In this thesis, problems of structural optimisation are approached through analytic and computational techniques. A particular focus is the effect of hierarchical design. The first chapter forms an introduction for the reader. Chapter 2 investigates the optimisation of elastic support on a buckling rod. A cost function is associated with the strength of the total elastic support provided to a beam of uniform cross-section supporting a compressive load. Through a perturbative method, it is found that for a low cost of support, a single, centrally placed support is optimal; furthermore it is found, using simulational and analytic methods, that the optimal support placement undergoes a series of bifurcations as the cost increases. The nature of these bifurcations is non-trivial and, although the analogy is not complete, there exist similarities between the solution to this problem and Landau theory of second-order phase transitions. In Chapter 3, the theme of hierarchical design is introduced. By analysing all possible failure modes, it is shown that a hierarchical design is highly efficient for withstanding external pressure loading in the limit of low applied pressures. By changing the level of hierarchy, the scaling law for volume of material required for structural stability against the applied external pressure can be changed systematically. For a given applied pressure, a particular level of hierarchy is shown to be optimal. This optimal level of hierarchy increases without bound as the pressure decreases. The Hausdorff dimension of the optimal structure and its dependence on applied pressure is found. Two example structures are presented, although the design is applicable to any convex shape. The fourth chapter of this thesis investigates the use of hierarchical geometry for a highly efficient interface between two surfaces. It is proposed that for a given strength of surface interaction, alterations to the geometry of the interface play a strong role in determining the force that is required to separate the surfaces. In particular the case of two surfaces with one being very much more rigid than the second is investigated. Increasing the hierarchical order of the design is seen to change the scaling relationship between the interface interaction strength and failure load. In Chapter 5, a hierarchical design for high mechanical efficiency under compressive loading is fabricated and mechanically tested. The particular design has previously been shown to be highly efficient under compressive loading. The scaling of material required to build a stable structure against a specific loading has previously been shown to be dependent on the level of hierarchy. A second order design is fabricated using rapid prototyping techniques. Additionally, a similar design based on hollow tubes rather than solid beams is proposed and is shown to make further savings on volume when compared to the original design. The final investigation presented in this thesis focuses on the role of imperfections in determining the buckling load of a hierarchical design. A two-dimensional design is proposed before simple, single beam, imperfections are added to the structure. The dependence of the structure on the magnitude of the imperfections is calculated analytically for the generation-1 and 2 designs. In the generation-1 structure, the magnitude of the imperfection is related to the reduction in failure load by a one-half power-law. The behaviour of a generation-2 frame with a single beam perturbed in thickness is found to be dominated by the behaviour of the generation-1 subframe. The behaviour found analytically is confirmed with finite element simulations for the generation-1 structure.

Surface plasmons (SP) are guided electromagnetic wave propagating along the surface of metal. The properties of SP are affected by the material attached to the metallic surface so they can be used as a very sensitive sensor capable of detecting the deposition of subnanometric layers of dielectric. SP has been widely investigated for biosensor applications and the theory is well established. Although SP sensors have been well studied, integrating the SP to a microscope is a relatively young field. Since the SPs are surface waves; microscopy techniques to optimise the SP microscope performance will require totally different techniques to non-surface wave microscopy. This thesis develops a theoretical framework to understand different types of SP microscope setups through the rigorous diffraction theory. The framework analyses the diffraction process through rigorous wave coupled analysis (RCWA) and a software package processes the diffracted orders to recover the microscope response for a range of different systems. In this thesis I will investigate the non-interferometric SP microscope, interferometric SP microscope and confocal SP microscope. I will show that the non-interferometric system exhibits a trade-off between lateral resolution and sensitivity, where an image obtained with a good contrast will have low lateral resolution. In order to get around the trade-off, the interferometric system can be employed; however, the main challenge for the interferometric setup is its optical alignment. I will show that a confocal SP microscope, which has been developed as a part of this thesis, can simplify the complexity of the interferometric system and give similar measurement performance. For the interferometric and confocal systems, the SP measurements are normally carried out through the interference signal, which is interference between a reference beam and the SP. I will suggest a method to extract SP propagation parameters from the interference signal by employing a spatial light modulator and also show that the SP propagation parameters do not only give us some insight to the SP effect for the interferometric system, but also gives us a new imaging mode to improve the resolution.

This thesis describes the recent improvements of the SMILETRAP Penning trap mass spectrometer and a number of interesting high precision mass measurements, which have been performed using the improved apparatus, and are relevant in todays fundamental physics problems. The mass of the hydrogen-like 24,26Mg ions as well as the masses of the hydrogen- and lithium-like 40Ca ions are presented in this work that are indispensable input values when evaluating g-factor measurements of the bound electron. In both cases the uncertainty in the masses was improved by one order of magnitude compared to the literature values known so far. The mass of 7Li has been measured and a new mass value has been obtained with an unprecedented relative uncertainty of 6.3x10-10. A large deviation of 1.1 μu (160ppb) compared to the literature value has been observed. In order to find the reason of this large deviation and to look for possible systematics we have measured the mass of 4He and 6Li and concluded that the 6Li(n,γ)7Li reaction Q-value used in the literature when calculating the 7Li mass is wrong by about 1 keV. The mass difference between 3He and 3H (Δ m (3H -3He)) is the Q-value of the tritium β-decay. An accurate knowledge of the tritium Q-value is of importance in the search for a finite rest mass of the electron neutrino. By adding a measurement of the mass of 3He1+ to previous mass measurement of 3H1+ and 3He2+ we have improved our previous Q-value by a factor of 2. At the moment our Q-value is the most accurate and more importantly it is based on the correct atomic mass values.

In this thesis, we present a theoretical study of the effects that a current-carrying Two-Dimensional Electron Gas (2DEG) produce on a neighbouring magnetically trapped Bose-Einstein Condensate of alkali atoms (BEC). We suggest that technology used for magnetic micro-controlling of cold gases could be improved by replacing or combining the metallic wires used in such structures with 2DEG-based conductors or quantum electronic devices. All calculations presented in this thesis consider parameters attainable with present technology, suggesting that experimental realization of the scenarios proposed here is already feasible. In Chapter one we present the general context in which this thesis is developed. It includes a definition of the Bose-Einstein condensate state, a description of the principles of magnetic trapping and a brief review of the developments in the area of micromanipulation of atomic BECs. Chapter two is devoted to describe the characteristics of the heterojunction considered in the thesis, and a simple model used to evaluate the electron flow in the 2DEG it contains. Chapter three shows in detail the properties of magnetic trapping configurations considered afterwards. We study two simple applications that can be developed by bringing a BEC near to a current carrying 2DEG. Firstly, in Chapter four, we demonstrate the feasibility of creating magnetic trapping potentials using a current-carrying 2DEG and an external magnetic field. We identify the advantages of such a 2DEG-based trap over traditional metal-based traps and the conditions needed for operability. Recently developed techniques of magnetic field microscopy with BECs motivate our second considered application, namely, using a Bose-Einstein Condensate to probe the electron transport in 2DEGs and structural characteristics of the heterojunction. In Chapter five, we demonstrate how the quantization of conductance occurring in Quantum Point Contacts (QPC) fabricated from the heterojunction in the 2DEG, can be detected through a localized depletion of the BEC caused by a small inhomogeneity of the magnetic field that originates from a current through the QPC. In addition, we show that the electron density fluctuations in the 2DEG can be measured by detecting the corresponding inhomogeneous magnetic field produced when current flows, via modulation of the BEC’s density. We establish the conditions under which a sensitive response of the BEC to the magnetic field is possible. We also derive a general relation between the modulation of the magnetic field affecting the BEC and the distribution of ionized donors in the heterojunction. Creating semiconductor-cold-atom hybrid systems, where electrons in the semiconductor and atoms in the BEC are coupled to each other, requires a full understanding of the properties of both systems. Since typical micro-traps have an elongated geometry, in Chapter six we present a study of phase correlations of BECs in such geometries. To do this, we use a recently proposed effective one-dimensional equation that takes into account the 3D character of the BEC. Finally, in Chapter seven, we conclude and identify directions for future work emerging from this thesis.

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