This thesis presents a very efficient hybrid method for analysis and design of optical and passive photonic devices. The main focus is on unbounded wave structures. This class of photonic systems are in general very large in terms of the wavelength of the driving optical sources. The size of the problem space makes the electromagnetic modelling of these structure a very challenging problem. Our approach and main contribution has been to combine or hybridize three methods that together can handle this class of photonic structures as a whole. <br /><br /> The basis of the hybrid method is a novel Gaussian Beam Tracing method GBT. Gaussian Beams (GB) are very suitable elementary functions for tracing and tracking purposes due to their finite extent and the fact that they are good approximations for actual laser beams. The GBT presented in this thesis is based on the principle of phase matching. This method can be used to model the reflection and refraction of Gaussian beams from general curved surfaces as long as the curvature of the surface is relatively small. It can also model wave propagation in free space. The developed GBT is extremely fast as it essentially uses simple algebraic equations to find the parameters of the reflected and refracted beams once the parameters of the incident beam is known. Therefore sections of the systems whose dimensions are large relative to the optical wavelength are simulated by the GBT method. <br /><br /> Fields entering a photonic system may not possess an exact Gaussian profile. For example if an aperture limits the input laser to the system, the field is no longer a GB. In these and other similar cases the field at some aperture plane needs to be expanded into a sum of GBs. Gabor expansion has been used for this purpose. This method allows any form of field distribution on a flat or curved surface to be expanded into a sum of GBs. The resultant GBs are then launched inside the system and tracked by GBT. Calculation of the coefficients of the Gabor series is very fast (1-2 minutes on a typical computer for most applications). <br /><br /> In some cases the dimensions or physical properties of structures do not allow the application of the GBT method. For example if the curvature of a surface is very large (or its radius of curvature is very small) or if the surface contains sharp edges or sub-wavelength dimensions GBT is no longer valid. In these cases we have utilized the Finite Difference Time Domain method (FDTD). FDTD is a rigorous and very accurate full wave electromagnetic solver. The time domain form of Maxwell's equations are discretized and solved. No matrix inversion is needed for this method. If the size of the structure that needs to be analyzed is large relative to the wavelength FDTD can become increasingly time consuming. Nevertheless once a structure is simulated using FDTD for a given input, the output is expanded using Gabor expansion and the resultant beams can then be efficiently propagated through any desired system using GBT. For example if a diffraction grating is illuminated by some source, once the reflection is found using FDTD, it can be propagated very efficiently through any kind of lens or prism (or other optical structures) using GBT. Therefore the overall computational efficiency of the hybrid method is very high compared to other methods.

Novel metamaterial and metamaterial-inspired structures and microwave/antenna applications thereof are proposed and studied in this thesis. Motivated by the challenge of extending the applicability of metamaterial structures into practical microwave solutions, the underlying objective of this thesis has been the design of low-cost, easily fabricated and deployable metamaterial-related devices and the development of computational tools for the analysis of those. For this purpose, metamaterials composed of tightly coupled resonators are chosen for the synthesis of artificial transmission lines and enabling antenna applications. Specifically, fully-printed double spiral resonators are employed as modular elements for the design of tightly coupled resonators arrays. After thoroughly investigating the properties of such resonators, they are used for the synthesis of artificial lines in either grounded or non-grounded configurations. In the first case, the supported backward waves are exploited for the design of microstrip-based filtering/diplexing devices and series-fed antenna arrays. In the second case, the effective properties of such structures are employed for the design of a novel class of self-resonant, low-profile folded monopoles, exhibiting low mutual coupling and robust radiating properties. Such monopoles are, in turn, used for the synthesis of different sub-wavelength antenna arrays, such as superdirective arrays. Finally, an in-home periodic FDTD-based computational tool is developed and optimized for the efficient and rigorous analysis of planar, metamaterial-based, high-gain antennas.

This thesis reports on my Ph.D. research in the area of microwave remote sensing of the Arctic. The main objective of this research is to reconstruct the dielectric profile of the snow-covered sea ice, and indirectly retrieve some of its geophysical and thermodynamic properties. To meet this objective, a nonlinear electromagnetic inverse scattering algorithm is developed that consists of forward and inverse solvers. The input to this algorithm is the normalized radar cross section (NRCS) data collected by radar systems from the snow-covered sea ice profile. The proposed inversion algorithm iteratively minimizes a discrepancy between the measured and simulated NRCS data to achieve an accurate reconstruction. Two main challenges associated with this inverse problem are its ill-posedness and its limited available scattering data. To tackle these, the utilization of appropriate regularization and weighting schemes as well as the incorporation of prior information into the inversion algorithm are employed. These include the utilization of (i) appropriate weighting factors for the misfit cost function, (ii) more sensitive NRCS data with respect to the unknown parameters, (iii) further parametrization of the profile based on the expected distribution, (iv) time-series NRCS data to better initialize the inversion process, and (v) NRCS data collected by the satellite and on-site scatterometer to be inverted simultaneously for profile reconstruction. The experimental data utilized are collected by the author in collaboration with the Centre for Earth Observation Science. These measurements are performed on (i) the artificially-grown sea ice in the Sea-ice Environmental Research Facility, located at the University of Manitoba during winter 2014, and (ii) the landfast sea ice located in the Arctic (Cambridge Bay, Nunavut) during May 2014. The measurement procedure includes NRCS data collection through an on-site C-band scatterometer and a spaceborne SAR satellite and physical sampling of the snow and sea ice. The proposed electromagnetic inverse scattering algorithm is utilized to invert these experimental data sets, as well as some synthetic data sets. It will be shown that the use of various techniques developed in this thesis in conjunction with the developed inversion algorithm results in reasonable snow-covered sea ice profile reconstruction. / October 2016

<p>Domain decomposition method provides a solution for the very large electromagnetic</p><p>system which are impossible for single domain methods. Discontinuous Galerkin</p><p>(DG) method can be viewed as an extreme version of the domain decomposition,</p><p>i.e., each element is regarded as one subdomain. The whole system is solved element</p><p>by element, thus the inversion of the large global system matrix is no longer necessary,</p><p>and much larger system can be solved with the DG method compared to the</p><p>continuous Galerkin (CG) method.</p><p>In this work, the DG method is implemented on a subdomain level, that is, each subdomain contains multiple elements. The numerical flux only applies on the</p><p>interfaces between adjacent subdomains. The subodmain level DG method divides</p><p>the original large global system into a few smaller ones, which are easier to solve,</p><p>and it also provides the possibility of parallelization. Compared to the conventional</p><p>element level DG method, the subdomain level DG has the advantage of less total</p><p>DoFs and fexibility in interface choice. In addition, the implicit time stepping is </p><p>relatively much easier for the subdomain level DG, and the total CPU time can be</p><p>much less for the electrically small or multiscale problems.</p><p>The hybrid of elements are employed to reduce the total DoF of the system.</p><p>Low-order tetrahedrons are used to catch the geometry ne parts and high-order</p><p>hexahedrons are used to discretize the homogeneous and/or geometry coarse parts.</p><p>In addition, the non-conformal mesh not only allow dierent kinds of elements but</p><p>also sharp change of the element size, therefore the DoF can be further decreased.</p><p>The DGTD method in this research is based on the EB scheme to replace the</p><p>previous EH scheme. Dierent from the requirement of mixed order basis functions</p><p>for the led variables E and H in the EH scheme, the EB scheme can suppress the</p><p>spurious modes with same order of basis functions for E and B. One order lower in</p><p>the basis functions in B brings great benets because the DoFs can be signicantly</p><p>reduced, especially for the tetrahedrons parts.</p><p>With the basis functions for both E and B, the EB scheme upwind </p><p>ux and</p><p>EB scheme Maxwellian PML, the eigen-analysis and numerical results shows the</p><p>eectiveness of the proposed DGTD method, and multiscale problems are solved</p><p>eciently combined with the implicit-explicit hybrid time stepping scheme and multiple</p><p>kinds of elements.</p><p>The EB scheme DGTD method is further developed to allow arbitrary anisotropic</p><p>media via new anisotropic EB scheme upwind </p><p>ux and anisotropic EB scheme</p><p>Maxwellian PML. The anisotropic M-PML is long time stable and absorb the outgoing</p><p>wave eectively. A new TF/SF boundary condition is brought forward to</p><p>simulate the half space case. The negative refraction in YVO4 bicrystal is simulated</p><p>with the anisotropic DGTD and half space TF/SF condition for the rst time with</p><p>numerical methods.</p> / Dissertation

<p>Electromagnetic forward modeling and inversion methods have extensive applications in geophysical exploration, and large-scale controlled-source electromagnetic method has recently drawed lots of attention. However, to obtain a rigorous and efficient forward solver for this large-scale three-dimensional problem is difficult, since it usually requires to solve for a large number of unknowns from a system of equations describing the complicate scattering behavior of electromagnetic waves that happened within inhomogeneous media. As for the development of an efficient inversion solver, because of the nonlinear, non-unique and ill-posed properties of the problem, it is also a very challenging task. </p><p>In the first part of this dissertation, a fast three-dimensional nonlinear reconstruction method is proposed for controlled-source electromagnetic method. The borehole-to-surface and airborne electromagnetic survey methods are investigated using synthetic data. In this work, it is assumed that there is only electric contrast between the inhomogeneous object and the layered background medium, for the reason that the electric contrast is much more dominant than magnetic contrast in most cases of the earth formation. Therefore, only the EFIE is needed to solve. While the forward scattering problem is solved by the stabilized bi-conjugate gradient FFT (BCGS-FFT) method to give a rigorous and efficient modeling, the Bore iterative method along with the multiplicative regularization technique is used in the inversion through frequency hopping. In the inversion, to speed up the expensive computation of the sensitivity matrix relating every receiver station to every unknown element, a fast field evaluation (FFE) technique is proposed using the symmetry property of the layered medium Green's function combined with a database strategy. The conjugate-gradient method is then applied to minimize the cost function after each iteration. Due to the benefits of using 3D FFT acceleration, the proposed FFE technique as well as the recursive matrix method combined with an interpolation technique to evaluate the LMGF, the developed inversion solver is highly efficient, and requires very low computation time and memory. Numerical experiments for both 3D forward modeling and conductivity inversion are presented to show the accuracy and efficiency of the method. </p><p>Some recent research on artificial nanoparticles have demonstrated the improved performance in geophysical imaging using magnetodielectric materials with enhanced electric and magnetic contrasts. This gives a promising perspective to the future geophysical exploration by infusing well-designed artificial magnetodielectric materials into the subsurface objects, so that a significantly improved imaging can be achieved. As a preparation for this promising application, the second part of the dissertation presents an efficient method to solve the scattering problem of magnetodielectric materials with general anisotropy that are embedded in layered media. In this work, the volume integral equation is chosen as the target equation to solve, since it solves for fields in inhomogeneous media with less number of unknowns than the finite element method. However, for complicated materials as magnetodielectric materials with general anisotropy, it is a very challenging task, because it requires to simultaneously solve the electric field integral equation (EFIE) and magnetic field integral equation (MFIE). Besides that, the numerous evaluation of the layered medium Green's function (LMGF) for the stratified background formation adds on the difficulty and complexity of the problem. To my knowledge, there is no existing fast solver for the similar problem. In this dissertation, using the mixed order stabilized biconjugate-gradient fast Fourier transform (mixed-order BCGS-FFT) method, a fast forward modeling method is developed to solve this challenging problem. Several numerical examples are performed to validate the accuracy and efficiency of the proposed method.</p><p> </p><p>Besides the above mentioned two topics, one-dimensional inversion method is presented in the third part to determine the tilted triaxial conductivity tensor in a dipping layered formation using triaxial induction measurements. The tilted triaxial conductivity tensor is described by three conductivity components and three Euler angles. Based on my knowledge, due to the highly nonlinear and ill-posed nature of the inverse problem, this study serves as the first work that investigates on the subject. There are six principal coordinate systems that can give the same conductivity tensor. Permutation is performed to eliminate the ambiguity of inversion results caused by the ambiguity of the principal coordinate system. Three new Euler angles after permutation for each layer can be found by solving a nonlinear equation. Numerical experiments are conducted on synthetic models to study the feasibility of determining triaxially anisotropic conductivity tensor from triaxial induction data. This project is accomplished during my internship in the Houston Formation Evaluation Integration Center (HFE) at Schlumberger, a world-leading oilfield service company.</p> / Dissertation

Over the past decades, phased arrays have played a significant role in the development of modern radar and communication systems. The availability of printed circuit technology and ease of integration with microwave components, as well as the development of low profile and low weight approaches, have also played an important role in their conformal adaptation. However, fabrication costs remain prohibitive for many emergent platforms, including 5G base stations and autonomous vehicles, when compared to a conventional mechanically steered passive array. Therefore, cost reductions in the fabrication and integration of modern phased arrays are essential to their adaptation for many upcoming commercial applications. Indeed, although phased array design methods are well-understood, even for wideband and wide-angle scanning applications, their fabrication is still based on high-cost, low-yield printed circuit technology. With this in mind, this dissertation focuses on a new planar aperture topology and low-cost techniques for phased array methodologies. The first part of the thesis presents new fabrication advancements using commercially available multi-layered printed circuit technologies. We discuss methods for low cost fabrication while still maintaining performance and design constraints for planar array apertures. The second part of the dissertation presents a novel Integrated Planar Array (IPA) at S-Band and discusses dramatic cost reductions for multi-function radar applications. Performance and cost benefits are presented, and fabrication techniques to exploit an emerging class of high-speed digital laminates are discussed. These are compatible with high-volume, high-yield production, while reducing aperture cost by 75% when compared to conventional approaches. Performance of a planar array employing a pin-fed dual-polarized antenna element with active VSWR Overall, this dissertation addresses several manufacturing and performance challenges in realizing affordable planar phased arrays using low cost fabrication without performance compromise. As commercial interest in phased array technology is anticipated to grow, the proposed approaches for phased array design and fabrication will enable quick turnaround times for mainstream adoption.

This thesis presents a very efficient hybrid method for analysis and design of optical and passive photonic devices. The main focus is on unbounded wave structures. This class of photonic systems are in general very large in terms of the wavelength of the driving optical sources. The size of the problem space makes the electromagnetic modelling of these structure a very challenging problem. Our approach and main contribution has been to combine or hybridize three methods that together can handle this class of photonic structures as a whole. <br /><br /> The basis of the hybrid method is a novel Gaussian Beam Tracing method GBT. Gaussian Beams (GB) are very suitable elementary functions for tracing and tracking purposes due to their finite extent and the fact that they are good approximations for actual laser beams. The GBT presented in this thesis is based on the principle of phase matching. This method can be used to model the reflection and refraction of Gaussian beams from general curved surfaces as long as the curvature of the surface is relatively small. It can also model wave propagation in free space. The developed GBT is extremely fast as it essentially uses simple algebraic equations to find the parameters of the reflected and refracted beams once the parameters of the incident beam is known. Therefore sections of the systems whose dimensions are large relative to the optical wavelength are simulated by the GBT method. <br /><br /> Fields entering a photonic system may not possess an exact Gaussian profile. For example if an aperture limits the input laser to the system, the field is no longer a GB. In these and other similar cases the field at some aperture plane needs to be expanded into a sum of GBs. Gabor expansion has been used for this purpose. This method allows any form of field distribution on a flat or curved surface to be expanded into a sum of GBs. The resultant GBs are then launched inside the system and tracked by GBT. Calculation of the coefficients of the Gabor series is very fast (1-2 minutes on a typical computer for most applications). <br /><br /> In some cases the dimensions or physical properties of structures do not allow the application of the GBT method. For example if the curvature of a surface is very large (or its radius of curvature is very small) or if the surface contains sharp edges or sub-wavelength dimensions GBT is no longer valid. In these cases we have utilized the Finite Difference Time Domain method (FDTD). FDTD is a rigorous and very accurate full wave electromagnetic solver. The time domain form of Maxwell's equations are discretized and solved. No matrix inversion is needed for this method. If the size of the structure that needs to be analyzed is large relative to the wavelength FDTD can become increasingly time consuming. Nevertheless once a structure is simulated using FDTD for a given input, the output is expanded using Gabor expansion and the resultant beams can then be efficiently propagated through any desired system using GBT. For example if a diffraction grating is illuminated by some source, once the reflection is found using FDTD, it can be propagated very efficiently through any kind of lens or prism (or other optical structures) using GBT. Therefore the overall computational efficiency of the hybrid method is very high compared to other methods.

A set of tools are proposed for the efficient modeling of several classes of problems related to periodic structures in microwave and optical regimes with Finite-Difference Time-Domain method. The first category of problems under study is the interaction of non-periodic sources and printed elements with infinitely periodic structures. Such problems would typically require a time-consuming simulation of a finite number of unit cells of the periodic structures, chosen to be large enough to achieve convergence. To alleviate computational cost, the sine-cosine method for the Finite-Difference Time-Domain based dispersion analysis of periodic structures is extended to incorporate the presence of non-periodic, wideband sources, enabling the fast modeling of driven periodic structures via a small number of low cost simulations. The proposed method is then modified for the accelerated simulation of microwave circuit geometries printed on periodic substrates. The scheme employs periodic boundary conditions applied at the substrate, to dramatically reduce the computational domain and hence, the cost of such simulations. Emphasis is also given on radiation pattern calculation, and the consequences of the truncated computational domain of the proposed method on the computation of the electric and magnetic surface currents invoked in the near-to-far-field transformation. It has been further demonstrated that from the mesh truncation point of view, the scheme, which has a unified form regardless dispersion and conductivity, serves as a much simpler but equally effective alternative to the Perfectly Matched Layer provided that the simulated domain is periodic in the direction of termination. The second category of problems focuses on the efficient characterization of nonlinear periodic structures. In Finite-Difference Time-Domain, the simulation of these problems is typically hindered by the fine spatial and time gridding. Originally proposed for linear structures, the Alternating-Direction Implicit Finite-Difference Time-Domain method, as well as a novel spatial filtering method, are extended to incorporate nonlinear media. Both methods are able to use time-step sizes beyond the conventional stability limit, offering significant savings in simulation time.

<p>As research in electromagnetics has expanded, it has given rise to the examination of metamaterials, which possess nontrivial electromagnetic material properties such as engineered permittivity and permeability. Aside from their application in the microwave industry, metamaterials have been associated with novel phenomena since their invention, including sub-wavelength focusing in negative refractive index slabs, evanescent wave amplification in negative index media, and invisibility cloaking and its demonstration at microwave frequency with controlled material properties in space.</p><p>Effective medium theory plays a key role in the development and application of metamaterials, simplifying the electromagnetic analysis of complex engineered metamaterial composites. Any metamaterial composite can be treated as a homogeneous or inhomogeneous medium, while every unit structure in the composite is represented by its permittivity and permeability tensor. Hence, studying an electromagnetic wave's interaction with complex composites is equivalent to studying the interaction between the wave and an artificial material.</p><p>This dissertation first examines the application of a magnetic metamaterial lens in wireless power transfer (WPT) technology, which is proposed to enhance the mutual coupling between two magnetic dipoles in the system. I examine and investigate the boundary effect in the finite sized magnetic metamaterial lens using a numerical simulator. I propose to implement an anisotropic and indefinite lens in a WPT system to simplify the lens design and relax the lens dimension requirements. The numerical results agree with the analytical model proposed by Smith et al. in 2011, where lenses are assumed to be infinitely large.</p><p>By manipulating the microwave properties of a magnetic metamaterial, the nonlinear properties come into the scope of this research. I chose split-ring resonators (SRR) loaded with varactors to develop nonlinear metamaterials. Analogous to linear metamaterials, I developed a nonlinear effective medium model to characterize nonlinear processes in microwave nonlinear metamaterials. I proposed both experimental and numerical methods here for the first time to quantify nonlinear metamaterials' effective properties. I experimentally studied three nonlinear processes: power-dependent frequency tuning, second harmonic generation, and three-wave mixing. Analytical results based on the effective medium model agree with the experimental results under the low power excitation assumption and non-depleted pump approximation. To overcome the low power assumption in the effective medium model for nonlinear metamaterials, I introduced general circuit oscillation models for varactor/diode-loaded microwave metamaterial structures, which provides a qualitative prediction of microwave nonlinear metamaterials' responses at relatively high power levels when the effective medium model no longer fits.</p><p>In addition to 1D nonlinear processes, this dissertation also introduces the first 2D microwave nonlinear field mapping apparatus, which is capable of simultaneously capturing both the magnitude and phase of generated harmonic signals from nonlinear metamaterial mediums. I designed a C-band varactor loaded SRR that is matched to the frequency and space limitation of the 2D mapper. The nonlinear field generation and scattering properties from both a single nonlinear element and a nonlinear metamaterial medium composite are experimentally captured in this 2D mapper, and the results qualitatively agree with numerical results based on the effective medium model.</p> / Dissertation

Tree-based methods have wide applications for solving large-scale problems in electromagnetics, astrophysics, quantum chemistry, fluid mechanics, acoustics, and many more areas. This thesis focuses on their applicability for solving large-scale problems in electromagnetics. The Barnes-Hut (BH) algorithm and the Fast Multipole Method (FMM) are introduced along with a survey of important previous work. The required theory for applying those methods to problems in electromagnetics is presented with particular emphasis on the capacitance extraction problem and broadband full-wave scattering. A novel single source approximation is introduced for approximating clusters of electrostatic sources in multi-layered media. The approximation is derived by matching the spectra of the field in the vicinity of the stationary phase point. Combined with the BH algorithm, a new algorithm is shown to be an efficient method for evaluating electrostatic fields in multilayered media. Specifically, the new BH algorithm is well suited for fast capacitance extraction. The BH algorithm is also adapted to the scalar Helmholtz kernel by using the same methodology to derive an accurate single source approximation. The result is a fast algorithm that is suitable for accelerating the solution of the Electric Field Integral Equation (EFIE) for electrically small structures. Finally, a new version of FMM is presented that is stable and efficient from the low frequency regime to mid-range frequencies. By applying analytical derivatives to the field expansions at the observation points, the proposed method can rapidly evaluate vectorial kernels that arise in the FMM-accelerated solution of EFIE, the Magnetic Field Integral Equation (MFIE), and the Combined Field Integral Equation (CFIE).