Review of spurious modes in the modal analysis of waveguides using finite elements

Orientador: Hugo Enrique Hernandez FigueroaDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Desde as primeiras formulações em elementos finitos para o eletromagnetismo, os modos espúrios sempre foram um empecilho para a analise modal. Inicialmente, elementos nodais foram utilizados a fim de resolver a equação vetorial de onda no domínio da frequência. Naquele momento, os modos espúrios eram encontrados espalhados pelo espectro, o que dificultava a distinção entre soluções físicas e soluções espúrias. Anos mais tarde, o problema foi aparentemente resolvido pelo uso de elementos de aresta na equação vetorial de onda. Essa nova aproximação não eliminava por completo as soluções não físicas, apenas as confinava em torno do autovalor zero, o que não era problema na maioria das situações na analise modal. Recentemente, novas aplicações de elementos finitos no domínio do tempo resultaram em soluções instáveis, devido a presença dos modos espúrios em baixas frequências. Na década de 1990, Demkowicz[1], propôs uma formulação diferente em elementos finitos usando uma combinação da equação vetorial de onda com a condição do divergente de Gauss, que eliminou por completo os modos espúrios. O objetivo deste trabalho e analisar o impacto da condição de Gauss no FEM e entender como a mesma elimina os modos espúriosAbstract: Since the earlier finite elements formulation for electromagnetics, spurious solutions have always plagued the modal analysis. Initially, nodal elements were employed to solve the vectorial wave equation in frequency domain. At that moment, the spurious modes could be found spread in frequency spectrum, and it was hard to distinguish between physical solutions and spurious ones. Some years later, this issue was apparently solved by using in the edge elements. This new approach does not completely eliminate non-physical solutions, but only confine them around to zero eigenvalue, which is not a problem in most of the situations in modal analysis. Recently, new applications of finite elements in time domain resulted to instable solution behavior, due to spurious modes at low frequencies. In the nineties, Demkowicz [1] proposed a different FEM formulation using a combination of vectorial wave equation and Gauss divergent equation, which definitely eliminates the spurious modes. The goal of this work is to analyze the impact of the Gauss condition in FEM and understand how it can eliminate spurious modesMestradoTelecomunicações e TelemáticaMestre em Engenharia Elétric

On-chip interrogator based on Fourier Transform spectroscopy

In this paper, the design and the characterization of a novel interrogator based on integrated Fourier transform (FT) spectroscopy is presented. To the best of our knowledge, this is the first integrated FT spectrometer used for the interrogation of photonic sensors. It consists of a planar spatial heterodyne spectrometer, which is implemented using an array of Mach-Zehnder interferometers (MZIs) with different optical path differences. Each MZI employs a 3$\times$3 multi-mode interferometer, allowing the retrieval of the complex Fourier coefficients. We derive a system of non-linear equations whose solution, which is obtained numerically from Newton's method, gives the modulation of the sensor's resonances as a function of time. By taking one of the sensors as a reference, to which no external excitation is applied and its temperature is kept constant, about 92$\%$ of the thermal induced phase drift of the integrated MZIs has been compensated. The minimum modulation amplitude that is obtained experimentally is 400 fm, which is more than two orders of magnitude smaller than the FT spectrometer resolution.Comment: 15 pages, 6 figure