Fractional order electromagnetics

The Maxwell equations constitute a formalism for the development of models describing electromagnetic phenomena. The four Maxwell laws have been adopted successfully in many applications and involve only the integer order differential calculus. Recently, a closer look for the cases of transmission lines, electrical motors and transformers, that reveal the so-called skin effect, motivated a new perspective towards the replacement of classical models by fractional-order mathematical descriptions. Bearing these facts in mind this paper addresses the concept of static fractional electric potential. The fractional potential was suggested some years ago. However, the idea was not fully explored and practical methods of implementation were not proposed. In this line of thought, this paper develops a new approximation algorithm for establishing the fractional order electrical potential and analyzes its characteristics.

Soluções cosmológicas e locais para uma eletrodinâmica modificada

The present work investigates some consequences that arise from the use of a modifed lagrangean for the eletromagnetic feld in two diferent contexts: a spatially homogeneous and isotropic universe whose dynamics is driven by a magnetic feld plus a cosmological parameter A, and the problem of a static and charged point mass (charged black hole). In the cosmological case, three diferent general solutions were derived. The first, with a null cosmological parameter A, generalizes a particular solution obtained by Novello et al [gr-qc/9806076]. The second one admits a constant A and the third one allows A to be a time-dependent parameter that sustains a constant magnetic feld. The first two solutions are non-singular and exhibit in ationary periods. The third case studied shows an in ationary dynamics except for a short period of time. As for the problem of a charged point mass, the solutions of the Einstein-Maxwell equations are obtained and compared with the standard Reissner-Nordstrom solution. Contrary to what happens in the cosmological case, the physical singularity is not removed

Efeitos da geometria e da composição no espectro de cristais fotônicos 2D

Pós-graduação em Ciência e Tecnologia de Materiais - FC

Influência de estruturas geológicas bidimensionais no campo geoeletromagnético na presença do eletrojato equatorial

A Terra atua como um grande magneto esférico, cujo campo assemelha-se àquele gerado por um dipolo magnético. Este campo apresenta mudanças de intensidade que variam com a localização e a hora local. A parte principal do campo geomagnético se origina no interior da Terra através de processos eletromagnéticos. Extensivos estudos mostraram ainda que existem contribuições de origem externa ao planeta, principalmente de origem solar. Dentre estas fontes há anomalias do campo magnético que surgem a partir de um aumento diurno da corrente elétrica em uma estreita faixa da ionosfera, de direção leste-oeste, centrada no equador magnético e denominada Eletrojato Equatorial (EEJ). Ocasionalmente estas correntes podem apresentar reversões de fluxo, sendo denominadas Contra-Eletrojato (CEJ). Vários autores têm estudado os efeitos do EEJ e CEJ sobre as observações geoeletromagnéticas. Eles estão interessados no efeito combinado do EEJ e estruturas geológicas condutivas 1-D e 2-D. Nestes trabalhos a estrutura 2-D sempre se apresentava paralela ao eletrojato, o que é uma hipótese bastante restritiva ao se modelar ambientes geológicos mais realistas, em que corpos bidimensionais podem ter qualquer strike em relação ao EEJ. Neste trabalho apresentamos a solução deste problema sem esta restrição. Assim, mostramos os campos geoeletromagnéticos devidos a estruturas bidimensionais que possuam strike oblíquo em relação ao EEJ, através de perfis dos campos elétrico e magnéticos calculados na superfície e formando direção arbitrária à heterogeneidade condutiva 2-D. Com esta resposta avaliamos ainda qual a influência que estruturas bidimensionais exercem sobre a resposta magnetotelúrica, sob influência do Eletrojato Equatorial. Durante o desenvolvimento deste trabalho, utilizamos o método de elementos finitos, tendo por fonte eletromagnética o EEJ e o CEJ, que por sua vez foram representados por uma combinação de distribuições gaussianas de densidade de corrente. Estas fontes foram decompostas nas direções paralela e perpendicular à estrutura 2-D, resultando nos modos de propagação TE₁ e TE₂ e TM acoplados, respectivamente. Resolvemos o modo acoplado aplicando uma Transformada de Fourier nas equações de Maxwell e uma Transformada Inversa de Fourier na solução encontrada. De acordo com os experimentos numéricos realizados em um modelo interpretativo da Anomalia Condutiva da Bacia do Parnaíba, formado por uma enorme estrutura de 3000 ohm-m dentro de um corpo externo condutivo (1 ohm-m), concluímos que a presença do CEJ causa uma inversão na anomalia, se compararmos com o resultado do EEJ. Concluímos também que para as frequências mais altas as componentes do campo elétrico apresentam menor influência da parte interna do corpo 2-D do que da parte externa. Já para frequências mais baixas este comportamento se observa com as componentes do campo magnético. Com relação à frequência, vimos os efeitos do “skin-depth”, principalmente nas respostas magnéticas. Além disso, quando a estrutura 2-D está paralela ao eletrojato, o campo elétrico é insensível à estrutura interna do modelo para todos os valores de frequência utilizados. Com respeito ao ângulo θ_h entre a heterogeneidade e a fonte, vimos que o modo TM se manifesta naturalmente quando θ_h é diferente de 0°. Neste caso, o modo TE é composto por uma parte devido à componente da fonte paralela à heterogeneidade e a outra devido à componente da fonte perpendicular, que é acoplada ao modo TM. Assim, os campos calculados têm relação direta com o valor de θ_h. Analisando a influência do ângulo entre a direção do perfil dos campos e o strike da heterogeneidade verificamos que, à medida que θ_h se aproxima de 90°, os campos primários tornam-se variáveis para valores de θ_p diferentes de 90°. Estas variações causam uma assimetria na anomalia e dão uma idéia da inclinação da direção do perfil em relação aos corpos. Finalmente, concluímos que uma das influências que a distância entre o centro do EEJ e o centro da estrutura 2-D, causa sobre as componentes dos campos está relacionado às correntes reversas do EEJ e CEJ, pois a 500 km do centro da fonte estas correntes têm máxima intensidade. No entanto, com o aumento da distância, as anomalias diminuem de intensidade. Nas sondagens MT, nós também usamos o EEJ e o CEJ como fonte primária e comparamos nossos resultados com a resposta da onda plana. Deste modo observamos que as componentes do campo geoeletromagnético, usadas para calcular a impedância, têm influência do fator de acoplamento entre os modos TE₂ e TM. Além disso, esta influência se torna maior em meios resistivos e nas frequências mais baixas. No entanto, o fator de acoplamento não afeta os dados magnetotelúricos em frequências maiores de 10^-2 Hz. Para frequências da ordem de 10^-4 Hz os dados MT apresentam duas fontes de perturbação: a primeira e mais evidente é devido à presença fonte 2-D (EEJ e CEJ), que viola a hipótese da onda plana no método MT; e a segunda é causada pelo acoplamento entre os modos TE₂ e TM, pois quando a estrutura bidimensional está obliqua à fonte 2-D temos correntes elétricas adicionais ao longo da heterogeneidade. Concluimos assim, que o strike de uma grande estrutura condutiva bidimensional relativamente à direção do EEJ ou CEJ tem de fato influência sobre o campo geomagnético. Por outro lado, para estudos magnetotelúricos rasos (frequências maiores de 10^-3 Hz) o efeito do ângulo entre a estrutura geológica 2-D e a direção do EEJ não é tão importante. Contudo, em estudos de litosfera frequências menores de 10^-3 Hz) o acoplamento entre os modos TE₂ e TM não pode ser ignorado.

Equilibrium topology for plasmas with reversed current density.

Actually, transition from positive to negative plasma current and quasi-steady-state alternated current (AC) operation have been achieved experimentally without loss of ionization. The large transition times suggest the use of MHD equilibrium to model the intermediate magnetic field configurations for corresponding current density reversals. In the present work we show, by means of Maxwell equations, that the most robust equilibrium for any axisymmetric configuration with reversed current density requires the existence of several nonested families of magnetic surfaces inside the plasma. We also show that the currents inside the nonested families satisfy additive rules restricting the geometry and sizes of the axisymmetric magnetic islands; this is done without restricting the equilibrium through arbitrary functions. Finally, we introduce a local successive approximations method to describe the equilibrium about an arbitrary reversed current density minimum and, consequently, the transition between different nonested topologies is understood in terms of the eccentricity of the toroidal current density level sets.

Nichtkommutative Geometrie und Quantisierung von Raumzeiten und Konfigurationsräumen

Das Standardmodell der Elementarteilchenphysik istexperimentell hervorragend bestätigt, hat auf theoretischerSeite jedoch unbefriedigende Aspekte: Zum einen wird derHiggssektor der Theorie von Hand eingefügt, und zum anderenunterscheiden sich die Beschreibung des beobachtetenTeilchenspektrums und der Gravitationfundamental. Diese beiden Nachteile verschwinden, wenn mandas Standardmodell in der Sprache der NichtkommutativenGeometrie formuliert. Ziel hierbei ist es, die Raumzeit der physikalischen Theoriedurch algebraische Daten zu erfassen. Beispielsweise stecktdie volle Information über eine RiemannscheSpinmannigfaltigkeit M in dem Datensatz (A,H,D), den manspektrales Tripel nennt. A ist hierbei die kommutativeAlgebra der differenzierbaren Funktionen auf M, H ist derHilbertraum der quadratintegrablen Spinoren über M und D istder Diracoperator. Mit Hilfe eines solchen Tripels (zu einer nichtkommutativenAlgebra) lassen sich nun sowohl Gravitation als auch dasStandardmodell mit mathematisch ein und demselben Mittelerfassen. In der vorliegenden Arbeit werden nulldimensionale spektraleTripel (die diskreten Raumzeiten entsprechen) zunächstklassifiziert und in Beispielen wird eine Quantisierungsolcher Objekte durchgeführt. Ein Problem der spektralenTripel stellt ihre Beschränkung auf echt RiemannscheMetriken dar. Zu diesem Problem werden Lösungsansätzepräsentiert. Im abschließenden Kapitel der Arbeit wird dersogenannte 'Feynman-Beweis der Maxwellgleichungen' aufnichtkommutative Konfigurationsräume verallgemeinert.

Advanced Numerical Simulation of Silicon-Based Solar Cells

Photovoltaic (PV) conversion is the direct production of electrical energy from sun without involving the emission of polluting substances. In order to be competitive with other energy sources, cost of the PV technology must be reduced ensuring adequate conversion efficiencies. These goals have motivated the interest of researchers in investigating advanced designs of crystalline silicon solar (c-Si) cells. Since lowering the cost of PV devices involves the reduction of the volume of semiconductor, an effective light trapping strategy aimed at increasing the photon absorption is required. Modeling of solar cells by electro-optical numerical simulation is helpful to predict the performance of future generations devices exhibiting advanced light-trapping schemes and to provide new and more specific guidelines to industry. The approaches to optical simulation commonly adopted for c-Si solar cells may lead to inaccurate results in case of thin film and nano-stuctured solar cells. On the other hand, rigorous solvers of Maxwell equations are really cpu- and memory-intensive. Recently, in optical simulation of solar cells, the RCWA method has gained relevance, providing a good trade-off between accuracy and computational resources requirement. This thesis is a contribution to the numerical simulation of advanced silicon solar cells by means of a state-of-the-art numerical 2-D/3-D device simulator, that has been successfully applied to the simulation of selective emitter and the rear point contact solar cells, for which the multi-dimensionality of the transport model is required in order to properly account for all physical competing mechanisms. In the second part of the thesis, the optical problems is discussed. Two novel and computationally efficient RCWA implementations for 2-D simulation domains as well as a third RCWA for 3-D structures based on an eigenvalues calculation approach have been presented. The proposed simulators have been validated in terms of accuracy, numerical convergence, computation time and correctness of results.

Different approaches to grounding resistance. Calculation for thin wire structures at low frequency energization

The present paper deals with the calculation of grounding resistance of an electrode composed of thin wires, that we consider here as perfect electric conductors (PEC) e.g. with null internal resistance, when buried in a soil of uniform resistivity. The potential profile at the ground surface is also calculated when the electrode is energized with low frequency current. The classic treatment by using leakage currents, called Charge Simulated Method (CSM), is compared with that using a set of steady currents along the axis of the wires, here called the Longitudinal Currents Method (LCM), to solve the Maxwell equations. The method of moments is applied to obtain a numerical approximation of the solution by using rectangular basis functions. Both methods are applied to two types of electrodes and the results are also compared with those obtained using a thirth approach, the Average Potential Method (APM), later described in the text. From the analysis performed, we can estimate a value of the error in the determination of grounding resistance as a function of the number of segments in which the electrodes are divided.

Sobre la solución del problema de la radiación electromagnética

En la presente Tesis Doctoral se propone una posible solución al problema de la radiación electromagnética. Este es un problema científico estudiado desde hace mucho tiempo y que hasta el presente no se ha podido resolver de manera exacta. Aquí se redefine nuevamente el problema y se plantea el mismo partiendo un modelo de radiador formado por dos placas planas circulares paralelas, que por sus reducidas dimensiones eléctricas, se considerará para el estudio como un dipolo simétrico corto. Se obtienen las soluciones de las ecuaciones de Maxwell para la radiación electromagnética emitida por dicho sistema y se propone lo que sería el principio de formación de las ondas electromagnéticas. Con ayuda de este se ofrece una posible interpretación de las soluciones obtenidas. Luego a dicho sistema se le realizan correcciones en sus dimensiones a fin de ajustar las soluciones obtenidas al caso real de un dipolo simétrico de media onda. Además se analiza de una manera sencilla la naturaleza material del campo electromagnético y su estructura, lo que hace posible proponer una manera original de comprender las ecuaciones de Maxwell. Finalmente se propone un modelo sencillo mediante el cual se puede demostrar la Ley de Planck para la distribución espectral de la energía electromagnética radiada por los cuerpos calientes, pero desde las soluciones de las ecuaciones de Maxwell para la radiación de las antenas. Con esto podría ser aceptado que la solución del problema de la radiación electromagnética es única tanto para las antenas como para los cuerpos calientes en la naturaleza. ABSTRACT In this PhD thesis a possible solution to the problem of electromagnetic radiation is proposed. This is a scientific problem which has been studied for a long time and to this day has still not exactly been solved. The problem is redefined here again and it is stated using a model formed by two flat parallel circular plates as the radiator system, which due to its reduced electrical dimensions will be considered for the study as a short symmetrical dipole. The solutions of the Maxwell equations for emitted electromagnetic radiation are obtained by said system, and that which could be the principle formation of electromagnetic waves is proposed. Using this principle, a possible interpretation of the solutions obtained is presented. Then the system will perform corrections in size to adjust the solutions obtained to the real case of a symmetrical half-wave dipole. The material nature of the electromagnetic field and its structure is also analyzed in a simple way, making it possible to present an original way to understand the Maxwell equations. Finally, a simple model which can be used to prove Planck's law of spectral distribution of the electromagnetic energy radiated by hot bodies is proposed, but parting from the solutions of the Maxwell equations for the antennas radiation. With this it could be accepted that the solution to the problem of electromagnetic radiation is unique to both antennas and hot bodies on nature.

Reflexión y refracción de la luz en medios anisótropos

En este proyecto se pretende estudiar el comportamiento de la luz al atravesar medios de diversos materiales, tanto isótropos como anisótropos uniáxicos. Para ello se requiere realizar un estudio previo de las condiciones de contorno aplicables a las ecuaciones de Maxwell en la interfase de dos medios que pueden ser isótropos o anisótropos. En el caso de dos materiales isótropos, la solución del problema son los conocidos coeficientes de Fresnel de reflexión y transmisión. En este trabajo se pretende generalizar el estudio al caso del paso de la luz desde un medio isótropo a otro anisótropo uniáxico (con su eje óptico en orientación arbitraria) y viceversa y al caso de dos materiales anisótropos uniáxicos con ejes ópticos en orientaciones arbitrarias. Es de especial interés el caso de un mismo material uniáxico en el que las dos partes tienen el eje óptico con distinta orientación. Una vez planteadas las condiciones de contorno específicas en cada caso, se obtendrá un conjunto de ecuaciones algebraicas cuya resolución permitirá obtener los coeficientes de reflexión y transmisión buscados. Para plantear el sistema de ecuaciones adecuado, será necesario tener una descripción de las características ópticas de los materiales empleados, la orientación de los ejes ópticos en cada caso, y los posibles ángulos de incidencia. Se realizará un tratamiento matricial de modo que el paquete MatLab permite su inversión de manera inmediata. Se desarrollará una interfaz sencilla, realizada con MatLab, que permita al usuario introducir sin dificultad los datos correspondientes a los materiales de los medios incidente y transmitido, la orientación en espacial del o de los ejes ópticos, de la longitud de onda de trabajo y del ángulo de incidencia del haz de luz, con los que la aplicación realizará los cálculos. Los coeficientes de reflexión y refracción obtenidos serán representados gráficamente en función del ángulo de incidencia. Así mismo se representarán los ángulos transmitidos y reflejados en función del de incidencia. Todo ello de esta forma, que resulte sencilla la interpretación de los datos por parte del usuario. ABSTRACT. The reason for this project is to study the behavior of light when light crosses different media of different materials, isotropic materials and uniaxial anisotropic materials. For this, a previous study is necessary where the boundary conditions apply to Maxwell equations at the interface between two media which can be isotropic and anisotropic. If both materials are isotropic, the Fresnel ccoefficients of reflection and refraction are used to solve the problem. The aim of this work is to generalize a study when light crosses from an isotropic media to a uniaxial anisotropic media, where its axis have arbitrary directions, and vicecersa. The system consisting of two materials with axis in arbitrary directions are also being studied. Once the specific boundary conditions are known in each case, a set of algebraic equations are obtained whose solution allows obtaining the reflection coefficients and refraction coefficients. It is necessary to have a description of the optical characteristics of the materials used; of the directions axis in each case and the possible angle of incidence. A matrix is proposed for later treatment in Matlab that allows the immediate inversion. A simple interface will de developed, manufactured with Matlab, that allows the user to enter data easily corresponding to the incident media and transmission media of the different materials, the special axis directions, the wavelength and the angle of incidence of the light beam. This data is used by the application to perform the necessary calculations to solve the problem. When reflection coefficients and refraction coefficients are obtained, the application draws the graphics in function of the angle of incidence. Also transmitted and reflected angles depending on the incidence are represented. This is to perform a data representation which is a simple interpretation of the user data.

Application of Mathieu functions for the study of non-slanted reflection gratings

In this work we prensent an analysis of non-slanted reflection gratings by using exact solution of the second order differential equation derived from Maxwell equations, in terms of Mathieu functions. The results obtained by using this method will be compared to those obtained by using the well known Kogelnik's Coupled Wave Theory which predicts with great accuracy the response of the efficieny of the zero and first order for volume phase gratings, for both reflection and transmission gratings.

An FDTD model for calculation of gradient-induced eddy currents in MRI system

In most magnetic resonance imaging (MRI) systems, pulsed magnetic gradient fields induce eddy currents in the conducting structures of the superconducting magnet. The eddy currents induced in structures within the cryostat are particularly problematic as they are characterized by long time constants by virtue of the low resistivity of the conductors. This paper presents a three-dimensional (3-D) finite-difference time-domain (FDTD) scheme in cylindrical coordinates for eddy-current calculation in conductors. This model is intended to be part of a complete FDTD model of an MRI system including all RF and low-frequency field generating units and electrical models of the patient. The singularity apparent in the governing equations is removed by using a series expansion method and the conductor-air boundary condition is handled using a variant of the surface impedance concept. The numerical difficulty due to the asymmetry of Maxwell equations for low-frequency eddy-current problems is circumvented by taking advantage of the known penetration behavior of the eddy-current fields. A perfectly matched layer absorbing boundary condition in 3-D cylindrical coordinates is also incorporated. The numerical method has been verified against analytical solutions for simple cases. Finally, the algorithm is illustrated by modeling a pulsed field gradient coil system within an MRI magnet system. The results demonstrate that the proposed FDTD scheme can be used to calculate large-scale eddy-current problems in materials with high conductivity at low frequencies.

Advanced fibre gratings and their applications

This thesis describes a detailed study of advanced fibre grating devices using Bragg (FBG) and long-period (LPG) structures and their applications in optical communications and sensing. The major contributions presented in this thesis are summarised below. One of the most important contributions from the research work presented in this thesis is a systematic theoretical study of many distinguishing structures of fibre gratings. Starting from the Maxwell equations, the coupled-mode equations for both FBG and LPG were derived and the mode-overlap factor was analytically discussed. Computing simulation programmes utilising matrix transform method based on the models built upon the coupled-mode equations were developed, enabling simulations of spectral response in terms of reflectivity, bandwidth, sidelobes and dispersion of gratings of different structures including uniform and chirped, phase-shifted, Moiré, sampled Bragg gratings, phase-shifted and cascaded long-period gratings. Although the majority of these structures were modelled numerically, analytical expressions for some complex structures were developed with a clear physical picture. Several apodisation functions were proposed to improve sidelobe suppression, which guided effective production of practical devices for demanding applications. Fibre grating fabrication is the other major part involved in the Ph.D. programme. Both the holographic and scan-phase-mask methods were employed to fabricate Bragg and long-period gratings of standard and novel structures. Significant improvements were particularly made in the scan-phase-mask method to enable the arbitrarily tailoring of the spectral response of grating devices. Two specific techniques - slow-shifting and fast-dithering the phase-mask implemented by a computer controlled piezo - were developed to write high quality phase-shifted, sampled and apodised gratings. A large number of LabVIEW programmes were constructed to implement standard and novel fabrication techniques. In addition, some fundamental studies of grating growth in relating to the UV exposure and hydrogenation induced index were carried out. In particular, Type IIa gratings in non-hydrogenated B/Ge co-doped fibres and a re-generated grating in hydrogenated B/Ge fibre were investigated, showing a significant observation of thermal coefficient reduction. Optical sensing applications utilising fibre grating devices form the third major part of the research work presented in this thesis. Several experiments of novel sensing and sensing-demodulating were implemented. For the first time, an intensity and wavelength dual-coding interrogation technique was demonstrated showing significantly enhanced capacity of grating sensor multiplexing. Based on the mode-splitting measurement, instead of using conventional wavelength-shifting detection technique, successful demonstrations were also made for optical load and bend sensing of ultra-high sensitivity employing LPG structures. In addition, edge-filters and low-loss high-rejection bandpass filters of 50nm stop-band were fabricated for application in optical sensing and high-speed telecommunication systems

On a Direct Approach to the Solution of Inverse Optical Problems

The evaluation from experimental data, of physical quantities, which enter into the electromagnetic Maxwell equations, is described as inverse optical problem. The functional relations between the dependent and independent variables are of transcendental character and numeric procedures for evaluation of the unknowns are largely used. Herein, we discuss a direct approach to the solution, illustrated by a specific example of determination of thin films optical constants from spectrophotometric data. New algorithm is proposed for the parameters evaluation, which does not need an initial guess of the unknowns and does not use iterative procedures. Thus we overcome the intrinsic deficiency of minimization techniques, such as gradient search methods, Simplex methods, etc. The price of it is a need of more computing power, but our algorithm is easily implemented in structures such as grid clusters. We show the advantages of this approach and its potential for generalization to other inverse optical problems.

Effective Cell-Centred Time-Domain Maxwell's Equations Numerical Solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed.