“均质化”的理由，是无法捕捉的基本特征之间的相互作用的介质和波运动在这种情况下，作为相关的平均技术的假设下工作的异质性的大小和相关的波长之间的小。该项目将为电磁复合材料提供一个定量框架，其中一些组件具有谐振特性，这导致宏观尺度上的非经典行为。我们将研究在介电介质中的导电（例如，金属）夹杂物的情况，这对于深入了解具有不寻常的，反直觉性质（如“负”折射）的复合材料的行为是很重要的，复合材料是许多有源器件和功能材料的关键成分。这种材料最近已经成为突破性物理实验的焦点（例如，那些涉及“开口环”谐振器的实验），尽管对它们的色散特性（即波速对频率的依赖性）知之甚少。虽然近年来已经取得了质的进展，定量的数学公式的复合材料与导电组件超出了现有的方法的数学理论的均匀化（用于分析复合材料的整体性能），由于导体的固有色散性质。 受算子理论应用于数学物理边界值问题渐近分析的最新进展以及随后对具有任意微观几何形状的复合材料的定量结果的启发，我们将为具有频率依赖界面条件的非均匀周期性介质开发一个数学框架，并将其用于明确推导具有导电夹杂物的复合材料的有效性质，具有对近似误差的精确控制。然后，我们将提供定量尖锐的公式，在光子晶体中的频率色散与导电夹杂物，开辟了新的途径，按需金属光子光纤的设计与显式控制的带隙传播的频率范围很广。

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.