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Asymptotic solutions of the plasmonic eigenvalue problem and applications

等离子体特征值问题的渐近解及其应用

基本信息

批准号：
EP/R041458/1
负责人：
Ory Schnitzer
金额：
$ 24.86万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FR041458%2F1
关键词：
Asymptotic solutions plasmonic eigenvalue problem

项目摘要

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.

应用物理学的一项重大努力，对生物传感、医疗、可再生能源和纳米技术产生了深远的影响，就是设计出在纳米尺度上有效操纵光的方法。关键的挑战是在与自由空间中的传播波长相比小的尺度上工作，这似乎与光学设备的基本界限相矛盾。近年来取得了显着进展，特别是在纳米等离子体领域，利用金属在可见频率下独特的光学性质来引导、限制和增强电磁能。该领域是成熟的应用数学，以促进基本的建模，我们提出了一个新的框架，首次利用和阐明的奇点等离子现象的基础。我们的目标是改革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，我们将为等离子体学开发的新数学技术将具有更广泛的适用性，并增强应用数学家和物理学家之间的协同作用。当表面等离子体激元，即集体电子电荷和电场振荡，在金属和电介质之间的界面处被激发时，发生等离子体激元现象。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。接近某些“自然频率”，外部辐射能够最佳地将能量转移到局部等离子体中，导致共振响应，其中吸收，散射和结构周围的电场被增强。这些增强可以通过使用接近奇异的纳米几何形状（例如，紧密间隔的颗粒和细长的纳米棒）而变得特别显著。在过去的二十年里，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，纳米结构的特设数值模拟，假设特定的几何形状，材料，频率和外部辐射源，往往缺乏洞察力，是低效的，当探索一个大的参数空间。或者，LSPR可以根据纳米结构支持的自然表面等离子体模式进行阐明和有效研究，类似于分析拉伸弦的驻波谐波。然而，与弦的类比不同，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受尺度和材料不变的“等离子体激元本征值问题”支配，只涉及结构的形状。因此，等离子体激元本征值问题是等离子体现象建模和解释的关键。然而，解析解是罕见的，通常是繁琐的，而它是很难计算推断的无限模式，特别是对于近奇异和多尺度几何无处不在的应用。该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“奇异摄动”技术，以精确解决传统方法难以解决的极端情况，或掩盖细节背后的主导物理学。具体来说，我们将获得“渐近”近似变得更加准确和简单的形式，几何形状变得多尺度或频谱变得密集。特别是，在前一个极限中，我们将推导出表征紧密间隔粒子和细长粒子的极端等离子体响应的基本标度和渐近公式（例如幂律定律）;在后一个极限中，我们将开发一种类似于射线光学的方法-一种新的局部等离子体的几何解释-产生类似于量子力学中产生的表面等离子体量子化规则。