An efficient integral equation method based on a method of moment (MoM) discretization of the Mixed-Potential Integral Equation (MPIE) for the analysis of 2.5D or 3D planar microwave circuits is presented. The robust Discrete Complex Image Method (DCIM) is employed to approximate the Greens functions in layered media for horizontal and vertical sources of fields, where closed-form formulations of the z-integrations are derived in the spectral domain. Meanwhile, an efficient and accurate numerical integration technique based on the Khayat-Wilton transform is used to integrate functions with 1/R singularities and near singularities. The fast iterative solver - Quadrature Sampled Pre-Corrected Fast Fourier Transform (QSPCFFT) - is associated with the MoM formulation to analyze electrically large, dense and complex microwave circuits.

Understanding physical configurations and how these can emerge from the underlying gauge theory is a fundamental problem in modern particle physics. This thesis investigates the study of these configurations primarily focussing on the need for gauge invariance in constructing the gauge invariant fields for any physical theory. We consider Wu’s approach to gauge invariance by identifying the gauge symmetry preserving conditions in quantum electrodynamics and demonstrate how Wu’s conditions for one-loop order calculations (under various regularisation schemes) leads to the maintenance of gauge invariance. The need for gauge invariance is stressed and the consequences discussed in terms of the Ward identities for which various examples and proofs are presented in this thesis. We next consider Zwanziger’s description of a mass term in Yang-Mills theory, where an expansion is introduced in terms of the quadratic and cubic powers of the field strength. Although Zwanziger introduced this expansion there is, however, no derivation or discussion about how it arises and how it may be extended to higher orders. We show how Zwanziger’s expansion in terms of the inverse covariant Laplacian can be derived and extended to higher orders. An explicit derivation is presented, for the first time, for the next to next to leading order term. The role of dressings and their factorisation lies at the heart of this analysis.

The two variable Fitzhugh Nagumo model behaves qualitatively like the four variable Hodgkin-Huxley space clamped system and is more mathematically tractable than the Hodgkin Huxley model, thus allowing the action potential and other properties of the Hodgkin Huxley system to be more readily be visualized. In this thesis, it is shown that the Carleman Embedding Technique can be applied to both the Fitzhugh Nagumo model and to Van der Pol's model of nonlinear oscillation, which are both finite nonlinear systems of differential equations. The Carleman technique can thus be used to obtain approximate solutions of the Fitzhugh Nagumo model and to study neural activity such as excitability.

This thesis focuses on theoretical study of charge transportproperties in molecular systems. The understanding of the transportprocess and mechanism in molecular systems is essential forthe design of new functional molecular materials and molecularelectronic devices. The molecular junctions and organic molecularcrystals have been used as the model systems to highlight the usefulnessof theoretical modelling. A molecular junction is a system that consists ofone or several molecules sandwiched between two electrodes.The charge transport in molecular junctions is a very complex processthat is affected by the interaction between molecules and electrodes,the surroundings, as well as electron-electron (e-e) andelectron-phonon (e-p) couplings. When the molecule-electrode couplingis strong, the transport process can be very quick. If the e-p couplingis weak, the inelastic tunneling has only negligible contributions to thetotal current and the elastic electron tunneling plays the dominant role.Furthermore, the hopping process becomes dominant in the case of strong e-pcoupling, for which the geometric relaxation of the molecule needsto be considered. In this thesis, we have examined these three kinds oftransport processes separately. The first studied system is a molecular junction consisting of aromaticallycoupled bimolecules. Its elastic electron tunneling property is simulatedusing Green's functional theory at density functional theory level.The dependence of the conductance of bimolecular junctions on the vertical distances,horizontal distances and the tilt angles has been systematically studied. Theinelastic electron tunneling spectra (IETS) of molecular junctions have beencalculated for several systems that were experimentally measured with conflictingresults and controversial assignments. Our calculations provide the reliableassignments for the experimental spectra and revealed unprecedented detailsabout the molecular conformations within the junctions under different conditions.It demonstrates that a combined theoretical and experimental IETS study is capableof accurately determining the structure of a single molecule inside the junction.The hopping process is a dominant charge transfer process in organic molecularcrystals. We have studied the charge transport ability of four kinds of n-typeorganic semiconductor materials to find out the related structure-to-propertyrelationship. It is done by adopting the quantum charge transfer rate equationcombined with the random walk approach. / QC 20120515

Photonics nowadays is one of the most rapidly developing areas of modern physics. Photonic chips are considered to be promising candidates for a new generation of high-performance systems for informational technology, as the photonic devices provide much higher information capacity in comparison to conventional electronics. They also offer the possibility of integration with electronic components to provide increased functionality. Photonics has also found numerous applications in various fields including signal processing, computing, sensing, printing, and others. Photonics, which traditionally covers lasing cavities, waveguides, and photonic crystals, is now expanding to new research directions such as plasmonics and nanophotonics. Plasmonic structures, namely nanoparticles, metallic and dielectric waveguides and gratings, possess unprecedented potential to guide and manipulate light at nanoscale. This Thesis presents the results of theoretical studies of light propagation in photonic and plasmonic structures, namely lasing disk microcavities, photonic crystals, metallic gratings and nanoparticle arrays. A special emphasis has been made on development of high-performance techniques for studies of photonic devices. The following papers are included: In the first two papers (Paper I and Paper II) we developed a novel scattering matrix technique for calculation of resonant states in 2D disk microcavities with the imperfect surface or/and inhomogeneous refraction index. The results demonstrate that the surface imperfections represent the crucial factor determining the $Q$ factor of the cavity. A generalization of the scattering-matrix technique to the quantum-mecha\-nical electron scattering has been made in Paper III. This has allowed us to treat a realistic potential of quantum-corrals (which can be considered as nanoscale analogues of optical cavities) and has provided a new insight and interpretation of the experimental observations. Papers IV and V present a novel effective Green's function technique for studying light propagation in photonic crystals. Using this technique we have analyzed surface modes and proposed several novel surface-state-based devices for lasing/sensing, waveguiding and light feeding applications. In Paper VI the propagation of light in nanorod arrays has been studied. We have demonstrated that the simple Maxwell Garnett effective-medium theory cannot properly describe the coupling and clustering effects of nanorods. We have demonstrated the possibility of using nanorod arrays as high-quality polarizers. In Paper VII we modeled the plasmon-enhanced absorption in polymeric solar cells. In order to excite a plasmon we utilized a grated aluminum substrate. The increased absorption has been verified experimentally and good agreement with our theoretical data has been achieved.

Hole-doped maganite with middle to narrow bandwidth La1-xCaxMnO3 was extensively studied because of its colossal magnetoresistance (CMR) characteristic under a magnetic field. These kind of materials show un- common magnetic and electric properties. The charge order phase only happens to the region x> 0.5, and along with decreasing temperature, its phase goes from para-insulator to charge-ordered then to antiferromagne- tism. In our studies, we apply correlation function of Green¡¦s function to LCMO and get susceptibility of charge and spin. Then we can get the cri- tical value of Coulomb repulsion inside the material by substituting the experimental values of phase transition temperature. This critical values is the key point of charge-ordered. Then we can also get the size of char- ge gap which decides the stability of charge-ordered phase. After know- ing the Coulomb repulsion and charge gap, we can picture the relation of inside and on-site Coulomb repulsion qualitatively while the transition happens. Here the on-site Coulomb repulsion means to the Hund¡¦s coupl- ing between d electrons. And by this we¡¦ll understand the physics inside CMR materials.

<p>A theoretical framework for calculating electron transport through molecular junctions is presented. It is based on scattering theory using a Green's function formalism. The model can take both elastic and inelastic scattering into account and treats chemical and physical bonds on equal footing. It is shown that it is quite reliable with respect to the choice of functional and basis set. Applications concerning both elastic and inelastic transport are presented, though the emphasis is on the inelastic transport properties. The elastic scattering application part is divided in two part. The first part demonstrates how the current magnitude is strongly related to the junction width, which provides an explanation why experimentalists get two orders of magnitude differences when performing measurements on the same type of system. The second part is devoted to a study of how hydrogenbonding affects the current-voltage (I-V) characteristics. It is shown that for a conjugated molecule with functional groups, the effects can be quite dramatic. This shows the importance of taking possible intermolecular interactions into account when evaluating and comparing experimental data. The inelastic scattering part is devoted to get accurate predictions of inelastic electron tunneling spectroscopy (IETS) experiments. The emphasis has been on elucidating the importance of various bonding conditions for the IETS. It is shown that the IETS is very sensitive to the shape of the electrodes and it can also be used to discriminate between different intramolecular conformations. Temperature dependence is nicely reproduced. The junction width is shown to be of importance and comparisons between experiment as well as other theoretical predictions are made.</p>

Elastic Wave propagation in structures with irregular boundaries is studied by transforming the plates with irregular surfaces to sinusoidal wave-guides. Guided elastic wave in a two-dimensional periodically corrugated plate is studied analytically. The plate material is considered as homogeneous, isotropic and linearly elastic. In a periodically corrugated wave-guide, all possible spectral orders of wave numbers are considered. The dispersion equation is obtained by applying the traction free boundary conditions at the two surfaces. The analysis is carried out in the wave-number domain for both symmetric and anti-symmetric modes. Non-propagating 'stop bands' and propagating 'pass bands' are investigated. Experimental analyses with two different pairs of transducers are also performed and compared with the results from the mathematical analysis. Newly developed semi-analytical DPSM technique has been also adopted in this dissertation to model the ultrasonic field in sinusoidally corrugated plate. Distributed Point Source Method (DPSM) is gradually gaining popularity in the field of Non-Destructive Evaluation (NDE). DPSM can be used to calculate the ultrasonic field (pressure, velocity and displacement in a fluid or stress and displacement in a solid) generated by ultrasonic transducers. So far the technique has been used to model ultrasonic field in homogeneous or multilayered fluid structures. In this dissertation the method is extended to model the ultrasonic field generated in both fluid and solid media. The Prime objective of using DPSM technique in this dissertation is to model the ultrasonic field generated in the corrugated wave guide. This method has never been used to model ultrasonic field in solids. Development of stress and displacement Green's functions in solids are presented. In addition to the wave propagation problem in the sinusoidal wave guide, a few unsolved problems such as ultrasonic field generated by bounded acoustic beams in multilayered fluid structures, near a fluid-solid interface and in flat solid isotropic plates are also presented in this dissertation.

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics.

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics.