In this report, we study various nonlinear wave equations arising in mathematical physics and investigate the existence of solutions to these equations using variational methods. In particular, we look for particle-like traveling wave solutions known as solitary waves. This study is motivated by the prevalence of solitary waves in applications and the rich mathematical structure of the nonlinear wave equations from which they arise. We focus on a semilinear perturbation of Maxwell's equations and the nonlinear Klein - Gordon equation coupled with Maxwell's equations. Physical ramifications of these equations are also discussed.

Traditional light microscopy suffers from the diffraction limit, which limits the spatial resolution to λ/2. The current trend in optical microscopy is the development of techniques to bypass the diffraction limit. Resolutions below 40 nm will make it possible to probe biological systems by imaging the interactions between single molecules and cell membranes. These resolutions will allow for the development of improved drug delivery mechanisms by increasing our understanding of how chemical communication within a cell occurs. The materials sciences would also benefit from these high resolutions. Nanomaterials can be analyzed with Raman spectroscopy for molecular and atomic bond information, or with fluorescence response to determine bulk optical properties with tens of nanometer resolution. Near-field optical microscopy is one of the current techniques, which allows for imaging at resolutions beyond the diffraction limit. Using a combination of a shear force microscope (SFM) and an inverted optical microscope, spectroscopic resolutions below 20 nm have been demonstrated. One technique, in particular, has been named tip enhanced near-field optical microscopy (TENOM). The key to this technique is the use of solid metal probes, which are illuminated in the far field by the excitation wavelength of interest. These probes are custom-designed using finite difference time domain (FDTD) modeling techniques, then fabricated with the use of a focused ion beam (FIB) microscope. The measure of the quality of probe design is based directly on the field enhancement obtainable. The greater the field enhancement of the probe, the more the ratio of near-field to far-field background contribution will increase. The elimination of the far-field signal by a decrease of illumination power will provide the best signal-to-noise ratio in the near-field images. Furthermore, a design that facilitates the delocalization of the near-field imaging from the far-field will be beneficial. Developed is a novel microscope design that employs two-photon non-linear excitation to allow the imaging of the fluorescence from almost any visible fluorophore at resolutions below 30 nm without changing filters or excitation wavelength. The ability of the microscope to image samples at atmospheric pressure, room temperature, and in solution makes it a very promising tool for the biological and materials science communities. The microscope demonstrates the ability to image topographical, optical, and electronic state information for single-molecule identification. A single computer, simple custom control circuits, field programmable gate array (FPGA) data acquisition, and a simplified custom optical system controls the microscope are thoroughly outlined and documented. This versatility enables the end user to custom-design experiments from confocal far-field single molecule imaging to high resolution scanning probe microscopy imaging. Presented are the current capabilities of the microscope, most importantly, high-resolution near-field images of J-aggregates with PIC dye. Single molecules of Rhodamine 6G dye and quantum dots imaged in the far-field are presented to demonstrate the sensitivity of the microscope. A comparison is made with the use of a mode-locked 50 fs pulsed laser source verses a continuous wave laser source on single molecules and J-aggregates in the near-field and far-field. Integration of an intensified CCD camera with a high-resolution monochromator allows for spectral information about the sample. The system will be disseminated as an open system design.

Thesis (Ph.D)--Electrical and Computer Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Smith, Glenn; Committee Member: Buck, John; Committee Member: Goldsztein, Guillermo; Committee Member: Peterson, Andrew; Committee Member: Scott, Waymond. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Orientador: Alexys Bruno Alfonso / Banca: Lucila Helena Deliesposte Cesgato / Banca: Clarissa de Almeida Olivati / O Programa de Pós Graduação em Ciência e Tecnologia de Materiais, PosMat, tem carater institucional e integra as atividades de pesquisa em materiais de diversos campi / Resumo: Os cristais fotônicos têm recebido grande atenção da comunidade científica nos últimos anos. As suas aplicações incluem a construção de guias de onda, chips ópticos e células solares, o controle de emissão de radiação e o gerenciamento de informações. Neste trabalho são calculados os modos de propagação de ondas eletromagnéticas em cristais fotônicos bidimensionais. Os cristais estudados são arranjos de fios circulares infinitos de um tipo de meio óptico, imersos numa matriz de outro meio óptico. As estruturas cristalinas consideradas são a quadrada simples, hexagonal simples e honeycomb. É analisada a propagação perpendicular aos fios e, portanto, são consideradas duas polarizações: a transversão elétricas e a transversal magnética. Os cálculos numéricos são realizados para diferentes combinações dos materiais que compõem o cristal fotônico. Isto permite investigar como o espectro fotônico depende da geometria do cristal e do contraste entre os meios ópticos que o compõem. Para resolver a equação de ondas, o campo eletromagnético é expresso como combinação linear de um conjunto de ondas harmônicas planas. Foi encontrado bom acordo entre as frequencia calculadas e resultados disponíveis na literatura do tema, corroborando a existência de gaps fotônicos parciais nas estrutuaras simples e gaps fotônicos completos na estrutura honeycomb. Além disso, verificou-se que o surgimento de gaps requer menor constrate de índice de refração no caso hexagonal, quando comparado com o quadrado / Abstract: Photoic crystals have received some attention from the scientific community in the las years. Their applications include the fabrication of waveguides, optical chips and solar cells, control of radiation emission and information management. In this work, the propagation of electromagnetic waves in two-dimensional photonic crystals is theoretically investigated. The structures under study are periodic arrangements of infinite circular wires of a given optical medium, immersed in a matrix of another optical medium. The considered cyrstalline structures are the simple square, the simple hexagonal and the honeycomb lattices. Only the propagation perpendicular to the wires is analyzed. Hence, we consider two polarizations: transverse electrical and transverse magnetic. The calculations are performed for different combinations of the materials composing the crystal. This allows the investigation of how the photonic spectrum depends on both the crystal geometry and the contrast between the refractuve index of the composing materials. The wave equation is solved by expressing the solution as a linear combination of a set of harmonic plane waves. Good agreement between the calculated frequencies and the results available in the literature is found, thus confirming the existence of partial photonic gaps in the simple lattices and complete photonic gaps in the honeycomb lattice. Moreover, it was verified that occurrence of gaps in the hexagonal case requires less refractive-index contrast, when compared with the square one / Mestre

Made available in DSpace on 2014-06-11T19:30:20Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-06-29Bitstream added on 2014-06-13T20:40:14Z : No. of bitstreams: 1 sereguetti_jm_me_bauru.pdf: 655401 bytes, checksum: f8b80337046a58d12f1aed1d4dc5a28c (MD5) / Os cristais fotônicos têm recebido grande atenção da comunidade científica nos últimos anos. As suas aplicações incluem a construção de guias de onda, chips ópticos e células solares, o controle de emissão de radiação e o gerenciamento de informações. Neste trabalho são calculados os modos de propagação de ondas eletromagnéticas em cristais fotônicos bidimensionais. Os cristais estudados são arranjos de fios circulares infinitos de um tipo de meio óptico, imersos numa matriz de outro meio óptico. As estruturas cristalinas consideradas são a quadrada simples, hexagonal simples e honeycomb. É analisada a propagação perpendicular aos fios e, portanto, são consideradas duas polarizações: a transversão elétricas e a transversal magnética. Os cálculos numéricos são realizados para diferentes combinações dos materiais que compõem o cristal fotônico. Isto permite investigar como o espectro fotônico depende da geometria do cristal e do contraste entre os meios ópticos que o compõem. Para resolver a equação de ondas, o campo eletromagnético é expresso como combinação linear de um conjunto de ondas harmônicas planas. Foi encontrado bom acordo entre as frequencia calculadas e resultados disponíveis na literatura do tema, corroborando a existência de gaps fotônicos parciais nas estrutuaras simples e gaps fotônicos completos na estrutura honeycomb. Além disso, verificou-se que o surgimento de gaps requer menor constrate de índice de refração no caso hexagonal, quando comparado com o quadrado / Photoic crystals have received some attention from the scientific community in the las years. Their applications include the fabrication of waveguides, optical chips and solar cells, control of radiation emission and information management. In this work, the propagation of electromagnetic waves in two-dimensional photonic crystals is theoretically investigated. The structures under study are periodic arrangements of infinite circular wires of a given optical medium, immersed in a matrix of another optical medium. The considered cyrstalline structures are the simple square, the simple hexagonal and the honeycomb lattices. Only the propagation perpendicular to the wires is analyzed. Hence, we consider two polarizations: transverse electrical and transverse magnetic. The calculations are performed for different combinations of the materials composing the crystal. This allows the investigation of how the photonic spectrum depends on both the crystal geometry and the contrast between the refractuve index of the composing materials. The wave equation is solved by expressing the solution as a linear combination of a set of harmonic plane waves. Good agreement between the calculated frequencies and the results available in the literature is found, thus confirming the existence of partial photonic gaps in the simple lattices and complete photonic gaps in the honeycomb lattice. Moreover, it was verified that occurrence of gaps in the hexagonal case requires less refractive-index contrast, when compared with the square one

Orientador: Edmundo Capelas de Oliveira / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T07:12:57Z (GMT). No. of bitstreams: 1 Rodrigues_FabioGrangeiro_M.pdf: 2473878 bytes, checksum: edf9dfdf8d0b07b971df24b68798f67f (MD5) Previous issue date: 2010 / Resumo: Neste trabalho é dada uma prova de equivalência entre diferentes formas de se escrever a lei de Faraday. a forma integral e a diferencial. Nosso objetivo é esclarecer alguns pontos relevantes da prova dessas equivalências que são normalmente apresentadas nos livros textos de Física apenas para casos muito particulares. Aqui apresentamos a derivação das ditas equivalências para uma situação geral. Incluímos também as ferramentas matemáticas necessárias para o tratamento rigoroso do nosso problema / Abstract: In this work we give a rigorous proof of the equivalence of some different forms of Faraday's law of induction clarifying some misconceptions on the subject and emphasizing that many derivations of this law appearing in textbooks and papers are only valid under very special circunstances and not satisfactory under a mathematical point of view. We include also the mathematical apparatus necessary for a rigorous presentation of our subject / Mestrado / Fisica-Matematica / Mestre em Matemática Universitária

Cette thèse s'inscrit dans un projet d'innovation duale RAPID financé par DGA/DS/MRIS et appelé GREAT faisant intervenir la société Axessim, l'ONERA, INRIA, l'IRMA et le CEA. Ce projet a pour but la mise en place d'une solution industrielle de simulation électromagnétique basée sur une méthode Galerkin Discontinue (GD) parallèle sur maillage hexaédrique. Dans un premier temps, nous établissons un schéma numérique adapté à un système de loi de conservation. Nous pouvons ainsi appliquer cette approche aux équations de Maxwell, mais également à tout système hyperbolique. Dans un second temps, nous mettons en place une parallélisation à deux niveaux de ce schéma. D'une part, les calculs sont parallélisés sur carte graphique au moyen de la bibliothèque OpenCL. D'autre part, plusieurs cartes graphiques peuvent être utilisées, chacune étant pilotée par un processus MPI. De plus, les communications MPI et les calculs OpenCL sont asynchronisés permettant d'obtenir une forte accélération. / This thesis is part of a dual innovation project funded by RAPID DGA/DS/MRIS and called GREAT involving Axessim company, ONERA, INRIA, IRMA and the CEA. This project aims at the establishment of an industrial solution of electromagnetic simulation based on a method Discontinuous Galerkin (DG) on parallel hexahedral mesh. First, we establish a numerical scheme adapted to a conservation law system. We can apply this approach to the Maxwell equations but also to any hyperbolic system. In a second step, we set up a two-level parallelization of this scheme. On the one hand, the calculations are parallelized on graphics card using the OpenCL library. On the other hand, multiple graphics cards can be used, each driven by a MPI process. In addition, MPI communications and OpenCL computations are launched asynchronously in order to obtain a strong acceleration.

We address the problem of the correct description of light-matter coupling for excitons and cavity photons in the case of systems with multiple photon modes or excitons, respectively. In the literature, two different approaches for the phenomenological coupling Hamiltonian can be found: Either one single Hamiltonian with a basis whose dimension equals the sum of photonic modes and excitonic resonances is used. Or a set of independent Hamiltonians, one for each photon mode, is chosen. Both are usually used equivalently for the same kind of multi-photonic systems which cannot be correct. However, identifying the suitable Hamiltonian is difficult when modeling experimental data. By means of numerical transfer matrix calculations, we demonstrate the scope of application of each approach: The first one holds only for the coupling of a single photon state to several excitons, while in the case of multiple photon modes, separate Hamiltonians must be used for each photon mode.

Maxwell's equations for the electromagnetic field are symmetrized by introducing magnetic charges into the formalism of electrodynamics. The symmetrized equations are solved for the fields and potentials of point particles. Those potentials, some of which are found to be singular along a line, are used to formulate the Lagrangian for a system of two dyons (particles with both electric and magnetic charge). The equations of motion are derived from the Lagrangian. It is shown that the dimensionality constants k and k * , which we r e introduced to define the units of the electromagnetic fields, have to be equal in order to avoid center of mass acceleration in the two dyon system.

We present a method for solving the Einstein-Maxwell equations in a five dimensional, asymptotically flat, black hole spacetime with three commuting Killing vector fields. In particular, we show that by reducing the dimension of the Einstein-Maxwell equations in a Kaluza-Klein like manner we can determine the components of the metric and vector potential which lie in the direction of the Killing vector fields. These components are determined by nine scalar fields each of which satisfy a partial differential equation in two variables. These equations take the form of an elliptic operator set equal to a nonlinear source. We find evidence that particular combinations of these fields satisfy Dirichlet boundary conditions, and are well suited to numerical solution using Green functions. Using this method we generate numerical solutions to the 4+1 Einstein-Maxwell equations corresponding to charged generalizations of the Myers-Perry solution. We also discover symmetry relations among the scalar equations which constrain their functional forms and posit the existence of two rigidity-theorem-like relations for electrovac spacetimes and sketch how their use generalizes our method to N+1 dimensions.