Los solitones ópticos son paquetes de luz (haces y/o pulsos) que no se dispersan gracias al balance entre difracción/dispersión y no linealidad. Al propagarse e interactuar los unos con los otros muestran propiedades que normalmente se asocian a partículas. Las propiedades de los solitones ópticos en fibras ópticas y cristales han sido investigadas en profundidad durante las últimas dos décadas. Sin embargo, los solitones en mallas, o redes, ópticas, que podrían ser usados para procesado y direccionamiento totalmente óptico de señales, se han convertido en una nueva área de investigación. El principal objetivo de esta tesis es el estudio de nuevas técnicas para controlar solitotes en medios no lineales en mallas ópticas.El capítulo 2 se centra en ciertas propiedades de los solitones ópticos en medios no lineales cuadráticos. La primera sección presenta en detalle la existencia y estabilidad de tres familias representativas de solitones espacio temporales en dos dimensiones en series de frentes de onda cuadráticos no lineales. Se asume, además de la dispersión temporal del pulso, la combinación de difracción discreta que surge debido al acoplamiento débil entre frentes de onda vecinos. La otra sección da cuenta de la existencia y estabilidad de vórtices de solitones multicolores en retículo, consistentes en cuatro jorobas principales dispuestas en una configuración cuadrada. También se investiga la posibilidad de generarlos dinámicamente a partir de haces de entrada Gaussianos con vórtices anidados. La técnica de inducción de mallas ópticas ofrece un sinfín de posibilidades para la creación de configuraciones de guía de ondas con varios haces de luz no difractantes. El capítulo 3 presenta el concepto de estructuras reconfigurables ópticamente inducidas por haces no difractantes de Bessel mutuamente incoherentes en medios no lineales de tipo Kerr. Los acopladores de dos nucleos son introducidos y se muestra cómo calibrar las propiedades de conmutación de estas estructuras variando la intensidad de los haces de Bessel. El capítulo también discute varios escenarios de conmutación para solitones lanzados al interior de acopladores direccionales multinucleares ópticamente inducidos por apropiadas series de haces de Bessel. Es más, la propagación de solitones es investigada en redes reconfigurables bidimensionales inducidas ópticamente por series de haces de Bessel no difractantes. Se muestra que los haces anchos de solitones pueden moverse a través de redes con diferentes topologías casi sin pérdidas por radiación. Finalmente, se estudian las propiedades de las uniones X, que se crean a partir de dos haces de Bessel intersectantes. La respuesta no local de los medios no lineales puede jugar un papel importante en las propiedades de los solitones. El capítulo 4 trata el impacto de la no localidad en las características físicas exhibidas por los solitones que permiten los medios no lineales de tipo Kerr con una retícula óptica integrada. El capítulo investiga propiedades de diferentes familias de solitones en mallas en medios no lineales no locales. Se muestra que la no localidad de la respuesta no lineal puede afectar profundamente la movilidad de los solitones. Las propiedades de los solitones de gap también se discuten en el caso de cristales fotorefractivos con una respuesta de difusión no local asimétrica y en presencia de una malla inducida.El capítulo 5 trata del impacto de la no localidad en la estabilidad de complejos de solitones en medios no lineales de tipo Kerr uniformes. En primer lugar, se muestra que la diferente respuesta no local de los materiales tiene distinta influencia en la estabilidad de los complejos de solitones en el caso escalar. En segundo lugar, se da cuenta de una serie de resultados experimentales sobre solitones multipolares escalares en medios no lineales fuertemente no locales en 2D, incluyendo solitones dipolares, tripolares y de tipo pajarita, organizados en series de puntos brillantes fuera de fase. Finalmente, el capítulo estudia la interacción entre la no linealidad no local y el acoplamiento vectorial, enfatizando especialmente la estabilización de efectos vectoriales en complejos de solitones en medios no lineales no locales.Por último, el capítulo 6 resume los principales resultados obtenidos en la tesis y discute algunas cuestiones abiertas. / Optical solitons are light packets (beams and/or pulses) that do not broaden because of the proper balance between diffraction/dispersion and nonlinearity. They propagate and interact with one another while displaying properties that are normally associated with real particles. The properties of optical solitons in optical fibers and crystals have been investigated comprehensively during the last two decades. However, solitons in optical lattices, which might be used for all-optical signal processing and routing have recently emerged a new area of research. The main objective of this thesis is the investigation of new techniques for soliton control in nonlinear media with/without an imprinted optical lattice. Chapter 2 focuses on properties of optical solitons in quadratic nonlinear media. The first section presents in detail the existence and stability of three representative families of two-dimensional spatiotemporal solitons in quadratic nonlinear waveguide arrays. It is assumed in addition to the temporal dispersion of the pulse, the combination of discrete diffraction that arises because of the weak coupling between neighboring waveguides. The other section reports on the existence and stability of multicolor lattice vortex solitons, which comprise four main humps arranged in a square configuration. It is also investigated the possibility of their dynamical generation from Gaussian-type input beams with nested vortices. The technique of optical lattice induction opens a wealth of opportunities for creation of waveguiding configurations with various nondiffracting light beams. Chapter 3 puts forward the concept of reconfigurable structures optically induced by mutually incoherent nondiffracting Bessel beams in Kerr-type nonlinear media. Two-core couplers are introduced and it is shown how to tune the switching properties of such structures by varying the intensity of the Bessel beams. The chapter also discusses various switching scenarios for solitons launched into the multi core directional couplers optically-induced by suitable arrays of Bessel beams. Furthermore, propagation of solitons is investigated in reconfigurable two-dimensional networks induced optically by arrays of nondiffracting Bessel beams. It is shown that broad soliton beams can move across networks with different topologies almost without radiation losses. Finally, properties of X-junctions are studied, which are created with two intersecting Bessel beams.Nonlocal response of nonlinear media can play an important role in properties of solitons. Chapter 4 treats the impact of nonlocality in the physical features exhibited by solitons supported by Kerr-type nonlinear media with an imprinted optical lattice. The chapter investigates properties of different families of lattice solitons in nonlocal nonlinear media. It is shown that the nonlocality of the nonlinear response can profoundly affect the soliton mobility. The properties of gap solitons are also discussed for photorefractive crystals with an asymmetric nonlocal diffusion response and in the presence of an imprinted optical lattice.Chapter 5 is devoted to the impact of nonlocality on the stability of soliton complexes in uniform nonlocal Kerr-type nonlinear media. First, it is shown that the different nonlocal response of materials has different influence on the stability of soliton complexes in scalar case. Second, experimental work is reported on scalar multi-pole solitons in 2D highly nonlocal nonlinear media, including dipole, tripole, and necklace-type solitons, organized as arrays of out-of-phase bright spots. Finally, the chapter addresses the interplay between nonlocal nonlinearity and vectoral coupling, specially emphasizing the stabilization of vector effects on soliton complexes in nonlocal nonlinear media.Finally, Chapter 6 summarizes the main results obtained in the thesis and discusses some open prospects.

Percutaneous transluminal balloon angioplasty followed by drug-eluting stent implantation has been of great benefit in coronary applications, whereas in peripheral applications, success rates remain low. Analysis of healing patterns in successful deployments shows that six months after implantation the artery has reorganized itself to accommodate the increase in caliber and there is no purpose for the stent to remain, potentially provoking inflammation and foreign body reaction. Thus, a fully biodegradable polymeric stent that fulfills the mission and steps away is of great benefit. Biodegradable polymers have a widespread usage in the biomedical field, such as sutures, scaffolds and implants. Degradation refers to bond scission process that breaks polymeric chains down to oligomers and monomers. Extensive degradation leads to erosion, which is the process of mass loss from the polymer bulk. The prevailing mechanism of biodegradation of aliphatic polyesters (the main class of biodegradable polymers used in biomedical applications) is random scission by passive hydrolysis and results in molecular weight reduction and softening. In order to understand the applicability and efficacy of biodegradable polymers, a two pronged approach involving experiments and theory is necessary. A constitutive model involving degradation and its impact on mechanical properties was developed through an extension of a material which response depends on the history of the motion and on a scalar parameter reflecting the local extent of degradation and depreciates the mechanical properties. A rate equation describing the chain scission process confers characteristics of stress relaxation, creep and hysteresis to the material, arising due to the entropy-producing nature of degradation and markedly different from their viscoelastic counterparts. Several initial and boundary value problems such as inflation and extension of cylinders were solved and the impacts of the constitutive model analyzed. In vitro degradation of poly(L-lactic acid) fibers under tensile load was performed and degradation and reduction in mechanical properties was dependent on the mechanical environment. Mechanical testing of degraded fibers allowed the proper choice of constitutive model and its evolution. Analysis of real stent geometries was made possible with the constitutive model integration into finite element setting and stent deformation patterns in response to pressurization changed dramatically as degradation proceeded.

<p>Nonlinear transmission lines (NLTLs) are of great interest for high power microwave (HPM) generation because they can sharpen pulses to create an electromagnetic shockwave to produce oscillations from 100 MHz to low GHz. NLTLs provide frequency agility, compactness, durability and reliability, providing a solid-state radiofrequency (RF) source for producing HPM. The essential component of NLTLs is the nonlinear material, typically a dielectric that varies with voltage or a magnetic material whose permeability varies with current, incorporated in the transmission line in various topologies. This thesis presents an alternative approach involving designing composites comprised of nonlinear dielectric inclusions (barium strontium titanate (BST)) and/or nonlinear inductive inclusions (nickel zinc ferrites (NZF)) in a polymer base host material, analogous to electromagnetic interference designs that incorporate stainless steel inclusions of various shapes in a plastic to tune the composite’s electromagnetic properties at GHz. Appropriately designing NLTL composites requires predicting these effective properties both in linear (for a fixed and low voltage and current) and nonlinear regions (permittivity and permeability become voltage dependent and current dependent, respectively) prior to designing HPM systems comprised of them. As a first step, this thesis evaluates and benchmarks composites models in the commercial software CST Microwave Studios (CST MWS) to various effective medium theories (EMTs) to predict the permittivity and permeability of composites of BST and/or NZF inclusions in the linear regime, compared with experimental measurements. The manufacturing and measurement of the nonlinear composites will be briefly discussed with an analysis of the homogeneity of a composite sample using 3D X-ray scan. Long-term application of these approaches to predicting the effective nonlinear composite permittivity and permeability and future work will be discussed.</p>

Este trabalho tem por objetivo a extensão de métodos numéricos baseados em diferenças finitas no domínio do tempo (FDTD) e no domínio da freqüência (FD-BPM) para a simulação da propagação de ondas eletromagnéticas em materiais com propriedades ópticas diversas, por exemplo, isotrópicos, anisotrópicos, lineares, não-lineares, bem como a combinação destes em uma mesma estrutura. Inicialmente foram elaborados formalismos bidimensionais (FDTD e FD-BPM), dos quais foram investigados modos com polarização TM (Magnético Transversal) que se propagam em estruturas planares magnetoópticas/não-lineares/lineares. Esta polarização foi escolhida tendo em vista o campo magnetostático dc adotado, o qual possibilitou a observação do fenômeno não-recíproco associado ao não-linear simultaneamente. Por outro lado, é bem sabido que o método FDTD é computacionalmente muito intensivo. Portanto, um grande esforço foi dedicado aos formalismos no domínio da freqüência, os quais foram implementados em duas e três dimensões. Este último foi estendido para um formalismo totalmente vetorial, capaz de simular modos híbridos ou até mesmo a transferência de energia entre modos de polarizações ortogonais. Isto nos permitiu investigar geometrias ainda mais complexas, tais como um isolador óptico baseado em um guia de onda tip rib utilizando material magnetooptico. Adicionalmente, fenômenos de natureza complexa, tais como a dinâmica dos condensados de luz em materiais com não-lineares do tipo Kerr com saturação, também conhecidos como meios não-lineares cúbico-qüínticos, foram investigados pela primeira vez com um formalismo vetorial. Finalmente, métodos numéricos capazes de considerar qualquer combinação de materiais com propriedades ópticas distintas (linear e/ou não-linear e/ou magnetoóptico) são uma ferramenta extraordinária para a comunidade científica para o projeto de novos dispositivos ópticos, bem como a investigação de novos efeitos físicos com vistas à aplicações em computação óptica, que podem resultar em um menor e mais eficiente número de componentes para sistemas de comunicações ópticos. / This work introduces three improved formalisms for the analysis of electromagnetic wave propagation through materials with distinct optical properties, i.e., isotropic, anisotropic, linear, nonlinear, or any combination of them. Two finite difference approaches were extensively investigated in this work for this purpose, namely the finite difference in time domain (FDTD), and the finite difference beam propagation method (2D and 3D FD-BPM), these in frequency domain. Initially, a TM (transverse magnetic) mode propagating through a planar magnetooptic/nonlinear/linear waveguide was investigated by way of a two-dimensional formalism (FDTD and FD-BPM). This mode polarization was chosen based on the orientation of the external magnetostatic field adopted, which favored the observation of non-reciprocal and nonlinear effects simultaneously. On the other hand, it is well known that FDTD formalisms are computationally intensives. Therefore, a great effort was dedicated to its frequency domain counterpart (FD-BPM), which was implemented in two and three dimensions. The later was further extended to a fully vectorial formalism, which is capable of simulating hybrid modes or even the energy transfer between orthogonal modes. This enabled us to investigate more complex geometries, such as an optical isolator based on magnetooptic rib waveguide. Additionally, complex phenomena, such as the dynamic of light condensates in bulk nonlinear Kerr media with saturation, also known as cubic-quintic nonlinear media, were investigated for the first time with a 3D vectorial formalism. Finally, numerical methods capable of handling any combination of materials with distinct optical properties (linear and/or nonlinear and/or magnetooptic) are an extraordinary tool for the scientific community for the design of new optical devices, as well as the investigation of new physical effects aimed for optical computing, that may result in fewer and more efficient components for optical communication systems.

Ces dernières années, la Méthode Intégrale de Volume (MIV) a reçu une attention particulière pour lamodélisation des problèmes électromagnétiques en basse fréquence. Son intérêt principal est l’absencedu maillage de la région air, ce qui rend la méthode légère et rapide. Associée aux méthodes decompression matricielle la MIV devient aujourd'hui une alternative compétitive à la méthode deséléments finis pour la modélisation de dispositifs électromagnétiques ayant un volume d'airprépondérant.Ce rapport porte sur le développement de deux formulations intégrales de volume pour la résolution deproblèmes magnétostatiques avec prise en compte des matériaux non linéaires, des aimants, desbobines, des circuits magnétiques avec ou sans entrefer et des régions minces magnétiques. Lapremière est une formulation en flux de mailles indépendantes basée sur l'interpolation par éléments defacette. La deuxième est une formulation en potentiel vecteur magnétique basée sur l'interpolation paréléments d'arête. L'application de ces formulations permet d’une part d'obtenir des résultats précismême en présence d’un faible maillage et d’autre part de résoudre aisément des problèmes nonlinéaires. Des méthodes de calcul de la force magnétique globale ainsi que du flux magnétique dansles bobines ont été également mises en oeuvre. Les développements informatiques ont été réalisés dansla plateforme MIPSE et ont été validés sur des problèmes académiques ainsi que sur quelquesdispositifs industriels. / In recent years, the Volume Integral Method (VIM) has been received particular attention formodeling of low frequency electromagnetic problems. The main advantage of this method is thatinactive regions do not to be discretized, which makes it light and rapid. Associated with matrixcompression methods, the VIM is a competitive alternative to the finite element method for modelingelectromagnetic devices containing a predominant air volume.This PhD thesis focuses on the development of two volume integral formulations for solvingmagnetostatic problems, in the presence of nonlinear materials, magnets, coils, multiply connectedmagnetic regions, and the presence of magnetic shielding. The first one is a mesh magnetic fluxformulation based on the interpolation of facet elements and the second one is a magnetic vectorpotential formulation based on the interpolation of edge elements. The application of theseformulations provides accurate results even with coarse meshes and allows solving straightforwardnonlinear magnetostatic problems. Methods for computing global magnetic force and magnetic fluxthrough a coil were also implemented as part of this work. Developments performed in the MIPSEplatform were validated on academic case-tests as well as some industrial devices.

In this thesis, we investigate the nonlinear optical properties of four different physical systems: tellurite glasses, castor oil, ionic liquids and colloids of gold nanoparticles. Using Zscan and I-scan techniques, it was possible to determine the values of the electronic (n2 e) and thermal (n2 t) contributions of nonlinear refractive index of these systems as well as evaluating their respective thermo-optical coefficients (dn/dT). We use the I-scan technique to characterize five tellurite glass samples with different compositions. In this experiment we employ a Ti:sapphire laser operating in the modelocked regime, tuned at , delivering pulses, with a repetition rate that was controlled by a pulse selector. These glasses presented an ultra-fast self-focusing nonlinearity. The figure of merit 810 nm 200 fs 1kHz max 0 W= &#916;n &#955;&#945; was evaluated, and the condition was obtained for four of the five studied samples, displaying the potentiality of these glasses for ultra-fast all-optical switching applications, for example. Castor oil is a natural organic compound with a wide range of applications in industry. In the nanotechnology field, this oil has been exploited as very efficient dispersant and stabilizer agent for metallic gold nanoparticles in colloidal systems. However, a lack of data in literature concerning nonlinear optical properties of this material exists. In this work, we use the Z-scan technique to measure the nonlinear optical response of castor oil for laser excitation at 514 and 810 nm. In the visible region, the measurements had been carried out in the CW regime, using an Argon laser. In the infrared region, a Ti:sapphire laser, operating in the modelocked regime, producing pulses of 200 fs, with low (1 kHz) and high (76 MHz) repetition rate was employed instead. The castor oil presented a self-defocusing nonlinear refraction for both the laser wavelengths. The influence of the electronic and thermal contributions for nonlinearity was evaluated and the results indicate that the thermal effects are the main responsible for the observed nonlinear refraction. The thermo-optical coefficient ( ) of this compound was also measured for both wavelengths. We observe that castor oil thermo-optical coefficient is approximately an order of magnitude larger for the excitation tuned at 514 nm than at 810 nm. The nonlinear optical properties of two kinds of ionic liquids, BMI.BF4 and BMI.PF6, had been investigated. These materials are organic salts that present a low melting temperature and negligible vapor pressure. Although they have interesting physical-chemistry properties, and have been used in several applications, their nonlinear optical properties had been little investigated. In this work, we use the Z-scan technique at W > 0.27 dn / dT 514 nm and . Both ionic liquids displayed high self-defocusing nonlinearity, of thermal origin. We observed that the change of anion by anion modify the optical properties of these compounds. The ionic liquids had also presented a dispersion behavior in their thermo-optical coefficients in the spectral range studied. Although thermo-optical nonlinearities are a problem for the development of ultra-fast photonic devices, they can present a nonlocal character as a consequence of the process of heat conduction. Nonlinear effects in nonlocal media have been investigated in diverse branches of the physics, in particular in phenomena such as light pulses nonlinear propagation, as well as in generation and interaction of spatial solitons. Thus, these results suggest that castor oil and ionic liquids are promising candidates for investigation of nonlinear effects in nonlocal media. In the characterization of the colloidal systems of gold nanoparticles dispersed in castor oil, we evaluate the nonlinear refractive index, nonlinear absorption coefficient, as well as the thermo-optical coefficient in function of the filling factor f. Using Z-scan technique, for the laser excitation tuned at , we observe that the colloids presented an ultra-fast selfdefocusing refractive nonlinear response. Using the generalized Maxwell-Garnett model for composite materials it was possible to explain the behavior of the nonlinear refractive index of the colloid as a function of the filling factor, as well as estimate the value of the real part of the gold nanoparticles third-order nonlinear susceptibility. We also observe that the presence of gold nanoparticles dispersed in castor oil increased the absolute value of the linear absorption coefficient, the nonlinear refraction index of thermal origin and the thermo-optical coefficient. Our results indicate that the presence of gold nanoparticles modifies significantly local and nonlocal nonlinearities of a colloidal system. Moreover, the amount of nanoparticles is an extremely important factor for the development of new nanostructured materials aiming ultra-fast optical and nonlocal applications. 810 nm &#8722; 4 BF &#8722; 6 PF 800 nm / Fundação de Amparo a Pesquisa do Estado de Alagoas / Nesta tese, investigamos as propriedades ópticas não lineares de quatro sistemas físicos distintos: vidros teluretos, óleo de mamona, líquidos iônicos e colóides de nanopartículas de ouro. Utilizando as técnicas de varredura Z (Z-scan) e varredura de intensidade (I-scan) foi possível determinar os valores das contribuições de origem eletrônica (n2e) e térmica (n2t) do índice de refração não linear dos sistemas estudados, bem como avaliar os seus respectivos coeficientes termo-ópticos (dn/dT). Usamos a técnica de varredura de intensidade para caracterizar cinco amostras de vidros teluretos com diferentes composições. Neste experimento utilizamos um laser de Titânio de Safira operando no regime modelocked, sintonizado em , emitindo pulsos de de duração, com a taxa de repetição ajustada em através de um seletor de pulsos. Os vidros apresentaram uma não linearidade auto-focalizadora ultra-rápida. A figura de mérito 810 nm 200 fs 1 kHz max 0 W= &#916;n &#955;&#945; foi avaliada, ea condição foi obtida para quatro das cinco amostras estudadas, demonstrando a potencialidade destes vidros para aplicações em chaveamento totalmente óptico ultra-rápido, por exemplo. O óleo de mamona é um composto orgânico natural com uma ampla gama de aplicações na indústria. No campo da nanotecnologia, este óleo tem sido explorado como um agente dispersante e estabilizante muito eficiente para sistemas coloidais de nanopartículas metálicas de ouro. Entretanto, existe uma carência de dados na literatura acerca das propriedades ópticas não lineares deste material. Neste trabalho, utilizamos a técnica de varredura Z para medir a resposta óptica não linear do óleo de mamona para excitações em 514 nm e 810 nm. Na região visível, as medidas foram realizadas no regime CW, utilizando um laser argônio. No infravermelho, um laser de Titânio de Safira, operando no regime modelocked, produzindo pulsos de , com baixa ( ) e alta ( ) taxa de repetição foi utilizado. O óleo de mamona apresentou uma refração não linear autodesfocalizadora, em ambos os comprimentos de onda. A influência das contribuições eletrônica e térmica para a não linearidade medida foi avaliada e os resultados obtidos indicam que os efeitos térmicos são os principais responsáveis pela refração não linear observada. O coeficiente termo-óptico (W > 0,27 200 fs 1kHz 76 MHz dn dT ) deste composto também foi medido para os dois comprimentos de onda. Observamos que o dn dT do óleo de mamona é aproximadamente uma ordem de magnitude maior para a excitação sintonizada em 514 nm que em 810 nm. As propriedades ópticas não lineares de dois tipos de líquidos iônicos, BMI.BF4 e BMI.PF6, também foram investigadas. Estes materiais são sais orgânicos que se caracterizam por apresentar uma baixa temperatura de fusão e pressão de vapor desprezível. Apesar de possuir propriedades físico-químicas interessantes, e serem usados em diversas aplicações, suas propriedades ópticas não lineares foram pouco investigadas. Neste trabalho, usamos a técnica de varredura Z para excitação em 514 nm e 810 nm. Ambos os líquidos iônicos apresentaram uma grande não linearidade auto-desfocalizadora, de origem térmica. Observamos que a mudança do ânion pelo ânion modifica as propriedades ópticas destes compostos. Os líquidos iônicos também apresentaram uma dispersão nos seus coeficientes termo-ópticos no intervalo espectral estudado. Apesar de ser um problema para o desenvolvimento de dispositivos fotônicos ultra-rápidos, não linearidades termo-ópticas podem apresentar um caráter de não localidade como uma conseqüência do processo de condução de calor. Efeitos não lineares em meios não locais vêm sendo abordados em diversos ramos da física, em particular em fenômenos de propagação não linear de pulsos de luz, e na geração e interação de sólitons espaciais. Os resultados obtidos sugerem que tanto o óleo de mamona, quanto os líquidos iônicos são candidatos promissores para investigação de efeitos não lineares não locais. Na caracterização dos sistemas coloidais de nanopartículas de ouro dispersas em óleo de mamona avaliamos o índice de refração não linear, coeficiente de absorção não linear, bem como o coeficiente termo-óptico em função do fator de preenchimento f. Fazendo uso da técnica de varredura Z, para o laser de excitação sintonizado em , observamos que os colóides apresentaram uma resposta refrativa não linear autodesfocalizadora ultra-rápida. Utilizando o modelo de Maxwell-Garnett generalizado para materiais compostos foi possível explicar o comportamento do índice de refração não linear do colóide em função do fator de preenchimento, bem como estimar o valor da parte real da susceptibilidade não linear de terceira ordem das nanopartículas de ouro. Observamos também que a presença de nanopartículas de ouro dispersas no óleo de mamona aumentou o valor absoluto do coeficiente de absorção linear, do índice refração não linear de origem térmica e do coeficiente termo-óptico. Nossos resultados indicam que a presença de nanopartículas de ouro altera significativamente as respostas não lineares locais e não locais de um sistema coloidal. Desta forma, a quantidade de nanopartículas é um fator extremamente importante para o desenvolvimento de novos materiais nanoestruturados visando aplicações ópticas tanto ultra-rápidas, quanto não locais.

Este trabalho tem por objetivo a extensão de métodos numéricos baseados em diferenças finitas no domínio do tempo (FDTD) e no domínio da freqüência (FD-BPM) para a simulação da propagação de ondas eletromagnéticas em materiais com propriedades ópticas diversas, por exemplo, isotrópicos, anisotrópicos, lineares, não-lineares, bem como a combinação destes em uma mesma estrutura. Inicialmente foram elaborados formalismos bidimensionais (FDTD e FD-BPM), dos quais foram investigados modos com polarização TM (Magnético Transversal) que se propagam em estruturas planares magnetoópticas/não-lineares/lineares. Esta polarização foi escolhida tendo em vista o campo magnetostático dc adotado, o qual possibilitou a observação do fenômeno não-recíproco associado ao não-linear simultaneamente. Por outro lado, é bem sabido que o método FDTD é computacionalmente muito intensivo. Portanto, um grande esforço foi dedicado aos formalismos no domínio da freqüência, os quais foram implementados em duas e três dimensões. Este último foi estendido para um formalismo totalmente vetorial, capaz de simular modos híbridos ou até mesmo a transferência de energia entre modos de polarizações ortogonais. Isto nos permitiu investigar geometrias ainda mais complexas, tais como um isolador óptico baseado em um guia de onda tip rib utilizando material magnetooptico. Adicionalmente, fenômenos de natureza complexa, tais como a dinâmica dos condensados de luz em materiais com não-lineares do tipo Kerr com saturação, também conhecidos como meios não-lineares cúbico-qüínticos, foram investigados pela primeira vez com um formalismo vetorial. Finalmente, métodos numéricos capazes de considerar qualquer combinação de materiais com propriedades ópticas distintas (linear e/ou não-linear e/ou magnetoóptico) são uma ferramenta extraordinária para a comunidade científica para o projeto de novos dispositivos ópticos, bem como a investigação de novos efeitos físicos com vistas à aplicações em computação óptica, que podem resultar em um menor e mais eficiente número de componentes para sistemas de comunicações ópticos. / This work introduces three improved formalisms for the analysis of electromagnetic wave propagation through materials with distinct optical properties, i.e., isotropic, anisotropic, linear, nonlinear, or any combination of them. Two finite difference approaches were extensively investigated in this work for this purpose, namely the finite difference in time domain (FDTD), and the finite difference beam propagation method (2D and 3D FD-BPM), these in frequency domain. Initially, a TM (transverse magnetic) mode propagating through a planar magnetooptic/nonlinear/linear waveguide was investigated by way of a two-dimensional formalism (FDTD and FD-BPM). This mode polarization was chosen based on the orientation of the external magnetostatic field adopted, which favored the observation of non-reciprocal and nonlinear effects simultaneously. On the other hand, it is well known that FDTD formalisms are computationally intensives. Therefore, a great effort was dedicated to its frequency domain counterpart (FD-BPM), which was implemented in two and three dimensions. The later was further extended to a fully vectorial formalism, which is capable of simulating hybrid modes or even the energy transfer between orthogonal modes. This enabled us to investigate more complex geometries, such as an optical isolator based on magnetooptic rib waveguide. Additionally, complex phenomena, such as the dynamic of light condensates in bulk nonlinear Kerr media with saturation, also known as cubic-quintic nonlinear media, were investigated for the first time with a 3D vectorial formalism. Finally, numerical methods capable of handling any combination of materials with distinct optical properties (linear and/or nonlinear and/or magnetooptic) are an extraordinary tool for the scientific community for the design of new optical devices, as well as the investigation of new physical effects aimed for optical computing, that may result in fewer and more efficient components for optical communication systems.

La réduction d'ordre de modèles d'un problème de contact demeure un sujet de recherche important en mécanique numérique des solides.Nous proposons une extension de l'hyper-réduction avec domaine d'intégration réduit à la mécanique du contact sans frottement s'écrivant à l'aide d'une formulation mixte.Comme la zone de contact potentiel se limite au domaine réduit, nous faisons le choix de prendre comme base réduite pour la variable duale (représentative des forces de contact) la base du modèle d'ordre plein restreinte.Nous obtenons ainsi un modèle hyper-réduit hybride avec une approximation de la variable primale par des modes empiriques et de la variable duale par les fonctions de base des éléments finis. Si nécessaire, la condition inf-sup de ce modèle peut être forcée par une approximation hybride la variable primale. Cela mène à une stratégie hybride combinant un modèle d'ordre hyper-réduit et un modèle d'ordre plein permettant l'obtention d'une meilleure approximation de la solution sur la zone de contact.Un post-traitement permettant la reconstruction des multiplicateurs de Lagrange sur l'ensemble de la zone de contact est également introduit.De manière à optimiser la sélection des snapshots, un indicateur d'erreur simple et efficace est avancé pour être couplé à un algorithme glouton. / The model order reduction of mechanical problems involving contact remains an important issue in computational solid mechanics.An extension of the hyper-reduction method based on a reduced integration domain to frictionless contact problems written by a mixed formulation is proposed.As the potential contact zone is naturally reduced through the reduced domain, the dual reduced basis is chosen as the restriction of the dual full-order model basis.A hybrid hyper-reduced model combining empirical modes for primal variables with finite element approximation for dual variables is then obtained.If necessary, the inf-sup condition of this hybrid saddle point problem can be enforced by extending the hybrid approximation to the primal variables. This leads to a hybrid hyper-reduced/full-order model strategy. By this way, a better approximation on the potential contact zone is furthermore obtained.A post-treatment dedicated to the reconstruction of the contact forces on the whole domain is introduced.In order to optimize the snapshots selection, an efficient error indicator is coupled to a greedy sampling algorithm leading to a robust reduced-order model.

This work aims at dimensional reduction of nonlinear material models in an asymptotically accurate manner. The three-dimensional(3-D) nonlinear material models considered include isotropic, orthotropic and dielectric compressible hyperelastic material models. Hyperelastic materials have potential applications in space-based inflatable structures, pneumatic membranes, replacements for soft biological tissues, prosthetic devices, compliant robots, high-altitude airships and artificial blood pumps, to name a few. Such structures have special engineering properties like high strength-to-mass ratio, low deflated volume and low inflated density. The majority of these applications imply a thin shell form-factor, rendering the problem geometrically nonlinear as well. Despite their superior engineering properties and potential uses, there are no proper analysis tools available to analyze these structures accurately yet efficiently. The development of a unified analytical model for both material and geometric nonlinearities encounters mathematical difficulties in the theory but its results have considerable scope. Therefore, a novel tool is needed to dimensionally reduce these nonlinear material models. In this thesis, Prof. Berdichevsky’s Variational Asymptotic Method(VAM) has been applied rigorously to alleviate the difficulties faced in modeling thin shell structures(made of such nonlinear materials for the first time in the history of VAM) which inherently exhibit geometric small parameters(such as the ratio of thickness to shortest wavelength of the deformation along the shell reference surface) and physical small parameters(such as moderate strains in certain applications). Saint Venant-Kirchhoff and neo-Hookean 3-D strain energy functions are considered for isotropic hyperelastic material modeling. Further, these two material models are augmented with electromechanical coupling term through Maxwell stress tensor for dielectric hyperelastic material modeling. A polyconvex 3-D strain energy function is used for the orthotropic hyperelastic model. Upon the application of VAM, in each of the above cases, the original 3-D nonlinear electroelastic problem splits into a nonlinear one-dimensional (1-D) through-the-thickness analysis and a nonlinear two-dimensional(2-D) shell analysis. This greatly reduces the computational cost compared to a full 3-D analysis. Through-the-thickness analysis provides a 2-D nonlinear constitutive law for the shell equations and a set of recovery relations that expresses the 3-D field variables (displacements, strains and stresses) through thethicknessintermsof2-D shell variables calculated in the shell analysis (2-D). Analytical expressions (asymptotically accurate) are derived for stiffness, strains, stresses and 3-D warping field for all three material types. Consistent with the three types of 2-D nonlinear constitutive laws,2-D shell theories and corresponding finite element programs have been developed. Validation of present theory is carried out with a few standard test cases for isotropic hyperelastic material model. For two additional test cases, 3-Dfinite element analysis results for isotropic hyperelastic material model are provided as further proofs of the simultaneous accuracy and computational efficiency of the current asymptotically-correct dimensionally-reduced approach. Application of the dimensionally-reduced dielectric hyperelastic material model is demonstrated through the actuation of a clamped membrane subjected to an electric field. Finally, the through-the-thickness and shell analysis procedures are outlined for the orthotropic nonlinear material model.