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Details of Grant 

EPSRC Reference: EP/S029486/1
Title: Plasmon resonances and the Neumann-Poincare operator for 3D domains with singularities
Principal Investigator: Perfekt, Dr K
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Louisiana State University Lund, University of University of Lorraine
Department: Mathematics and Statistics
Organisation: University of Reading
Scheme: New Investigator Award
Starts: 23 September 2019 Ends: 01 May 2021 Value (£): 231,308
EPSRC Research Topic Classifications:
Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
27 Feb 2019 EPSRC Mathematical Sciences Prioritisation Panel February 2019 Announced
Summary on Grant Application Form
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.
Key Findings
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