# Plasmon resonances and the Neumann-Poincare operator for 3D domains with singularities

Lead Research Organisation:
University of Reading

Department Name: Mathematics and Statistics

### Abstract

A surface plasmon resonance is an oscillation of conduction electrons at the boundary between two different media, stimulated by incident light. Surface plasmons are responsible for the vivid colours of stained glass. Nanoparticles of an impurity such as gold are suspended in the glass, which leads to the plasmonic absorption of certain wavelengths of light, thus changing its colour. Surface plasmon resonances are utilized to enhance, confine, and guide electromagnetic fields, with applications in sensing, spectroscopy, microscopy, and display technology.

The colour of a metallic nanoparticle depends on its size and its shape. For instance, spherical silver nanoparticles are usually associated with yellow stained glasses, but 100 nanometer triangular silver particles give off a red hue instead. The sharp corners of triangular particles give rise to unique plasmonic phenomena. Certain constellations of particles with sharp features exhibit very strong effects of field enchancement and confinement.

The proposed research aims to lay the mathematical foundations of surface plasmon resonances for 3D nanoparticles with corners and edges. The mathematical theory of such particles is very subtle; an unusual feature is that the size and smoothness imposed on electromagnetic fields has drastic impact on the theory's conclusions. The presence of corners and edges determines fundamental aspects of the so-called spectral theory of the plasmonic problem, irrespective of the overall shape of the particle. Even smooth input leads to wildly oscillating and singular fields at corners, related to the formation of surface plasmon waves on the particle. For the simplest type of singularity in 3D, a rotationally symmetric corner point, we have already shown that for certain wavelengths and materials, it is possible to count the number of resonant solutions (eigensolutions) of a certain size and smoothness, solely based on the existence of a corner.

To develop the mathematics of the plasmonic problem, I will consider a reformulation in terms of an integral operator known as the Neumann-Poincare (NP) operator. The NP operator has been studied for over a century, but, to this day, continues to generate mathematical novelties and surprises. One advantage of using the NP operator is that it allows for direct use of the very extensive theory of singular integral operators that has been developed since the 1970s. The NP operator is also very interesting in and of itself, serving as a prominent example in non-selfadjoint spectral theory.

To study the spectral theory for a polyhedron, such as a cube or octahedron, I will begin by studying a model problem formed by considering a single corner and extending it to infinity. The first objective of the proposal is to develop the spectral theory for such polyhedral cones, as well as for models of multi-particle systems with touching edges. This naturally leads to the second objective, which explores how model results can be transferred to the spectral theory for polyhedral particles and systems.

The third objective sets out to develop a mathematical framework for the ignition of surface plasmon waves along a particle with corners or edges. The hypothesis is that a 3D corner acts as a sort of waveguide for incoming light. The fourth and final objective investigates the spectral theory for particles featuring a high amount of symmetries. I expect to exhibit particles whose spectral pictures feature infinitely many resonances embedded in a continuum, but also to find other new and exciting phenomena.

The colour of a metallic nanoparticle depends on its size and its shape. For instance, spherical silver nanoparticles are usually associated with yellow stained glasses, but 100 nanometer triangular silver particles give off a red hue instead. The sharp corners of triangular particles give rise to unique plasmonic phenomena. Certain constellations of particles with sharp features exhibit very strong effects of field enchancement and confinement.

The proposed research aims to lay the mathematical foundations of surface plasmon resonances for 3D nanoparticles with corners and edges. The mathematical theory of such particles is very subtle; an unusual feature is that the size and smoothness imposed on electromagnetic fields has drastic impact on the theory's conclusions. The presence of corners and edges determines fundamental aspects of the so-called spectral theory of the plasmonic problem, irrespective of the overall shape of the particle. Even smooth input leads to wildly oscillating and singular fields at corners, related to the formation of surface plasmon waves on the particle. For the simplest type of singularity in 3D, a rotationally symmetric corner point, we have already shown that for certain wavelengths and materials, it is possible to count the number of resonant solutions (eigensolutions) of a certain size and smoothness, solely based on the existence of a corner.

To develop the mathematics of the plasmonic problem, I will consider a reformulation in terms of an integral operator known as the Neumann-Poincare (NP) operator. The NP operator has been studied for over a century, but, to this day, continues to generate mathematical novelties and surprises. One advantage of using the NP operator is that it allows for direct use of the very extensive theory of singular integral operators that has been developed since the 1970s. The NP operator is also very interesting in and of itself, serving as a prominent example in non-selfadjoint spectral theory.

To study the spectral theory for a polyhedron, such as a cube or octahedron, I will begin by studying a model problem formed by considering a single corner and extending it to infinity. The first objective of the proposal is to develop the spectral theory for such polyhedral cones, as well as for models of multi-particle systems with touching edges. This naturally leads to the second objective, which explores how model results can be transferred to the spectral theory for polyhedral particles and systems.

The third objective sets out to develop a mathematical framework for the ignition of surface plasmon waves along a particle with corners or edges. The hypothesis is that a 3D corner acts as a sort of waveguide for incoming light. The fourth and final objective investigates the spectral theory for particles featuring a high amount of symmetries. I expect to exhibit particles whose spectral pictures feature infinitely many resonances embedded in a continuum, but also to find other new and exciting phenomena.

### Planned Impact

The proposal describes a theoretical project in pure mathematics, the immediate impact of which is primarily academic, affecting wide groups of researchers in mathematical and numerical analysis, mathematical physics, electromagnetic theory, materials science, and plasmonics. At the same time, the discovery and design of highly resonant plasmonic structures is important to applications with industrial potential, in the areas of imaging, sensing, and display technology. Particles with edges and corners are often suggested to enable such applications, owing to the extraordinary levels of field enhancement that they can present. I therefore envision that the proposed foundational research will serve as guidance in future design of plasmonic devices.

1. Numerical modelling of surface plasmon resonances

For most applications, further numerical simulations are likely needed for the design of appropriate plasmonic particles and structures, both within academic and industrial contexts. The outcomes of the research will be useful for the construction of robust numerical solvers for the integral equations associated with the plasmonic problem, for particles and systems with corners and edges. For permitivitties associated with plasmon resonances, the sought electromagnetic fields are highly oscillatory towards singularities, and thus difficult to resolve. The proposed research will yield an a priori description of the behaviour of these fields, and thus of bases of solutions for the plasmonic problem. This information can be encoded into future solvers, to improve their efficiency, accuracy, and capability.

2. Surface-enhanced Raman spectroscopy

Surface-enhanced Raman spectroscopy (SERS) is enabled by a metallic surface which is nanoscale rough, created for example by deposition of metal nanoparticles. According to the plasmonic explanation of SERS, surface plasmon resonances are responsible for field enchancements which are so strong that SERS can detect even single molecules. Therefore, SERS has the potential to be used for detection of early cancer biomarkers, embedding nanotags into bank notes, and food quality analysis, to mention just a few applications. The proposed research is well positioned to have long term influence on the community of physicists designing SERS substrates, as it explains the underlying mechanisms by which particles with corners and edges exhibit dramatic effects of field enhancement.

3. Transparent monitors

Another appealing application is to the construction of low-cost transparent monitors by embedding nanoparticles in a transparent medium. The goal is to construct particles which convert ultraviolet or infrared light into the visible spectrum. An ultraviolet or infrared source of light is thus invisible to the human eye, but the light scattered by the nanoparticles is not. To create coloured displays reconcilable with modern fabrication tolerances, it has recently been suggested by researchers in electromagnetic theory to use nanoparticles in shapes approaching that of a cube, for their ability to carry surface plasmon resonances at longer wavelengths. For particles which have a smoothed "corner" or "edge" which is very close to being sharp, the proposed research will yield a very good approximation of the resonant wavelengths that arise from the corner or edge feature alone, assuming a good model for the wavelength-dependent permittivity of the particle material. This information plays a part in the design of nanoparticles purposed for the construction of coloured displays with wide viewing angles and a high degree of transparency, at a very low material cost.

Furthermore, the problems posed by contemporary plasmonics research are often very mathematical in nature. It is therefore of relevance to build a mathematics community which has the right expertise, is aware of modern developments, and is connected to the wider plasmonic community. The project will be a step in this direction.

1. Numerical modelling of surface plasmon resonances

For most applications, further numerical simulations are likely needed for the design of appropriate plasmonic particles and structures, both within academic and industrial contexts. The outcomes of the research will be useful for the construction of robust numerical solvers for the integral equations associated with the plasmonic problem, for particles and systems with corners and edges. For permitivitties associated with plasmon resonances, the sought electromagnetic fields are highly oscillatory towards singularities, and thus difficult to resolve. The proposed research will yield an a priori description of the behaviour of these fields, and thus of bases of solutions for the plasmonic problem. This information can be encoded into future solvers, to improve their efficiency, accuracy, and capability.

2. Surface-enhanced Raman spectroscopy

Surface-enhanced Raman spectroscopy (SERS) is enabled by a metallic surface which is nanoscale rough, created for example by deposition of metal nanoparticles. According to the plasmonic explanation of SERS, surface plasmon resonances are responsible for field enchancements which are so strong that SERS can detect even single molecules. Therefore, SERS has the potential to be used for detection of early cancer biomarkers, embedding nanotags into bank notes, and food quality analysis, to mention just a few applications. The proposed research is well positioned to have long term influence on the community of physicists designing SERS substrates, as it explains the underlying mechanisms by which particles with corners and edges exhibit dramatic effects of field enhancement.

3. Transparent monitors

Another appealing application is to the construction of low-cost transparent monitors by embedding nanoparticles in a transparent medium. The goal is to construct particles which convert ultraviolet or infrared light into the visible spectrum. An ultraviolet or infrared source of light is thus invisible to the human eye, but the light scattered by the nanoparticles is not. To create coloured displays reconcilable with modern fabrication tolerances, it has recently been suggested by researchers in electromagnetic theory to use nanoparticles in shapes approaching that of a cube, for their ability to carry surface plasmon resonances at longer wavelengths. For particles which have a smoothed "corner" or "edge" which is very close to being sharp, the proposed research will yield a very good approximation of the resonant wavelengths that arise from the corner or edge feature alone, assuming a good model for the wavelength-dependent permittivity of the particle material. This information plays a part in the design of nanoparticles purposed for the construction of coloured displays with wide viewing angles and a high degree of transparency, at a very low material cost.

Furthermore, the problems posed by contemporary plasmonics research are often very mathematical in nature. It is therefore of relevance to build a mathematics community which has the right expertise, is aware of modern developments, and is connected to the wider plasmonic community. The project will be a step in this direction.