Asymptotic solutions of the plasmonic eigenvalue problem and applications

对生物传感、医疗、可再生能源和纳米技术产生深远影响的应用物理学的一项重大努力是设计出在纳米长度尺度上有效操纵光的方法。关键的挑战是在比在自由空间中传播的波长更小的尺度上工作，似乎与光学设备的基本界限相矛盾。近年来取得了显著的进展，特别是在纳米等离子体领域，金属在可见光频率下的独特光学性质被用来引导、限制和增强电磁能量。该领域对于应用数学对基本模型的贡献已经成熟，我们提出了一个新的框架，它首次利用和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，并开发新的工具来解释实验、初步设计和优化。反过来，我们将为等离子体激元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，即集体电荷和电场振荡，在金属和介质之间的界面上被激发时，就会产生等离子体现象。金属纳米颗粒和纳米结构允许将这种振荡定位到纳米级体积。在接近某些“自然频率”的情况下，外部辐射能够以最佳方式将能量转移到局域等离子体中，从而产生共振响应，从而增强结构周围的吸收、散射和电场。通过使用近乎奇异的纳米金属几何结构，例如紧密间隔的颗粒和细长的纳米棒，这些增强可以变得特别显著。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)现象极为丰富。因此，对纳米金属结构的特别数值模拟，假设特定的几何、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并且效率低下。或者，LSPR可以通过纳米结构支持的自然表面等离子激元模式来阐明和有效地研究，类似于根据其驻波谐波来分析拉伸的弦的声音。然而，与弦类比不同的是，表面等离子体激元的频率几乎与尺寸无关；事实上，表面等离子体激元模式是由尺度和材料不变的“等离子体本征值问题”控制的，只涉及结构的形状。因此，等离子体本征值问题是模拟和解释等离子体现象的关键。然而，解析解很少见，而且通常很繁琐，而且很难通过计算来推断无穷大的振型，特别是对于应用中普遍存在的近奇异和多尺度几何。这个项目提供了一种全新的理论方法；我们建议从应用数学中创新“奇异微扰”技术，以准确地解决那些传统方法难以解决的极端情况，或者掩盖细节背后的主导物理学。具体地说，当几何形状变得多尺度或谱变得密集时，我们将获得形式更加精确和简单的“渐近”近似。具体地说，在前一个极限中，我们将得到描述近距离粒子和拉长粒子的极端等离子体响应的基本标度和渐近公式(例如，幂定律)；在后一个极限中，我们将发展一种类似于光线光学的方法--一种对局域等离子体产生表面等离子体量化规则的新的几何解释，类似于量子力学中出现的那些规则。

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.