The interaction of light and matter at nanoscale is the subject of study of this dissertation. Particularly, the coupling of light to surface plasmons and their applications in the fields of spectroscopy and sensing is the focus of this work. In terms of spectroscopy, the simple reason of using light to study the chemical structures of different materials is the fact that the energy of light lies in the range of vibrational and electronic transitions of matter. Further, the ability to squeeze light to subwavelength dimensions opens up new possibilities of designing nano-optical devices. In this work we explore surface plasmons for two major applications: (i) Directivity enhanced Raman spectroscopy and (ii) Chemical/biological sensing. Here a new enhancement phenomenon has been demonstrated experimentally in regards to Raman spectroscopy. Typically, Raman enhancement is considered in terms of local fields only. Here we show the use of directive nanoantennas to provide additional enhancement of two orders of magnitude. The nanoantenna design is optimal in the sense that almost all of the scattered light is coupled into the numerical aperture of the collecting lens. It is shown that the additional enhancement from directivity pushes the sensitivity to single molecule regime. Further, the out of plane radiation and simplicity of the design makes it an ideal candidate for use with typical commercial microscope setups. Extra ordinary transmission through nanohole arrays in metallic films is studied for refractive index sensing. Bulk resolution of 6×10-7 is demonstrated by optimizing array dimensions, wavelength of operation, noise reduction and consideration of sensitivity of the detecting CCD camera. Self-assembled nanostructures are investigated for spectroscopic applications. Time dependent studies of nanorods assembled in end-to-end and side-by-side configurations are conducted. The end-to-end configuration results in higher local field enhancements whereas; the side-by-side configuration shows a reduction in local fields because of the cancellation of radial field components between the neighbouring nanorods. It should be noted that higher fields are desirable for Raman spectroscopy. Grating structures have been analysed using reduced coupled mode theory. In most cases, only three lowest order modes prove to be sufficient for accurate description of the system response. Here we present design guidelines for broadband operation and optimization of high quality factor resonators. Finally the complex reflection coefficient from arbitrary terminated nanorods has been investigated. Phase of reflection plays an important role in the determination of resonance wavelength of nanoantennas. It is shown that the localized surface plasmon resonance of nanoparticles can be considered in terms of propagating surface plasmons along a nanorod of similar geometry where the length of the nanorod approaches zero accompanied with π degrees of phase of reflection. The contributions made in this work can prove useful in the fields of analytical chemistry and biomedical sensing. The directive nanoantenna can find applications in a number of areas such as light emitting devices, photovoltaics, single photon sources and high resolution microscopy. Our work related to EOT based sensing is already approaching the resolution of commercially available refractive index sensors with the added advantage of multiplexed detection. / Graduate

Public-Private-Partnership (P3) is gaining momentum as the delivery method for the development of public infrastructure. These projects, however, are exposed to economic risks. If the private parties are not comfortable with the level of the risks, they would not participate in the project and, as a result, the infrastructure will most likely not be realized. As an incentive for participation in the P3 project, private parties are sometimes offered guarantees against unfavorable economic risks. Therefore, the valuation of these guarantees is essential for deciding whether or not to participate in the project. While previous works focused on the valuation of guarantees, the incorporation of credit risk in the value of the P3 projects and the guarantees has been neglected. The effect of credit risk can be taken into account by using the rigorous Credit Value Adjustment method (CVA). CVA is a computationally demanding method that the valuation methods currently in the literature are not capable of handling. This research offers a novel approach for the valuation of guarantees and P3 projects which is computationally superior to the existing methods. Because of this computational efficiency, CVA can be implemented to account for credit risk. For the development of this method, a continuous stochastic differential equation (SDE) is derived from the forecasted curve of an economic risk. Using the SDE, the partial differential equation (PDE) governing the value of the guarantees will be derived. Then, the PDE will be solved using Finite Difference Method (FDM). A new feature for this method is that it obtains exercise strategies for the Australian guarantees. The present work extends the literature by providing a valuation method for the cases that multiple risks affect P3 projects. It also presents an approach for the valuation of the Asian style guarantee, a contract which reimburses the private party based on the average of risk factor. Finally, a hypothetical case study illustrates the implementation of the FDM-based valuation method and CVA to obtain the value of the P3 project and the guarantees adjusted for the counterparty credit risk.

Public-Private-Partnership (P3) is gaining momentum as the delivery method for the development of public infrastructure. These projects, however, are exposed to economic risks. If the private parties are not comfortable with the level of the risks, they would not participate in the project and, as a result, the infrastructure will most likely not be realized. As an incentive for participation in the P3 project, private parties are sometimes offered guarantees against unfavorable economic risks. Therefore, the valuation of these guarantees is essential for deciding whether or not to participate in the project. While previous works focused on the valuation of guarantees, the incorporation of credit risk in the value of the P3 projects and the guarantees has been neglected. The effect of credit risk can be taken into account by using the rigorous Credit Value Adjustment method (CVA). CVA is a computationally demanding method that the valuation methods currently in the literature are not capable of handling. This research offers a novel approach for the valuation of guarantees and P3 projects which is computationally superior to the existing methods. Because of this computational efficiency, CVA can be implemented to account for credit risk. For the development of this method, a continuous stochastic differential equation (SDE) is derived from the forecasted curve of an economic risk. Using the SDE, the partial differential equation (PDE) governing the value of the guarantees will be derived. Then, the PDE will be solved using Finite Difference Method (FDM). A new feature for this method is that it obtains exercise strategies for the Australian guarantees. The present work extends the literature by providing a valuation method for the cases that multiple risks affect P3 projects. It also presents an approach for the valuation of the Asian style guarantee, a contract which reimburses the private party based on the average of risk factor. Finally, a hypothetical case study illustrates the implementation of the FDM-based valuation method and CVA to obtain the value of the P3 project and the guarantees adjusted for the counterparty credit risk.

In order to handle turbulent flow problems, one equation turbulence models are implemented in to a previously developed explicit, Reynolds averaged Navier-Stokes solver. Discretization of Navier-Stokes solver is based on cell-vertex finite volume formulation combined with single step Lax-Wendroff numerical method which is second order accurate in space. Turbulent viscosity is calculated by using one equation Spalart-Allmaras and Baldwin-Barth turbulence transport equations. For the discretization of Spalart-Allmaras and Baldwin-Barth equations, both finite volume scheme which is used for Navier-Stokes equation in this work and explicit finite difference discretization method are used. In order to increase the convergence rate of the solver, local time stepping technique is applied. Stabilization of non-physical oscillations resulting from the numerical scheme is maintained by adding second and fourth order artificial smoothing terms. Three test cases are considered. In order to validate the accuracy of the Navier-Stokes solver, solver is tested over a laminar flat plate. The results are compared with analytical solutions. Later, in order to check the performance of the turbulence models, turbulent flow over flat plate and turbulent transonic flow over NACA-0012 airfoil are handled. For turbulent flow over flat plate obtained results are compared with analytical and empirical solutions, whereas for transonic turbulent flow obtained results are compared with numerical and experimental solutions.

Power distribution networks (PDNs) are conducting structures employed in semiconductor systems with the aim of providing circuits with reliable and constant operating voltage. This network has non-neglible electrical parasitics. Consequently, when digital circuits inside the chip switch, the supply voltage delivered to them does not remain ideal and exhibits spatial and temporal voltage fluctuations. These fluctuations in the supply voltage, known as the power-supply noise (PSN), can affect the functionality and the performance of modern microprocessors. The design of this PDN in the chip is an important part in ensuring power integrity. Modeling and simulation of the PSN in on-chip PDNs is important to reduce the cost of processors. These PDNs have irregular geometries, which affect the PSN. As a result, they have to be modeled. The problem sizes encountered in this simulation are usually large (on the order of millions), necessitating computationally efficient simulation approaches. Existing approaches for this simulation do not guarantee at least one of the following three required properties: computationally efficiency, accuracy, and numerically robustness. Therefore, there is a need to develop accurate, numerically robust, and efficient algorithms for this simulation. For many interconnects (e.g., transmission lines, board connectors, package PDNs), only their frequency responses and SPICE circuits (e.g., nonlinear switching drivers, equivalent circuits of interconnects) terminating them are known. These frequency responses are usually available only up to a certain maximum frequency. Simulating the electrical behavior of these systems is important for the reliable design of microprocessors and for their faster time-to-market. Because terminations can be nonlinear, a transient simulation is required. There is a need for a transient simulation of band-limited frequency-domain data characterizing a multiport passive system with SPICE circuits. The number of ports can be large (greater than or equal to 100 ports). In this simulation, unlike in traditional circuit simulators, normal properties like stability and causality of transient results are not automatically met and have to be ensured. Existing techniques for this simulation do not guarantee at least one of the following three required properties: computationally efficiency for a large number of ports, causality, and accuracy. Therefore, there is a need to develop accurate and efficient time-domain techniques for this simulation that also ensure causality. The objectives of this Ph.D. research are twofold: 1) To develop accurate, numerically robust, and computationally efficient time-domain algorithms to compute PSN in on-chip PDNs with irregular geometries. 2) To develop accurate and computationally efficient time-domain algorithms for the causal cosimulation of band-limited frequency-domain data with SPICE circuits.

In this research, a new technique for modeling periodic structures in the finite-difference time-domain (FDTD) method is developed, and the technique is applied to the study of structural color in insects. Various recent supplements to the FDTD method, such as a nearly-perfect plane-wave injector and convolutional perfectly matched layer boundary condition, are used. A method for implementing the FDTD method on a parallel, distributed-memory computer cluster is given. To model a periodic structure, a single periodic cell is terminated by periodic boundary conditions (PBCs). A new technique for incorporating PBCs into the FDTD method is presented. The simplest version of the technique is limited to two-dimensional, singly-periodic geometries. The accuracy is demonstrated by comparing to independent results calculated with a frequency-domain, mode-matching method. The periodic FDTD method is then extended to the more general case of three-dimensional, doubly-periodic problems. This extension requires additional steps and imposes new limitations. The computational cost and limitations of the method are presented. Certain species of butterflies exhibit structural color, which is caused by quasi-periodic structures on the scales covering the wings. Numerical experiments are performed to develop a technique for modeling quasi-periodic structures using the periodic FDTD method. The observed structural color of butterflies is then calculated from the electromagnetic data using colorimetric theory. Three types of butterflies are considered. The first type are from the Morpho genus. These are typically a brilliant, almost metallic, blue color. The second type is the Troides magellanus, which exhibits an interplay of structural and pigmentary color, but the structural color is only visible near grazing incidence. The final type is the Ancyluris meliboeus, which exhibits iridescence on the ventral side. For all cases, the effects of changing the dimensions of various structural elements are considered. Finally, some earlier work on the design of TEM horn antennas is presented. The TEM horn is a simple and popular antenna, but only limited design information is available in the literature. A parametric study was performed, and the results are given. A complete derivation of the characteristic impedance of the basic antenna is also presented.

In this thesis, we have analyzed the accuracy and stability aspects of weak boundary and interface conditions (WBCs) for high order finite difference methods on Summations-By-Parts (SBP) form. The numerical technique has been applied to wave propagation and flow problems. The advantage of WBCs over strong boundary conditions is that stability of the numerical scheme can be proven. The boundary procedures in the advection-diffusion equation for a boundary layer problem is analyzed. By performing Navier-Stokes calculations, it is shown that most of the conclusions from the model problem carries over to the fully nonlinear case. The work was complemented to include the new idea of using WBCs on multiple grid points in a region, where the data is known, instead of at a single point. It was shown that we can achieve high accuracy, an increased rate of convergence to steady-state and non-reflecting boundary conditions by using this approach. Using the SBP technique and WBCs, we have worked out how to construct conservative and energy stable hybrid schemes for shocks using two different approaches. In the first method, we combine a high order finite difference scheme with a second order MUSCL scheme. In the second method, a procedure to locally change the order of accuracy of the finite difference schemes is developed. The main purpose is to obtain a higher order accurate scheme in smooth regions and a low order non-oscillatory scheme in the vicinity of shocks. Furthermore, we have analyzed the energy stability of the MUSCL scheme, by reformulating the scheme in the framework of SBP and artificial dissipation operators. It was found that many of the standard slope limiters in the MUSCL scheme do not lead to a negative semi-definite dissipation matrix, as required to get pointwise stability. Finally, high order simulations of shock diffracting over a convex wall with two facets were performed. The numerical study is done for a range of Reynolds numbers. By monitoring the velocities at the solid wall, it was shown that the computations were resolved in the boundary layer. Schlieren images from the computational results were obtained which displayed new interesting flow features.

This thesis consists of two different parts, related to two different fields in mathematical physics: wave propagation and kinetic theory of gases. Various mathematical and computational problems for equations from these areas are treated.   The first part is devoted to high order finite difference methods for the Helmholtz equation and the wave equation. Compact schemes with high order accuracy are obtained from an investigation of the function derivatives in the truncation error. With the help of the equation itself, it is possible to transfer high order derivatives to lower order or to transfer time derivatives to space derivatives. For the Helmholtz equation, a compact scheme based on this principle is compared to standard schemes and to deferred correction schemes, and the characteristics of the errors for the different methods are demonstrated and discussed. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous media.   The second part addresses some problems related to kinetic equations. A direct simulation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and numerical tests are performed to verify the accuracy of the algorithm. A formal derivation of the method from the Boltzmann equation with grazing collisions is performed. The linear and linearized Boltzmann collision operators for the hard sphere molecular model are studied using exact reduction of integral equations to ordinary differential equations. It is demonstrated how the eigenvalues of the operators are found from these equations, and numerical values are computed. A proof of existence of non-zero discrete eigenvalues is given. The ordinary diffential equations are also used for investigation of the Chapman-Enskog distribution function with respect to its asymptotic behavior.

One of the most significant goals in coal mining technology research is the automation of underground coal mining machinery. A current challenge with automating underground coal mining machinery is measuring and maintaining a coal mining horizon. The coal mining horizon is the horizontal path the machinery follows through the undulating coal seam during the mining operation. A typical mining practice is to leave a thin remnant of coal unmined in order to maintain geological stability of the cutting face. If the remnant layer is too thick, resources are wasted as the unmined coal is permanently unrecoverable. If the remnant layer is too thin, the product is diluted by mining into the overburden and there is an increased risk of premature roof fall which increases danger. The main challenge therefore is to develop a robust sensing method to estimate the thickness of thin remant coal layers. This dissertation addresses this challenge by presenting a pattern recognition methodology to estimate thin remnant coal layer thickness using ground penetrating radar (GPR). The approach is based upon a novel feature vector, derived from the bispectrum, that is used to characterise the early-time segment of 1D GPR data. The early-time segment is dominated by clutter inherent in GPR systems such as antenna crosstalk, ringdown and ground-bounce. It is common practice to either time-gate the signal, disregard the clutter by rendering the early-time segment unusable, or configure the GPR equipment to minimise the clutter effects which in turn reduces probing range. Disregarding the early-time signal essentially imposes a lower thickness limit on traditional GPR layer thickness estimators. The challenges of estimating thin layer thickness is primarily due to these inherent clutter components. Traditional processing strategies attempt to minimise the clutter using pre-processing techniques such as the subtraction of a calibration signal. The proposed method, however, treats the clutter as a deterministic but unknown signal with additive noise. Hence the proposed approach utilises the energy from the clutter and monitors change in media from subtle changes in the signal shape. Two complementary processing methods important to horizon sensing have been also proposed. These methods, near-surface interface detection and antenna height estimation, may be used as pre-validation tools to increase the robustness of the thickness estimation technique. The proposed methods have been tested with synthetic data and validated with real data obtained using a low power 1.4 GHz GPR system and a testbed with known conditions. With the given test system, it is shown that the proposed thin layer thickness estimator and near-surface interface detector outperform the traditional matched filter based processing methods for layers less than 5 cm in thickness. It is also shown that the proposed antenna height estimator outperforms the traditional height estimator for heights less than 7 cm. These new methods provide a means for reliably extending layer thickness estimation to the thin layer case where traditional approaches are known to fail.

Some normally consolidated soft soils manifest strength sensitivity, ie these soil manifest strain softening when shear in an undrained mode. These soils, referred to as sensitive soft soils, have the typical features of strain hardening in drained shearing and strain softening in undrained shearing. The consolidation lines of these soils are also curved (concave upwards) in the semi-log space. However, under high consolidation stress or upon large shearing, these soils re-gain the features of re-constituted soil. Ground improvement methods like stone columns were reported as not effective when installed in the sensitive soft clays. But mechanism of the un-effectiveness of the stone columns remains unknown because of lack of a suitable and simple model for simulating the stress-strain behaviours of sensitive soft soils. Although these soils have a meta-stable micro-structure, models that developed for simulating structured firm soils are not suitable for simulating sensitive soft soil features. Thus, a new model was formulated. The new model can degenerate back to a Modified Cam Clay model. The ability of new model in simulating a range of behaviour was verified by using the finite difference (FD) method in solving the partial differential equations of the soil model for a range of tri-axial test conditions. The model was further implemented in coupled analysis formulation and coded into FEM program AFENA. Various cases with different soil parameters were then simulated and compared with the FD solutions for various triaxial tests so as to check the stability of the FEM code. The coupled FEA was then used to simulate the performance of geosynthetic-encased stone columns. A new stone column element and a geo-encasement element were developed and coded into AFENA. The stone column simulations were then done for both non-sensitive soils (represented by Modified Cam Clay model) and sensitive soft soil (represented by the new model). Parametric study was conducted to examine the performance of the geo-encased stone columns in both types of soils. Furthermore, two different installation methods: wished-in installation and full displacement installation were studied numerically. Cross comparison was done to investigate how the sensitive soft soil features interact with the installation method in affecting the performance of the geo-encased stone columns. A range of factors that influence the geosynthetic-encased stone columns performance installed in soft soils were also made clear.