The challenge of density functional theory is the useful approximation of the exchange - correlation energy. This energy can be approximated with the local electron density and the gradient of the density. Many different generalized gradient approximations (GGA) have been made recently and there is controversy over the best overall functional. Recent Monte Carlo simulations give evidence that the Laplacian of the density might be a better starting place than the gradient to correct the local density approximation. We have tested several Laplacian based GGA models for exchange for small atoms. We use known constraints on the exchange energy used in current GGA’s. In many models unphysical oscillations occur in the potential when using the Laplacian, and understanding and eliminating them is part of the focus of this research. We also find that mixing gradient and Laplacian seems to give a better result than only using one or the other. / Department of Physics and Astronomy

Electroencephalography (EEG) is a valuable tool for clinical and cognitive applications. EEG allows for measuring and imaging of scalp potentials emitted by brain activity and allows researchers to draw conclusions about underlying brain activity and function. However EEG is limited by poor spatial resolution due to various factors. One reason is the fact that EEG electrodes are separated from current sources in the brain by cerebrospinal fluid (CSF), the skull, and the scalp. Unfortunately the conductivities of these tissues are not yet well known which limits the spatial resolution of EEG.Based on prior research, spatial resolution of the EEG can be improved via use of various mathematical techniques that provide increased accuracy of the representation of scalp potentials. One such method is the surface Laplacian. It has been shown to be a direct approach to improving EEG spatial resolution. Yet this approach depends on a geometric head model and much work has been done on assuming the human head to be spherical.In this project, we will develop a mathematical model for ellipsoidal head geometry based on surface Laplacian calculations by Law [1]. The ellipsoidal head model is more realistic to the human head shape and can therefore improve accuracy of the EEG imaging calculations. We will construct a computational program that utilizes the ellipsoidal head geometry in hopes to provide a more accurate representation of data fits compared to the spherical head models. Also, we will demonstrate that the spline surface Laplacian calculations do indeed increase the spatial resolution thereby affording a greater impact to the clinical and cognitive study community involving EEG. / Department of Physics and Astronomy

High Harmonic Generation (HHG) enables the production of bursts of coherent soft x-rays with attosecond pulse duration. This process arrises from the nonlinear interaction between intense infrared laser pulses and an ionizing gas medium. Soft x-ray photons are used for spectroscopy of inner-shell electron correlation and exchange processes, and the availability of attosecond pulse durations will enable these processes to be resolved on their natural time scales. The maximum or cutoff photon energy in HHG increases with both the intensity as well as the wavelength of the driving laser. It is highly desirable to increase the harmonic cutoff as this will allow for the generation of shorter attosecond pulses, as well as HHG spectroscopy of increasingly energetic electronic transitions. While the harmonic cutoff increases with laser wavelength, there is a corresponding decrease in harmonic yield. The first part of this thesis describes the experimental measurement of the wavelength scaling of HHG efficiency, which we report as lambda^(-6.3) in xenon, and lambda^(-6.5) in krypton. To increase the HHG cutoff, we have developed a 1.8 um source, with stable carrier envelope phase and a pulse duration of <2 optical cycles. The 1.8 um wavelength allowed for a significant increase in the harmonic cutoff compared to equivalent 800 nm sources, while still maintaing reasonable harmonic yield. By focusing this source into neon we have produced 400 eV harmonics that extend into the x-ray water window. In addition to providing a source of photons for a secondary target, the HHG spectrum caries the signature of the electronic structure of the generating medium. In krypton we observed a Cooper minimum at 85 eV, showing that photoionization cross sections can be measured with HHG. Measurements in xenon lead to the first clear observation of electron correlation effects during HHG, which manifest as a broad peak in the HHG spectrum centred at 100 eV. This thesis also describes several improvements to the HHG experiment including the development of an ionization detector for measuring laser intensity, as well as an investigation into the role of laser mode quality on HHG phase matching and efficiency.

This thesis describes a new approach to accelerate the simulation of the steady-state response of nonlinear circuits using the Harmonic Balance (HB) technique. The approach presented in this work focuses on direct factorization of the sparse Jacobian matrix of the HB nonlinear equations using a Graphics Processing Unit (GPU) platform. This approach exploits the heterogeneous structure of the Jacobian matrix. The computational core of the proposed approach is based on developing a block-wise version of the KLU factorization algorithm, where scalar arithmetic operations are replaced by block-aware matrix operations. For a large number of harmonics, or excitation tones, or both the Block-KLU (BKLU) approach effectively raises the ratio of floating-point operations to other operations and, therefore, becomes an ideal vehicle for implementation on a GPU-based platform. Motivated by this fact, a GPU-based Hybrid Block KLU framework is developed to implement the BKLU. The proposed approach in this thesis is named Hybrid-BKLU. The Hybrid-BKLU is implemented in two parts, on the host CPU and on the graphic card’s GPU, using the OpenCL heterogeneous parallel programming language. To show the efficiency of the Hybrid-BKLU approach, its performance is compared with BKLU approach performing HB analysis on several test circuits. The Hybrid-BKLU approach yields speedup by up to 89 times over conventional BKLU on CPU.

The linear and nonlinear optical response of plasmon hybridized systems is the subject of study of this thesis. Plasmonic silver nanoprisms are able to confine light to a sub-wavelength volume, which provides local field enhancement. This confined field is promising for achieving an enhanced nonlinear optical response. For many of plasmon nanoparticles, however, the plasmonic resonance is not at the near-infrared wavelengths of a Ti:Sapphire laser, the most common source used for ultra-fast measurements. To achieve resonance at these wavelengths, a tuning mechanism is required. The plasmon hybridization between silver nanoprisms and a thin gold film provides this tuning mechanism, which allows for enhanced optical second harmonic generation. Overlapping the plasmon resonance of the system with excitation source, by varying the spacer layer between the nanoprisms and the gold film, enhances the second harmonic counts by approximately three orders of magnitude. The finite-difference time-domain calculations agree to within a factor of two with the experimental findings in terms of the predicted enhancement factor. This plasmon hybridization approach is promising for future applications, including enhanced multi-photon lithography and nonlinear sensing using metal nanoparticles. / Graduate

This Thesis is concerned with the prediction, synthesis, characterization and testing of inorganic materials for Second Harmonic Generation (SHG). Chapter One describes the fundamentals of non-linear optics, and poses the problems, and some of their solutions which confront the synthetic chemist and the theoretical prediction of the second order hyperpolarizability constant β using CNDOVSB calculations. Chapter Two describes the design, implementation and calibration of an apparatus for measurements of the second harmonic generating efficiency of solids based on the Kurtz powder technique, and a solvatochromic method for the determination of β. Novel compounds with potential chirality due to atropisomerism, asymmetric octahedral structures, and asymmetric tetrahedral symmetry of metal centers are discussed in Chapters Three to Five. Chapter Three surveys the use of pentane-2,4-dionato- ligands and their coordination compounds as possible NLO active materials. The single crystal X-ray structures of bis(triphenylphosphine)(4-nitrobenzoylacetonato)palladium(II) tetrafluoroborate and tris(triphenylphosphine)[3-(2,4-dinitrophenyl)-pentane-2,4-dionato]palladium(II) tetrafluoroborate were determined. Chapter Four describes the syntheses, characterization and SHG properties of trans-β-ionylidenecyanoacetic acid (2-cyano-3-methyl-5-(2,6,6-trimethyl-l-cyclohexen-1- yl)-2,4-pentadienoic acid) and some of its metal and non-metal salts. Chapter Five describes the synthesis, characterization and second harmonic generation properties of some platinum(II) and palladium(II) complexes of β- ionylidenecyanoacetic acid. Chapter Six describes the use of conventional asymmetric carbon centers to introduce chirality into centrosymmetric compounds. The chiral compound (L)-N-[2-cyano- 3-methyl-5-(2,6,6-trimethyl-1 -cyclohexene-1 -yl)-2,4-pentadiene-1 -one]-L-proline and some of its salts were synthesized from β-ionylidenecyanoaeetic acid and tested for SHG.

No description available.

An accurate, intuitive and tractable transform as been identified and developed from which the spatial harmonics of reticle patterns defined in polar coordinates can be obtained. A description of reticles and generic methods for describing them mathematically are presented along with some background on general harmonic analysis. Focus then turns to candidate transforms for analysis of reticle patterns and the most promising are investigated in more detail. A fast linear algorithm is devised to overcome some problems with implementation of a fast transform and this is followed by analysis of the transform basis functions to assist with interpretation of the transform in azimuth and radius. A sampling guideline is presented so that aliasing can be avoided and, finally, the transforms of some representative reticle patterns are shown along with some insight into their interpretation. / Thesis (MSc(AppliedPhysics))--University of South Australia, 2005.

A large part of the structure of the objects in the theory of Dooley and Wildberger [Funktsional. Anal. I Prilozhen. 27 (1993), no. 1, 25-32] and that of Rouviere [Compositio Math. 73 (1990), no. 3, 241-270] can be described by considering a connected, finite-dimentional symmetric space G/H (as defined by Rouviere), with ???exponential map???, Exp, from L G/L H to G/H, an action, ???: K ??? Aut??(G) (where Aut?? (G) is the projection onto G/H of all the automorphisms of G which leave H invariant), of a Lie group, K, on G/H and the corresponding action, ???# , of K on L G/L H defined by g ??? L (???g), along with a quadruple (s, E, j, E#), where s is a ???# - invariant, open neighbourhood of 0 in L G/L H, E is a test-function subspace of C??? (Exp s), j ?? C??? (s), and E# is a test-function subspace of C??? (s) which contains { j.f Exp: f ?? E }. Of interest is the question: Is the function ???: ?? ??? ????, where ??: f ??? j.f Exp, a local associative algebra homomorphism from F# with multiplication defined via convolution with respect to a function e: s x s ??? C, to F, with the usual convolution for its multiplication (where F is the space of all ??? - invariant distributions of E and F# is the space of all ???# - invariant distributions of E#)? For this system of objects, we can show that, to some extent, the choice of the function j is not critical, for it can be ???absorbed??? into the function e. Also, when K is compact, we can show that ??? ker ?? = { f ?? E : ???k f (???g) dg = 0}. These results turn out to be very useful for calculations on s2 ??? G/H, where G = SO(3) and H??? SO(3) with H ??? SO(2) with ??? : h ??? Lh, as we can use these results to show that there is no quadruple (s, E, j, E#) for SO(3)/H with j analytic in some neighbourhood of 0 such that ??? is a local homomorphism from F# to F. Moreover, we can show that there is more than one solution for the case where s, E and E# are as chosen by Rouviere, if e is does not have to satisfy e(??,??) = e(??,??).

An accurate, intuitive and tractable transform as been identified and developed from which the spatial harmonics of reticle patterns defined in polar coordinates can be obtained. A description of reticles and generic methods for describing them mathematically are presented along with some background on general harmonic analysis. Focus then turns to candidate transforms for analysis of reticle patterns and the most promising are investigated in more detail. A fast linear algorithm is devised to overcome some problems with implementation of a fast transform and this is followed by analysis of the transform basis functions to assist with interpretation of the transform in azimuth and radius. A sampling guideline is presented so that aliasing can be avoided and, finally, the transforms of some representative reticle patterns are shown along with some insight into their interpretation. The transformations presented provide a first step toward raising the resolution and harmonic content required in simulation image scenes that will ultimately result in optimal use of computing resources for the simulation of reticle seekers. / thesis (MSc(AppliedPhysics))--University of South Australia, 2005.