Asymptotic solutions of the plasmonic eigenvalue problem and applications

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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.