In this thesis we provide several different systematic methods for constructing complex root spaces that remain invariant under an antilinear transformation. The first method is based on any element of the Weyl group, which is extended to factorizations of the Coxeter element and a reduced Coxeter element thereafter. An antilinear deformation method for the longest element of the Weyl group is given as well. Our last construction method leads to an alternative construction for q-deformed roots. For each of these construction methods we provide examples. In addition, we show a method of construction that for some special cases leads to rotations in the dual space and vice versa, starting from a rotation we find the root space involved. We then continue to apply these deformations to a generalized Calogero model and Affine Toda field theory. We provide a general solution for the ground state wave function of the Calogero model that is independent of a root representation and we extend this to the deformed case. An important property of this deformed Calogero model is that the amount of singularities in its potential is significantly reduced. We find that the exchange of particles in this model then leads to anyonic exchange factors. Following this we solve the model and find the ground state eigenvalues and eigenfunctions for the deformed Calogero model. We apply the q-deformed roots to an Affine Toda field theory and find that one may formulate a classical theory respecting the mass renormalisation of the quantum case.

This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.

Optical resonators – two mirrors facing each other, separated by a distance – are a very well studied technology. However, even such a well understood technology can sometimes present surprises. The first part of this thesis investigates the surprising properties of some canonical optical resonators. The basic properties of resonators are introduced. The imaging properties of stable and unstable resonators are examined. The second part of this thesis examines the potential use of grating- coupled cavities in gravitational wave detectors and describes an exper- iment carried out on a 10 m prototype of such a cavity.

With non-invasive properties and high sensitivities, portable optical biosensors are extremely desirable for point-of-care (POC) applications. Lab-on-a-chip technology such as microfluidics has been treated as an ideal approach to integrate complex sample processing and analysis units with optical detection elements. Spectroscopic sensing (such as fluorescence, Raman and absorption spectroscopy) remains the most highly developed, widely applied, optical technique. However, conventional spectroscopic sensing systems still rely on bulky and expensive dispersive components such as spectrophotometers in a well established laboratory. The work in this thesis is to develop an integrated dispersive component in combination with a microfluidic chip, providing a portable and inexpensive platform for on-chip spectroscopic sensing. In this study, an arrayed waveguide grating (AWG) design developed for telecommunication is re-engineered and utilized to realise a compact, dispersive optical component operating in the visible spectral region. The AWG devices operating in the visible region (λ_c=680 nm) are designed and fabricated with flame hydrolysis deposited (FHD) silica waveguide material. The micro-spectrometer in this proof of concept study has a small (1 cm x 1 cm) footprint and 8 output channels centred on different wavelengths. A series of fabrication issues and challenges are investigated and discussed for the specific AWG device. Subsequently, a sample cuvette is formed by using lithographic technique and dry etching process. Following this, a PDMS chip with microfluidic channels is bonded with the AWG device, leading to an integrated AWG-microfluidic platform. To the best of the author’s knowledge, this is the first work to integrate a visible AWG device and a microfluidic chip towards spectroscopic sensing. The monolithic integrated AWG microspectrometer–microfluidic platform is demonstrated for fluorescence spectroscopic analysis. Signals from the output channels detected on a camera chip can be used to re-create the complete fluorescence spectrum of an analyte. By making fluorescence measurements of (i) mixed quantum dot solutions, (ii) an organic fluorophore (Cy5) and (iii) the propidium iodide (PI)-DNA assay, the results obtained illustrate the unique advantages of the AWG platform for simultaneous, quantitative multiplex detection and its capability to detect small spectroscopic shifts. Although the current system is designed for fluorescence spectroscopic analysis, in principle, it can be implemented for other types of analysis, such as Raman spectroscopy. Fabricated using established semiconductor industry methods, this miniturised platform holds great potential to create a handheld, low cost biosensor with versatile detection capability. Also, the AWG device design is modified with focusing properties that enable localised spectroscopic measurements. Micro-beads based, multiplexed fluorescence detection is performed with the AWG + CCD system and the results have demonstrated capabilities of using the adapted AWG device for localised, multiplexed fluorescence detections, opening up potential applications in the field of cell sorting and single cell analysis. Furthermore, the AWG-microfluidic device is investigated for absorption spectroscopy measurement. As a test system, the pH dependence of the absorption spectra of bromophenol blue is measured to illustrate how an AWG device could be used as a colorimetric pH sensor. Overall, it is believed that the AWG technology holds great potential to realise a compact, integrated spectroscopic biosensor for point-of-care applications.

In recent years, there has been much interest in the possible existence of the N*(1685) narrow nucleon resonance, as predicted by the chiral soliton model. Several eta photoproduction experiments have been performed that rely on extracting neutron observables from deuteron target data. These have shown some evidence of narrow structure, however, no structure was observed in cross section measurements on the proton channel. Within the A2 collaboration at the MAMI accelerator in Mainz, Germany, a more detailed study has been undertaken using eta photoproduction on a liquid hydrogen target, and is the subject for this thesis. The study utilised the high resolution and high precision possible with the A2 detector setup, in an attempt to overcome the predicted low photocoupling between the N*(1685) and the proton. The MAMI electron beam was used to produce bremsstrahlung photons, the energies of which are determined by detecting the recoil electrons in the Glasgow-Mainz Tagged Photon Spectrometer. The reaction products from the interaction of the photon beam and liquid hydrogen were detected by the Crystal Ball and TAPS calorimeters. The gamma proton to eta proton reaction channel was identified via the 3 neutral pion decay branch. The reaction cross section was measured between E = 707 and 1477 MeV. Both differential and total cross sections have been presented with a particular emphasis on the centre of mass energy range around 1685 MeV. The peak structure seen on the neutron channel is not observed in the case of the proton. However, a dip in the cross section is observed, which is more prominent at forward polar angles. The leading models are not fully able to reproduce the data in this energy region. As such, given the high energy resolution of the measurement the results can be used to constrain future models and in doing so aid the understanding of the nucleon's resonance spectrum.

The LHCb experiment is a high energy physics detector at the Large Hadron Collider. The experiment has been designed and built to search for new physics in the b hadron sector. This thesis discusses a contribution to the detector construction and preparatory studies for a rare decay analysis. Quality assurance of the silicon modules of LHCb vertex detector One of the critical components of the LHCb experiment is the silicon vertex locator (VELO), which is used to measure the decay distance of the $b$-hadrons and is a principal component of the tracking of the experiment. This thesis describes the quality assurance tests of the VELO silicon modules. A facility was designed to operate the VELO modules in vacuum and thermally stress the modules. To verify the suitability of the modules for the experiment a range of studies were performed including measurements of the silicon leakage current and the identification of bad channels through a noise analysis. A full set of 42 modules (and spares) suitable for use in the experiment were successfully tested. Analysis of the $B_d\rightarrow K^* \mu^{+}\mu^{-}$ decay The $B_d\rightarrow K^* \mu^{+}\mu^{-}$ decay is a flavour changing neutral current process which occurs only via loop diagrams. This is a rare process with a measured branching ratio of \\$\rm 1.10^{+0.29}_{-0.26} \times 10^{-6}$. The $B_d\rightarrow K^* \mu^{+}\mu^{-}$ rare decay is sensitive to new physics (NP) effects. Through the measurement of the so-called forward backward asymmetry distribution a clear signature of NP can be found in this channel. The estimated number of signal events expected per year in LHCb ($\rm 2 fb^-1$) was estimated to be $\rm 7.0\pm0.1\times 10^{3}$. The sensitivity to the zero point of the forward backward asymmetry was calculated as $\rm 0.4~GeV^{2}/c^{4}$, assuming the estimated annual yield. Degradation of the sensitivity due to background events was estimated to be $\rm \sim10-15\%$. Potential systematic effects due to acceptance and background mismeasurement are also presented. The results on the forward backward asymmetry were obtained using a non-parametric unbinned method.

Einstein's General Theory of Relativity (1916) predicted the existence of gravitational waves. These waves can be considered as fluctuations, or ripples, in the curvature of space-time. Until now there has been only indirect evidence, produced by Hulse and Taylor, of their existence. However, for many years various groups of scientists around the world have been developing ultra-sensitive instruments and techniques which are expected to be capable of detecting gravitational wave signals. The direct detection of these waves will provide new information about the astrophysical processes and sources which produce them. Gravitational radiation is quadropole in nature, producing orthogonal stretching and squeezing of space. The resulting fluctuations in distance are, however, very small, with gravitational waves emmitted from violent astrophysical phenomena expected to produce strains in space of the order ~10 [superscript -22] over relevant timescales. One technique for detecting such strains is based on a Michelson Interferometer. The Institute for Gravitational Research at the University of Glasgow under the leadership of Professor James Hough, has been an active contributor of research targeted towards the detection of gravitational waves for over 35 years. A strong collaboration exists with the Albert-Einstein-Institut in Hanover and Golm, the University of Hanover and the University of Cardiff. This collaboration has developed and constructed a laser interferometer, with arms of 600 m length, in Germany named GEO600. The research presented in this thesis details experiments undertaken on materials and techniques used in current interferometric detectors and for proposed future detectors. The aim of this research is to investigate methods of reducing the levels of mechanical loss associated with the detector optics and thereby minimise the impact of thermal noise on the overall sensitivity of detectors.

In this thesis, two, seemingly different, classes of problems are discussed: locating gas sources from downwind gas concentration measurements and designing diffractive optics (i.e. computer generated holograms), which on illumination will produce a desired light beam in the far field. The similarity between these problems is that they are both “inverse problems” and we discuss the use of inverse techniques to solve them. In many instances within science, it is possible to calculate accurately a set of consequences which result from defined events. In most cases, however, it is math- ematically impossible to analytically calculate the unique set of events which led to the observed consequences. Such problems are termed “inverse problems”. Taking the example of gas dispersion, one sees that a known source leads to a calculable set of downwind concentrations. However, given a single concentration measurement it is impossible to distinguish a specific source and location from a larger, more distant source that would have given the same measured concentration. This is an example of the same consequence resulting from two, or more, different events. Key to solving inverse problems are iterative algorithms which randomly trial different possible events to find those which best describe the observed consequences. Such algorithms use a search method to postulate possible events, apply a forward model to calculate the anticipated consequences and then use a cost function to compare the postulated with the known consequences. The process is iterated until the optimum value of the cost function is found, at which point the current set of postulated events are taken to be the best estimate of the real events. In this thesis I apply similar iterative algorithms to solve the two classes of problem. The current demand on the world’s oil resources have encouraged the development of new prospecting techniques. LightTouch is one such solution which is discussed in this thesis and was developed with Shell Global Solutions. LightTouch uses the fact that oil reserves, through microseepages, leak hydrocarbons to their surface. Detection of these hydrocarbons can indicate the presence of oil reserves. LightTouch measures Ethane to sub-part-per-billion sensitivity at multiple positions across a survey area. Locating the source of the Ethane from the sparse downwind concentration measurements is an inverse problem and we deploy algorithms of the type discussed above to locate the Ethane sources. The algorithm is written in LabView and the software, Recon, is currently used by Shell Global Solutions to solve this problem. In appendix B the Recon user interface is shown. We investigate both the impact of choice of cost function (chapter 3) and forward model (chapter 4), which in this inverse problem is a gas dispersion model, on the algorithm’s ability to locate the gas sources. We find that the choice of cost function is more important to the success of the algorithm than the choice of forward model. Optical tweezers trap and manipulate particles with light beams. In order to manipulate the particles in a desired way it is necessary for the shape and position of the light beam to be controlled. One way of achieving a desired light beam is to use a spatial light modulator (SLM) which displays a phase pattern (referred to as a computer generated hologram), off which the light is diffracted. Calculating the phase pattern which will result in the desired light beam is an inverse problem and is referred to as holographic light shaping. The forward model in this case is a Fourier transform. In this thesis we use an algorithm similar to that used to solve the gas location problem and the Gerchberg-Saxton algorithm to calculate phase patterns with applications in optical tweezers. Within an optical tweezers system the highest trap resolution (the smallest distance between neighboring traps) that can be achieved is conventionally dictated by the diffraction limit. In this thesis we investigate two possible ways of beating the diffraction limit: superresolution and evanescent waves. In chapter 5 we investigate the application of inverse techniques to calculating phase patterns which produce superresolution optical traps. We calculate theoreti- cally the improvements to both relative trap stiffness and trap resolution using the superresolution optical traps. Although both are improved it comes at a cost to trap strength. In chapter 7 we simulate evanescent wave fields and demonstrate shaping three dimensional evanescent optical traps. Similar light shaping techniques are used in chapter 6 to shape light beams which after being disturbed will self-reconstruct.

Future generations of gravitational wave detectors will require significant progress in the reduction of all forms of noise affecting the system. One form of noise it is critical to reduce is thermal noise, which can be described as the consequence of the atoms which make up the measurement optics experiencing vibrations because of their non-zero temperature. The dielectric multilayer coatings of the mirror in an interferometric gravitational wave detector are known to contribute significantly to the overall levels of thermal noise. The next generation of gravitational wave detectors may need to use exotic coatings, cryogenic operating temperatures and silicon mirror substrates in an effort to mitigate the effects of thermal noise. Chapter Two describes thermal noise in detail, and introduces the concepts of substrate noise, coating noise, thermoelastic dissipation, mechanical loss and the formulae used to calculate them. Chapter Three describes the current state of research on the factors affecting mechanical loss in dielectric coatings. The technique of probing the structure and dissipation characteristics of materials by assessing the shape and position of the low temperature excess loss feature known as a Debye peak is introduced. The cryogenic mechanical loss measurement apparatus used in Chapters Four, Five and Six is described and characterised. Chapter Four concerns the variation of mechanical loss of ion-beam sputtered silica coatings with temperature and investigates the effects of heat-treatment upon them. The low-temperature Debye peak was found in some modes of a sample heat treated at 300oC and an Arrhenius analysis provided a characteristic energy for the dissipation process of (17.3 ± 2.3)meV. Further heat treatment of silica at 600oC and 800oC appears to narrow the Debye peak, which is thought to be indicative of the narrowing of the distribution of bond angles in the amorphous silica network. Hafnia is investigated as an alternative coating material in Chapter Five. The mechanical loss of hafnia heat-treated at 300oC was measured and two excess loss features were discovered, one below 100K and one above 200K. Electron scattering measurements indicate that this sample may already have developed polycrystalline regions which are known to be connected to high levels of mechanical loss. The mechanical loss of an un-heat-treated hafnia coating is also measured and an extremely low coating loss of 1.87 × 10−5 is found at 20K. Chapter Six describes an experiment to find the mechanical loss of a hydroxycatalysis bond between silicon cantilevers at temperatures between 10K and 300K.This new technique for the measurement of the mechanical loss of bond material produced a minimum upper limit of the bond loss of (0.13 ± 0.03) occurring in the fundamental mode at 80K and upper limit of the bond loss of (0.19 ± 0.07) occurring in the third mode at 15K. Chapter Seven describes the development and testing of a nodal support system to enable cryogenic measurements of cylindrical bulk mirror substrates to be made. The efficacy of the support varied significantly with the frequency of the mode and the cryogenic measurements were partially successful. The major results in this work are the successful measurements of the mechanical loss of amorphous hafnia coatings at low temperatures and the use of a structure made from hydroxy-catalysis bonded silicon cantilevers to obtain an upper limit for the mechanical loss of the bond material. These results may inform technological advances that reduce the level of thermal noise experienced in future gravitational wave detectors.

Deeply Virtual Compton Scattering (DVCS) is the simplest interaction that allows access to Generalised Parton Distributions (GPDs), a theoretical framework describing nucleon structure. The strong interest in GPDs results from the fact that they can be used to determine the total angular momentum of quarks inside the nucleon and provide a 3-dimensional picture of nucleon structure. The measurement of the DVCS process is facilitated by the interference with a competing interaction known as the Bethe-Heitler process which has the same final state. DVCS information is obtained from the asymmetrical in distribution of the real photon around the azimuthal angle $\phi$ at HERMES. Beam charge and beam helicity asymmetries, extracted from DVCS events with an unpolarised hydrogen target recorded during the 2006-2007 and 1996-2007 data taking periods, are presented in this thesis. The asymmetry amplitudes are presented over the range of HERMES kinematic acceptance, with their dependence on kinematic variables t, xB and Q^2 also shown and compared to a phenomenological model.