We give an explicit construction of a fundamental solution for an arbitrary non-degenerate partial differential equation with smooth coefficients.

Strongly localised plasmons in metallic nano-structures offer exciting characteristics for guiding and focusing light on the nano-scale, opening the way for the development of new types of sensors, circuitry and improved resolution of optical microscopy. The work presented in this thesis focuses on two major areas of plasmonics research - nano-focusing structures and nano-sized waveguides. Nano-focusing structures focus light to an area smaller than the wavelength and will find applications in sensing, efficiently coupling light to nano-scale devices, as well as improving the resolution of near field microscopy. In the past the majority of nano-focusing structures have been nano-scale cones or tips, which are capable of focusing light to a spot of nano-scale area whilst enhancing the light field. The alternatives are triangular nano-focusing structures which have received far less attention, and only one type of triangular nano-focusing structure is known – a sharp V-groove in a metal substrate. This structure focuses light to a strip of nano-scale width, which may lead to new applications in microscopy and sensing. The difficulty with implementing the V-groove is that the structure is not robust and is quite difficult to fabricate. This thesis aims to develop new triangular nano-focusing devices which will overcome these difficulties, whilst still producing an intense light source on the nano-scale. The two proposed structures presented in this thesis are a metallic wedge submerged in uniform dielectric and a tapered metal film lying on a dielectric substrate, the latter being the easier to fabricate and the more structurally sound and robust. The investigation is performed using the approximation of continuous electrodynamics, the geometrical optics approximation and the zero-plane method. The second aim of this thesis is to investigate plasmonic waveguides and couplers for the development of nano-optical circuitry, more compact photonic devices and sensors. The research will attempt to fill the gaps in the current knowledge of the V-groove waveguide, which consists of a sharp triangular groove in a metal substrate, and the gap plasmon waveguide, which consists of a rectangular slot in a thin metal film. The majority of this work will be performed using the author’s in house finite-difference time-domain algorithm and FEMLAB as well as the effective medium method and geometric optics approximation. The V-groove may be used as either a nano-focusing or waveguiding device. As a waveguide the V-groove is one of the most promising plasmonic waveguides in the optical regime. However, there exist quite a number of gaps in the current knowledge of V-groove waveguides which this thesis will attempt to fill. In particular, the effect of rounded groove tip on plasmon propagation has been assessed for the V-groove. The investigation of rounded groove tip is important, as due to modern fabrication processes it’s not possibly to produce an infinitely sharp groove, and the existing literature has not considered the impact of this problem. The thesis will also investigate the impacts of the inclusion of dielectric filling in the groove on plasmon propagation parameters. This research will be important for optimising the propagation characteristics of the mode for certain applications, but it may also lead to easier methods of fabricating the V-groove device and prevent oxidation of the metal film. The gap plasmon waveguide is easier to fabricate than the V-groove, and is a new type of sub-wavelength waveguide which displays many advantages over other types of plasmon waveguides, including ease of fabrication, almost 100% transmission around sharp bends, sub-wavelength localisation and long propagation distances of the guided mode, etc. This waveguide may prove invaluable in the development of compact photonic devices. In the past the modes supported by this structure were not thoroughly analysed and the possibility of using this structure to develop sub-wavelength couplers for sensing and nano-optical circuits was not considered in detail. This thesis aims to resolve these issues. In conclusion, the results of this thesis will lead to a better understanding of Vgroove and gap plasmon waveguide devices for the development of nano-optical circuits, compact photonic devices and sensors. This thesis also proposes two new nano-focusing structures which are easier to fabricate than the V-groove structure and will lead to applications in sensing, coupling light efficiently into nano-scale devices and improving the resolution of near-field microscopy.