| Elastic bounds incorporating symmetry Composite materials (made from two or more substances) can have properties vastly different from any of the constituents. Detailed prediction of their properties often involves tedious numerical computation. One approach to understanding their properties has been to calculate rigorously the minimum and maximum values these properties can have based on very limited information of the microstructural geometry. The elastic properties of microstructures are of interest in variety of fields, and bounds are available based only on the relative volume fractions of the constituents. These bounds can be made tighter by including two-point and three-point correlation functions. In this project, I want to try an alternative approach and include information about the global symmetry of the composite: in particular, relating the effective properties to one of the known crystallographic symmetry groups. This project requires some familiarity with tensors and a little bit of group theory. The project will follow on from earlier student projects with Daniel Ackland and Michelle Rigozzi. |
Level set theory for topological optimization
Many interesting problems in physics and engineering require determining a shape or geometry that optimizes some physical process: for example, the size, shape and locations of holes in a special purpose microstructured optical fibre. There may be in addition geometric constraints: how close two surfaces can come, the maximum or minimum curvature allowed etc. A simple approach is to choose in advance some basic geometric building blocks: circles, wedges, ellipses and then optimize the numerical parameters of these objects. A much more interesting approach allows even the topology or connectedness (number of holes) to be varied. The geometry is chosen from the contours (or level sets) of some characteristic function. As this function is varied, the contours move around, and peaks, ridges and valleys can appear, merge or vanish is a smooth way. These ideas have already been successfully applied in other fields such as architecture. Optimization of structural topology requires a combination of techniques to vary the level sets but also to recompute the solutions to the relevant differential equations (or other equations) using perturbation theory. Depending on the department and needs of the student this project can go in a number of directions from a more computer science algorithms approach, to a physics-computational science approach to studying perturbation theory of the differential equations themselves. |
High bandwidth performance of GIMP Fibres
Optical fibres have already revolutionized long haul telecommunications. However, for shorter distances there are a number of important applications that remain electronic: short distance communications such as local area networks [LAN], fibre to the home and even chip to chip communication. For these short distances the attenuation of the fibre is less important than issues such as how easily the fibres can be connected. The most important aspect for connectivity is the size of the core, where the light is confined. Fibres with much larger cores are difficult to make in glass, while polymer fibres can easily be made with cores 1 mm in diameter whilst still being flexible. For this reason most people believe that short distance, high-speed connections will probably be made using polymer optical fibres. Such fibres are massively multi-mode, and in order to control the intermodal dispersion, are generally made with a graded index core. Producing an appropriate graded index profile by chemical doping is however not a straightforward process, and for this reason we have begun exploring to possibility of making Graded index microstructured polymer optical fibres, or GImPOFs. These fibres have been fabricated, and show bandwidths that are competitive with conventional graded index fibres. However, the unique optical properties of microstructured fibres, make it difficult to understand the fundamental limits to bandwidth performance. Indeed, it is clear that a simple application of the formulae for conventional fibres produces results that are catastrophically wrong. In this project we will attempt to gain a deeper understanding of the GIMP, a study that will involve both experimental and theoretical studies. |
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Algebraic varieties in phylogenetics
Phylogenetics is the reconstruction of the evolutionary tree relating species from the degree of similarity of their genetic sequences. Many different algorithms existing for inferring, approximating and searching for the correct tree topology and the correct lengths of the branches of the evolutionary tree. Algebraic varieties are a branch of pure mathematics related to simultaneous polynomial equations. It turns out that certain algebraic varieties are invariant with respect to changes in the branch lengths but can be used as a test of the correct tree topology. This project will survey the field and also explore how these invariants can be used to enhance other algorithms used to search for the "best" evolutionary tree. A rare combination of pure and applied mathematics with important benefits in the emerging field of bio-informatics. |
Representation in genetic algorithms
Genetic algorithms are a popular technique used to search highly multidimensional spaces for the maxima or minima of complicated functions. The ability of a genetic algorithm to efficiently search the so called "fitness landscape" for both global and local optima can depend crucially on the representation of the search space itself. This project will look at optimization problems that can be simultaneously represented by different structures (trees, arrays, permutations, etc.) to explore how the choice of representation interacts with the genetic algorithm. This field is vast and this project could suit someone who wants to do an essay based project, but is also suitable for some original research. |
Optical fibres for atom guiding
One of the most intriguing developments in physics in the last decade has been the development of atom optics. Using the wave properties of atoms opens significant new frontiers including atom lithography, atom trapping and Bose Einstein condensates and possibly even an atom laser. Control and transport of atoms is critical for these applications, and one of the possible means of doing this is by using hollow fibres. These fibres allow atoms to be transported in the centre of the fibre, with interactions with the sides of the fibre being prevented by means of the interaction light, guided in the core of the fibre. The light atom interactions are of two types, the spontaneous force, which uses the momentum of the photons absorbed by the atom, and the gradient or dipole force. This effect relies on the AC Stark effect, in which the electric field of the light changes the energy state of the atom. This effect is proportional to the intensity of the light, so a gradient force is generated if the field is not uniform. The force can be repulsive or attractive depending on the detuning of the light from the atomic transmission .This project will explore the possibilities of using microstructured fibres for atom guiding. It will entail both theoretical and experimental components, to be carried out in conjunction with Professor Ken Baldwin at the ARC Centre of Excellence for Quantum-Atom Optics , Australian National University [http://www.acqao.org/index.htm]. The project will follow on from an earlier student project with James Griffin. |
| Adaptive recombination in genetic algorithms The recombination operator is an essential ingredient of any genetic algorithm allowing useful building blocks in the parent solutions to be combined into larger building blocks in the child solutions. The choice and design of the recombination operator is often the hardest part to get right, and an operator that is useful in the early stages of eveolution may be detrimental in later stages (the same operator that puts building blocks together, will also pull them apart...). In this project we look at changing the recombination operator with each generation. This involves developing a measure for the effectiveness of particular different reombination operators and also deciding how to change it. | Manipulating leaky modes Are you the type of student that enjoys abstract mathematical concepts such as vector spaces and orthogonality but at the same time is not afraid of writing a serious computer program? This project is certainly not for the faint-hearted. From a Physics point of view, we want to find many different modes of microstructured fibres and then be able to calculate how they couple to each other in the presence of perturbations. From a Mathematics point of view, we want to apply recently discovered methods for normalizing and orthogonalising eigenfunctions that satisfy outward propagating boundary conditions to derive and apply rigorous first-order perturbation formulae. From a Computer Science point of view, we must be able to do this in an automated way, over and over again, for many different fibre designs with no human intervention. There’s a lot more you need to know too, so if you haven’t fled in terror come and ask. |
Exotic mPOFs: tailoring optical performance through novel
materials
An appealing feature of microstructured fibres is that the holes can be exploited to incorporate novel materials which can affect optical performance. Such materials could include chiral materials which are optically active, and liquid crystals, whose state can be switched by the application of heat or voltage. We have developed a method by which electrodes can be incorporated into the fibre during the draw. This allows us to apply an electric field to produce either a transient or long-term alignment of molecules or nano-particles. A particularly appealing area, which has not been explored at all to date, is magneto-optic effects in microstructured fibres. The development of a strong magneto-optical effect is crucial for the development of in-fibre isolators, and is also of fundamental interest. The inclusion of cobalt nano-particles in the polymers has been used to create a material with a strong magneto-optical effect and other high Verdet constant materials that may be suitable for inclusion are the rare earths and iron garnet. Other approaches that could be used to produce a magneto-optic effect include the use of a helical conducting core fibre, and the use of magnetic fluids in the core. |
| Rugged Fitness Landscapes Kauffman developed a simple model for describing fitness functions that resemble rugged mountain ranges with multiple peaks, variation on different scales. This model, the NK model, consists of a total of N interacting parts which depend on at most K other parts of the system. Thus by varying the ratio of N and K different "levels" of complexity can be emulated. K=0 corresponds to models where independent causes produce independent effects, and K=N-1 corresponds to the absence of any correlations. The NK model is a powerful tool to use as a test of the effectiveness of any particular proposed genetic algorithm. In this project I would like to explore how some basic genetic algorithms cope with NK fitness functions and which attributes of the genetic algorithm need to be changed as the value of K increases. | Self-similarity in photosenstivity The nonlinear partial differential equation that describes photosenstivity in certain types of doped glass is similar to the famous nonlinear Schrodinger equation. However, the photosenstivity equation appears to be non-integrable and describes a cumulative rather than transient phenomenon. Some years ago, we found self-similar solutions for non-saturating models of photosensitivity were found using Lie group techniques (and even compared them succesfully to experiment, see Tanya Monro's Ph.D. project). Real systems however exhibit saturation and I recently found a saturating model that admits some self-similar solutions. The same trick also allows generalisation to multiple wavelengths and/or multiple beams of light. This project would look at the properties of these self-similar solutions and perhaps compare them to full numerical solutions of the original PDE's. |
Naturally occurring optical fibre bundles
An optical fibre bundle is a set of adjacent optical waveguide running in parallel and usually adhered together. They are used for transmitting light to brighten an area as well as transmitting intact images. Such structures also naturally, such as in the crystal ulexite (hydrated sodium calcium borate hydroxide) also known as “TV Rock” because it displays an image on one polished surface of any surface adjacent to the other side. A biological example of a similar structure occurs within the eyes of trilobites. Unlike optical fibres which are made from amorphous materials such as silica or polymer, these natural structures consist of crystalline material which is anisotropic. The collimating properties of such fibre-bundle structures may also play a role in the iridescence suppression mechanism used above the 3D photonic crystals found in the scales of some butterflies. Depending on the skills and interest of the student this project could encompass: theoretical modelling and comparison of these different optical systems, or experimental work such as fabricating a larger scale (e.g. sub-millimetre) version of these same periodic structures. Measurement or calculation of the numerical aperture and iridescence transforming properties of these or equivalent structures is also of considerable interest for endoscopy, astronomical imaging as well as flexible liquid crystal displays in addition to our basic fundamental curiosity. |
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