Quantitative tools for upscaling the micro-geometry of resonant media

在物理、工程和材料科学的许多背景下，人们面临着理解和量化大量单独材料成分的混合物（“复合材料”）的行为的任务，这些成分可能具有，例如，声学或电磁特性。为了解决这个问题，已经开发了许多分析和计算方法，在某些条件下，复合材料组成部分的几何和材料特性工作得很好，但当这些条件放松时，提供的近似值很差。扩大适用于分析方法的媒介范围的挑战已经推动了数学同质化这一更广泛的学科领域，在这一领域，必须开发新的方法来处理不属于现有框架的问题。需要新方法的一种情况是，具有与介质中发生的波动“共振”的成分的复合介质，即当介质的某些组成元素的波长（波速乘以时间周期）与这些元素的实际尺寸相当时。现有的近似，基于标准的“均质化”理论，在这种情况下无法捕捉到介质和波动之间相互作用的基本特征，因为相关的平均技术是在假设非均质尺寸和相关波长之间较小的情况下工作的。该项目将为电磁复合材料提供一个定量框架，其中一些组件表现出共振特性，从而导致宏观尺度上的非经典行为。我们将研究介电介质中导电（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）的复合材料的行为非常重要，这是许多有源器件和功能材料的关键成分。这些材料最近成为突破性物理实验的焦点（例如，那些涉及“分裂环”谐振器的实验），尽管人们对它们的色散特性（即波速与频率的依赖关系）知之甚少。尽管近年来在定性方面取得了进展，但由于导体固有的色散性质，具有导电成分的复合材料的定量数学公式超出了均匀化数学理论（用于分析复合材料的整体性能）的现有方法的范围。受算子理论应用于数学物理边值问题渐近分析的最新进展以及具有任意微观几何形状的复合材料的后续定量结果的启发，我们将开发具有频率依赖界面条件的非均匀周期介质的数学框架，并将使用它来显式推导具有导电内含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，我们将提供具有导电内含物的光子晶体中频率色散的定量精确公式，为按需金属光子光纤设计开辟新的途径，明确控制宽频率范围内的带隙传播。

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.