As part of my work in the TSM CDT, I have recently collaborated with Antonio Fernandez-Dominguez, Andrew Horsfield, John Pendry and Stefan Maier to develop a nonlocal wave solution for nanofocusing of optical energy on metallic tips, work that has been published this week in Nano Letters. 
Localisation of optical energy into sub-wavelength volumes is currently an active area of research in the field of nano-optics. The challenge arises from the fundamental limitation that a beam of visible light, propagating in free space, can not be focused in a spot that is smaller than its wavelength (several hundreds of nanometers). On the other hand, many chemical processes occur at molecular length scales (which span several tens of nanometers or less). From an experimental point of view, a focusing scheme for delivery of optical energy into these nanoscale volumes would be highly desirable. If achieved, such a scheme could also be reversed, so that it would act as a sensor for electromagnetic signals with nanometer accuracy.
From the point of view of multi scale materials theory and simulation, the problem of nano-focusing poses an interesting challenge. Transport of energy begins at a macroscopic scale, where Maxwellian electrodynamics applies, however it then proceeds to length scales spanning only a few atoms, a regime where quantum and nonlocal effects govern the propagation of electromagnetic waves.
Figure 1: Schematic illustration of a metallic tip geometry suitable for nanofocusing. The contour plot shows the radial electric field component at the sharp end of the tip.
Research into geometries suitable for nanofocusing has progressed rapidly over the past decades. In 1997, Nerkararyan  showed that light waves propagating on the surface of a metal wedge can be concentrated in a nanoscopic ‘hot spot’ at the sharp end of a metallic wedge. In 2004, Stockman  proposed a similar scheme based on the use of a metallic tip (see schematic illustration in Figure 1). This tip structure has since seen rapid experimental uptake, for example De Angelis et al. have used it in 2009  to construct an atomic force microscopy (AFM) tip that not only provides information about the three dimensional morphology of an object, but simultaneously acts as a chemical sensor that provides information about the local chemical makeup of its surface.
Despite this rapid progress, many experimental and theoretical questions in the field of nanofocusing remain unanswered. For example, previous theoretical descriptions of nanofocusing cones were based on local solutions of Maxwell’s equations, which restricted the validity of the solutions to distances many tens of nanometers away from the sharp end of metallic tips. Thus, the need for a nonlocal description to the problem was recognised as early as 2004 , as such a description would extend the applicability of the solution to the sub-nanometer lengtscales which are crucial for nanofocusing. Another unanswered question relates to the effects of surface roughness. Experimentally produced metallic surfaces are never perfectly flat, but instead show an inevitable surface modulation. However, when electromagnetic surface waves propagate towards the sharp end of a metallic tip, their wavelength compresses to such an extent that it begins to approach the dimension of this surface roughness. The sub nanometer lengscale involved in the description of such surfaces again necessitates a nonlocal treatment.
The multi-scale nature of the above questions meant that the doctoral training centre in theory and simulation of materials was ideally placed to make a contribution. Thus, CDT student Aeneas Wiener was able to draw from the expertises of three research groups at Imperial College London, who each offered crucial ingredients towards a solution of the above questions. Andrew Horsfield provided knowledge of electronic structure theory, the group of Stefan Maier contributed experience in computational electrodynamics) and John Pendry, together with postdoctoral researcher Antonio Fernandez-Dominguez, offered their expertise in analytical solutions of Maxwell’s equations in complex geometries.
In a recent study  these authors have developed a solution to the problem of wave propagation on metallic tips which takes into account the effects of nonlocality. This means that the solution is valid even for sub-nanometer length scales, which makes it suitable for the investigation of surface roughness effects. In the local description of Maxwell’s equations, surface roughness drastically reduces the intensity of the hot spot that can be achieved at the end of a metallic tip. Interestingly, the authors find that effects of nonlocality mitigate this adverse influence of surface roughness, restoring the amplitude of the hot spot to a larger value than would be expected from local solutions of Maxwell’s equations. These insights into the effects of nonlocality on electromagnetic surface waves are not limited to metallic tips, but apply equally to other devices which rely on the propagation of electromagnetic energy on metallic surfaces.
 De Angelis, F., Das, G., Candeloro, P., Patrini, M., Galli, M., Bek, A., Lazzarino, M., et al. (2009). Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons. Nature Nanotechnology, 5(1), 67–72. doi:10.1038/nnano.2009.348