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Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (p. [81]-83).

Thesis--Bonn. / Extra t.p. with thesis statement inserted. Includes bibliographical references (p. [87-90]).

Thesis--Bonn. / Extra t.p. with thesis statement inserted. Includes bibliographical references (p. [87-90]).

Thesis (Ph. D.)--Duke University, 2007.

Over the past several decades our understanding and meticulous characterization of the transient and spatial properties of materials evolved rapidly. The results present an exciting field for discovery, and craft materials to control and reshape light that we are just beginning to fathom. State-of-the-art nano-deposition processes, for example, can be utilized to build stratified waveguides made of thin dielectric layers, which put together result in a material with effective abnormal dispersion. Moreover, materials once deemed well known are revealing astonishing properties, v.gr. chalcogenide glasses undergo an atomic reconfiguration when illuminated with electrons or photons, this ensues in a temporal modification of its permittivity and permeability which could be used to build new Photonic Integrated Circuits.. This work revolves around the characterization and model of heterogeneous and time-varying materials and their applications, revisits Maxwell's equations in the context of nonlinear space- and time-varying media, and based on it introduces a numerical scheme that can be used to model waves in this kind of media. Finally some interesting applications for light confinement and beam transformations are shown.

The numerical solutions of Maxwell’s equations have been the cornerstone of photonic design for over a century. In recent years, the field of photonics has witnessed a surge in interest in inverse design, driven by the potential to engineer nonintuitive photonic structures with remarkable properties. However, the conventional approach to inverse design, which relies on fully discretized numerical simulations, faces significant challenges in terms of computational efficiency and scalability. This thesis delves into an alternative paradigm for inverse design, leveraging the power of semi-analytical methods. Unlike their fully discretized counterparts, semi-analytical methods hold the promise of enabling simulations that are independent of the computational grid size, potentially revolutionizing the design and optimization of photonic structures. To achieve this goal, we put forth a more generalized formalism for semi-analytical methods and have developed a comprehensive differential theory to underpin their operation. This theoretical foundation not only enhances our understanding of these methods but also paves the way for their broader application in the field of photonics. In the final stages of our investigation, we illustrate how the semi-analytical simulation framework can be effectively employed in practical photonic design scenarios. We demonstrate the synergy of semi-analytical methods with ray tracing techniques, showcasing their combined potential in the creation of large-scale optical lens systems and other complex optical devices.

We study two finite element methods for solving time-harmonic electromagnetic and acoustic problems: the discontinuous Petrov-Galerkin (DPG) method and the hybrid discontinuous Galerkin (HDG) method. The DPG method for the Helmholtz equation is studied using a test space normed by a modified graph norm. The modification scales one of the terms in the graph norm by an arbitrary positive scaling parameter. We find that, as the parameter approaches zero, better results are obtained, under some circumstances. A dispersion analysis on the multiple interacting stencils that form the DPG method shows that the discrete wavenumbers of the method are complex, explaining the numerically observed artificial dissipation in the computed wave approximations. Since the DPG method is a nonstandard least-squares Galerkin method, its performance is compared with a standard least-squares method having a similar stencil. We study the HDG method for complex wavenumber cases and show how the HDG stabilization parameter must be chosen in relation to the wavenumber. We show that the commonly chosen HDG stabilization parameter values can give rise to singular systems for some complex wavenumbers. However, this failure is remedied if the real part of the stabilization parameter has the opposite sign of the imaginary part of the wavenumber. For real wavenumbers, results from a dispersion analysis for the Helmholtz case are presented. An asymptotic expansion of the dispersion relation, as the number of mesh elements per wave increase, reveal values of the stabilization parameter that asymptotically minimize the HDG wavenumber errors. Finally, a dispersion analysis of the mixed hybrid Raviart-Thomas method shows that its wavenumber errors are an order smaller than those of the HDG method. We conclude by presenting some contributions to the development of software tools for using the DPG method and their application to a terahertz photonic structure. We attempt to simulate field enhancements recently observed in a novel arrangement of annular nanogaps.

M.Ing. / This work comprises the implementation and comparison of five design techniques for the design of gradient-index thin film optical filters: classical rugate, inverse Fourier transform, a wavelet-based design procedure, as well as the flip-flop and the genetic optimization techniques. Designs for a high-reflectance filter, a beamsplitter, a discrete level filter, a distributed filter, and an anti-reflection coating were used to compare the various filter synthesis techniques. The optical thickness of the various examples was maintained below 30 and the refractive index excursion limits were between 1.5 and 3.2. The overall performance of a specific design was evaluated by a weighted merit function. The classical rugate filter uses a sinusoidal refractive index modulation that produces a single reflection band. More complex filters are realized by linear superposition of these elementary profiles. Sidelobe and ripple suppression are obtained by applying quintic windowing functions to the refractive index profile and adding matching layers at the edges of the filter. This filter design procedure has the best figure of merit of 3.73 for the discrete level filter, and the second best of 3.09 for the high-reflectance filter. The inverse Fourier transform links the refractive index profile and reflection spectrum of an optical filter by an approximate relation. It is self-correcting and iterative in nature. It produces filters with the highest optical density. The procedure excels in the design of the distributed filter with a figure of merit of 4.17. Mortlett's wavelet is used as the basis of the wavelet design technique. A single wavelet yields a single reflection band, similar to the classical rugate filter. Sidelobe suppression is an inherent property of the method, but matching layers are needed for passband ripple suppression. The optical density of the high reflection filter is larger for a filter designed with this method than for the equivalent classical rugate filter. The figure of merit of 1.75 for the high-reflectance filter is the best for any of the designs. Flip-flop refinement is a brute force approach to filter design. The layers of a starting design are flipped between two values of refractive index, the change in figure of merit evaluated and the best case saved. This process is repeated for a fixed number of iterations. It is computationally intensive and lacks ripple suppression characteristics. The flip-flop method does not compare well with any of the other techniques. It yields filters with the worst figures of merit for most of the design examples. However, it was applied successfully to the anti-reflection coating. The peak ripple for the anti-reflection filter in the 400 nm to 1100 nm wavelength band is 9.62 % compared to the inverse Fourier transform's 57.30 %. The genetic algorithm operates on the principle of "survival of the fittest". It is a stochastic procedure and yields quasi-random refractive index profiles. It excels with the antireflection coating. The peak ripple in the passband of the anti-reflection coating is 3.29%. The figure of merit for the anti-reflection coating designed with the genetic algorithm is 2.09.

In this thesis, we consider four different stochastic partial differential equations. Firstly, we study stochastic Helmholtz equation driven by an additive white noise, in a bounded convex domain with smooth boundary of Rd (d = 2, 3). And then with the help of the perfectly matched layers technique, we also consider the stochastic scattering problem of Helmholtz type. The second part of this thesis is to investigate the time harmonic case for stochastic Maxwell's equations driven by an color noise in a simple medium, and then we expand the results to the stochastic Maxwell's equations in case of dispersive media in Rd (d = 2, 3). Thirdly, we study stochastic parabolic partial differential equation driven by space-time color noise, where the domain O is a bounded domain in R2 with boundary ∂O of class C2+alpha for 0 < alpha < 1/2. In the last part, we discuss the stochastic wave equation (SWE) driven by nonlinear noise in 1D case, where the noise 626x6t W(x, t) is the space-time mixed second-order derivative of the Brownian sheet. / Many physical and engineering phenomena are modeled by partial differential equations which often contain some levels of uncertainty. The advantage of modeling using so-called stochastic partial differential equations (SPDEs) is that SPDEs are able to more fully capture interesting phenomena; it also means that the corresponding numerical analysis of the model will require new tools to model the systems, produce the solutions, and analyze the information stored within the solutions. / One of the goals of this thesis is to derive error estimates for numerical solutions of the above four kinds SPDEs. The difficulty in the error analysis in finite element methods and general numerical approximations for a SPDE is the lack of regularity of its solution. To overcome such a difficulty, we follow the approach of [4] by first discretizing the noise and then applying standard finite element methods and discontinuous Galerkin methods to the stochastic Helmholtz equation and Maxwell equations with discretized noise; standard finite element method to the stochastic parabolic equation with discretized color noise; Galerkin method to the stochastic wave equation with discretized white noise, and we obtain error estimates are comparable to the error estimates of finite difference schemes. / We shall focus on some SPDEs where randomness only affects the right-hand sides of the equations. To solve the four types of SPDEs using, for example, the Monte Carlo method, one needs many solvers for the deterministic problem with multiple right-hand sides. We present several efficient deterministic solvers such as flexible CG method and block flexible GMRES method, which are absolutely essential in computing statistical quantities. / Zhang, Kai. / Adviser: Zou Jun. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3552. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 144-155). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.