Asymptotic solutions of the plasmonic eigenvalue problem and applications

应用物理学的一项重大努力是设计在纳米长度尺度上有效操纵光的方法，这对生物传感、医疗、可再生能源和纳米技术具有深远的影响。关键的挑战是要在比自由空间中传播波长更小的尺度上工作，这似乎与光学仪器的基本界限相矛盾。近年来取得了显著的进展，特别是在纳米等离子体学领域，利用金属在可见频率下独特的光学特性来引导、限制和增强电磁能。该领域已经成熟，应用数学可以为基础建模做出贡献，我们提出了一个新的框架，首次利用并阐明了支撑等离子体现象的奇点。我们的目标是改革对等离子体共振的基本理解，并为实验解释、初步设计和优化开发新的工具。反过来，我们将为等离子体学开发的新数学技术将具有更广泛的适用性，并增加应用数学家和物理学家之间的协同作用。当表面等离子体，即集体电子-电荷和电场振荡，在金属和介电介质之间的界面被激发时，就会发生等离子体现象。金属纳米粒子和纳米结构允许将这种振荡定位到纳米尺度的体积。在接近某些“固有频率”时，外部辐射能够将能量最佳地转移到局部等离子激元中，从而产生共振响应，从而增强结构周围的吸收、散射和电场。通过使用近乎单一的纳米金属几何形状，例如紧密间隔的颗粒和细长的纳米棒，可以使这些增强效果特别显著。近二十年来广泛的实验和理论研究表明，局域表面等离子体共振（LSPR）现象非常丰富。因此，纳米金属结构的临时数值模拟，假设特定的几何形状、材料、频率和外部辐射源，在探索大参数空间时往往缺乏洞察力和效率低下。或者，LSPR可以根据纳米结构支持的自然表面等离子体模式来解释和有效地研究，类似于根据其驻波谐波来分析拉伸弦的声音。然而，与弦的类比不同，表面等离子激元的频率几乎与尺寸无关；事实上，表面等离子体模式是由一个尺度和材料不变的“等离子体特征值问题”控制的，只涉及结构的形状。因此，等离子体特征值问题是模拟和解释等离子体现象的关键。然而，解析解是罕见的，通常是繁琐的，而计算推断模态的无限是困难的，特别是对于应用中普遍存在的近奇异和多尺度几何。这个项目提供了一个全新的理论方法；我们建议从应用数学中创新“奇异摄动”技术，以精确解决那些传统方法难以解决的极端情况，或者掩盖了细节背后的主要物理。具体地说，我们将获得“渐近”近似，随着几何形状变得多尺度或谱变得密集，在形式上变得更加精确和简单。特别地，在前一种极限中，我们将推导出基本的标度和渐近公式（例如幂律）来表征紧密间隔粒子和细长粒子的极端等离子体响应；在后一种情况下，我们将开发一种类似于射线光学的方法——局域等离子体的一种新的几何解释——产生类似于量子力学中产生的表面等离子体量子化规则。

A major endeavour in applied physics, carrying far-reaching ramifications for bio-sensing, medical treatment, renewable energy and nanotechnology, is to devise methods to effectively manipulate light on nanometric length scales. The key challenge is to work on scales small compared to the wavelength of propagation in free space, seemingly in contradiction with fundamental bounds on optical apparatus. Remarkable progress has been made in recent years, particularly in the field of nanoplasmonics, where the unique optical properties of metals at visible frequencies are exploited to guide, confine and enhance electromagnetic energy. The field is ripe for applied mathematics to contribute to fundamental modelling and we propose a novel framework that for the first time exploits and elucidates the singularities underpinning plasmonic phenomena. Our goal is to reform fundamental understanding of plasmonic resonance and develop new tools for interpretation of experiments, preliminary design and optimisation. In turn, the new mathematical techniques we will develop for plasmonics will have wider applicability and increase synergy between applied mathematicians and physicists. Plasmonic phenomena occur when surface plasmons, namely collective electron-charge and electric-field oscillations, are excited at an interface between a metal and a dielectric. Metallic nanoparticles and nanostructures allow localising such oscillations to nanoscale volumes. Close to certain "natural frequencies", external radiation is able to optimally transfer energy into the localised plasmons, resulting in a resonant response where absorption, scattering and the electric field around the structure are enhanced. These enhancements can be made particularly significant by using near-singular nanometallic geometries, e.g. closely spaced particles and elongated nanorods. Extensive experimental and theoretical research over the last two decades has demonstrated that the phenomenon of localised-surface-plasmon resonance (LSPR) is extremely rich. Accordingly, ad hoc numerical simulations of nanometallic structures, which assume specific geometries, materials, frequencies and external sources of radiation, often lack insight and are inefficient when exploring a large parameter space. Alternatively, LSPR can be elucidated and efficiently studied in terms of the natural surface-plasmon modes supported by the nanostructure, akin to analysing the sound of a stretched string in terms of its standing-wave harmonics. Unlike in the string analogy, however, surface-plasmon frequencies are nearly independent of size; in fact, surface-plasmon modes are governed by a scale- and material-invariant "plasmonic-eigenvalue problem", involving just the structure's shape.The plasmonic eigenvalue problem is therefore key to modelling and interpretation of plasmonic phenomena. Nevertheless, analytical solutions are rare and typically cumbersome, while it is difficult to computationally infer the infinity of modes, especially for the near-singular and multiple-scale geometries ubiquitous in applications. This project offers a completely new theoretical approach; we propose to innovate "singular-perturbation" techniques from applied mathematics to resolve exactly those extreme situations where conventional methods struggle, or mask the dominant physics behind details. Specifically, we will obtain "asymptotic" approximations becoming more accurate and simple in form as the geometry becomes multi-scale or as the spectrum becomes dense. In particular, in the former limit we will derive fundamental scalings and asymptotic formulae (e.g. power laws) characterising the extreme plasmonic response of closely spaced particles and elongated particles; in the latter limit we will develop a method akin to ray-optics - a new geometric interpretation of localised plasmons - yielding surface-plasmon quantisation rules analogous to those arising in quantum mechanics.