Eigenmode-based capacitance calculations with applications in passivation layer design

The design of high-speed metallic interconnects such as microstrips requires the correct characterization of both the conductors and the surrounding dielectric environment, in order to accurately predict their propagation characteristics. A fast boundary integral equation approach is obtained by modeling all materials as equivalent surface charge densities in free space. The capacitive behavior of a finite dielectric environment can then be determined by means of a transformation matrix, relating these charge densities to the boundary value of the electric potential. In this paper, a new calculation method is presented for the important case that the dielectric environment is composed of homogeneous rectangles. The method, based on a surface charge expansion in terms of the Robin eigenfunctions of the considered rectangles, is not only more efficient than traditional methods, but is also more accurate, as shown in some numerical experiments. As an application, the design and behavior of a microstrip passivation layer is treated in some detail

A high frequency analysis of electromagnetic plane wave scattering by perfectly-conducting semi-infinite parallel plate and rectangular waveguides with absorber coated inner walls

An approximate but sufficiently accurate high frequency solution which combines the uniform geometrical theory of diffraction (UTD) and the aperture integration (AI) method is developed for analyzing the problem of electromagnetic (EM) plane wave scattering by an open-ended, perfectly-conducting, semi-infinite hollow rectangular waveguide (or duct) with a thin, uniform layer of lossy or absorbing material on its inner wall, and with a planar termination inside. In addition, a high frequency solution for the EM scattering by a two dimensional (2-D), semi-infinite parallel plate waveguide with a absorber coating on the inner walls is also developed as a first step before analyzing the open-ended semi-infinite three dimensional (3-D) rectangular waveguide geometry. The total field scattered by the semi-infinite waveguide consists firstly of the fields scattered from the edges of the aperture at the open-end, and secondly of the fields which are coupled into the waveguide from the open-end and then reflected back from the interior termination to radiate out of the open-end. The first contribution to the scattered field can be found directly via the UTD ray method. The second contribution is found via the AI method which employs rays to describe the fields in the aperture that arrive there after reflecting from the interior termination. It is assumed that the direction of the incident plane wave and the direction of observation lie well inside the forward half space tht exists outside the half space containing the semi-infinite waveguide geometry. Also, the medium exterior to the waveguide is assumed to be free space

Edge waves and localisation in lattices containing tilted resonators

The paper presents the study of waves in a structured geometrically chiral solid. A special attention is given to the analysis of the Bloch-Floquet waves in a doubly periodic high-contrast lattice containing tilted resonators. Dirac-like dispersion of Bloch waves in the structure is identified, studied and applied to wave-guiding and wave-defect interaction problems. The work is extended to the transmission problems and models of fracture, where localisation and edge waves occur. The theoretical derivations are accompanied with numerical simulations and illustrations

Convection diffusion and reaction : bridging scales in science and engineering

Tese de doutoramento. Engenharia Química e Biológica. Faculdade de Engenharia. Universidade do Porto. 201

Collective electronic effects in scanning probe microscopy

The surface plasmon dispersion relations are calculated for a metal coated dielectric probe above a dielectric half space with and without metal coating. Employing prolate spheroidal coordinate system this configuration was modeled as confocal single-sheeted hyperboloids of revolution superimposed on planar domains. The involved media are characterized by frequency dependent, spatially local dielectric functions. Due to subwavelength dimensions of the region of interest, nonretarded electrodynamics is utilized to derive exact analytical expressions describing the resonant surface modes. The dis-persion relations are studied as functions of the parameter that defines the hyperboloidal boundaries of the tip and the corresponding coating, and as functions of the involved coating thicknesses. Both parallel and perpendicular polarizations are considered. The results are simulated numerically and limiting cases are discussed with comparison to the Cartesian thin foil case. Using this new type of probe-substrate configuration, the surface plasmon coupling mechanism is investigated experimentally utilizing a scanning probe microscope, and the signal strength acquired by the probe is measured as a function of the distance between the probe and the sample. This is repeated at three different wavelengths of the incident p-polarized photons used to stimulate surface plasmons in the thin metal foil. The results are compared with the theory. Utilizing the prolate spheroidal coordinate system, the related and relevant problem of the Coulomb interaction of a dielectric probe tip with a uniform field existing above a semiinfinite, homogeneous dielectric substrate was studied. This is of interest in atomic force microscopy when the sample surface is electrically charged. The induced polarization surface charge density and the field distribution at the bounding surface of the dielectric medium with the geometry of a single-sheeted hyperboloid of revolution located above the dielectric half space interfaced with a uniform surface charge density is calculated. The force density on the hyperboloidal probe medium is calculated as a function of the probe tip shape. The potential and the field distributions are calculated in the neighborhood of the apex of the tip. The calculation is based on solving Laplace\u27s equation and employing a newly derived integral expansion for the vanishing dielectric limit of the potential. The integral expansion is analytically proved and numerically verified. The involved numerical simulations comprise the evaluation of infinite double integrals involving conical functions similar to those arising in the Mehler-Fock integral transform

Effective Boundary Conditions for the Fisher-KPP Equation on a Domain with 3-dimensional Optimally Aligned Coating

We consider the Fisher-KPP equation on a three-dimensional domain surrounded by a thin layer whose diffusion rates are drastically different from that in the bulk. The bulk is isotropic, while the layer is considered to be anisotropic and ``optimally aligned", where the normal direction is always an eigenvector of the diffusion tensor. To see the effect of the layer, we derive effective boundary conditions (EBCs) by the limiting solution of the Fisher-KPP equation as the thickness of the layer shrinks to zero. These EBCs contain some exotic boundary conditions including the Dirichlet-to-Neumann mapping and the Fractional Laplacian. Moreover, we emphasize that each EBC keeps effective indefinitely, even as time approaches infinity.Comment: 19 pages, 1 figure. arXiv admin note: text overlap with arXiv:2301.1365

Computational Electromagnetism and Acoustics

The challenge inherent in the accurate and efficient numerical modeling of wave propagation phenomena is the common grand theme in both computational electromagnetics and acoustics. Many excellent contributions at this Oberwolfach workshop were devoted to this theme and a wide range of numerical techniques and algorithms were mustered to tackle this challenge

Time-dependent electromagnetic scattering from thin layers

The scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a second step the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell’s equations. A key role in the well-posedness of the time-dependent boundary integral equations and the stability of the numerical discretization is taken by the coercivity of the Calderón operator for the time-harmonic Maxwell’s equations with frequencies in a complex half-plane. This entails the coercivity of the full boundary operator that includes the impedance operator. The system of time-dependent boundary integral equations is discretized with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. The full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. The theoretical results are illustrated by numerical experiments