Divergence Boundary Conditions for Vector Helmholtz Equations with Divergence Constraints

The idea of replacing a divergence constraint by a divergence boundary condition is investigated. The connections between the formulations are considered in detail. It is shown that the most common methods of using divergence boundary conditions do not always work properly. Necessary and sufficient conditions for the equivalence of the formulations are given

Analysis of the geometric parameters influence in PCB fixtures for 2D multipole magnetization patterning of thin layer micro-magnets

Magnetic actuators, magnetic gears, vibrational energy harvesters and other micro-electromagnetic devices requires micro-magnetic rotors with alternant magnetizations to optimize their performance. Different approaches have been used for multipole magnetization of micro-magnets like fixed micro-fixtures, thermomagnetic patterning or laser machining. The main limitation of the previous techniques is that the inversion of the magnetic polarizations is only partially done. In this work, a concept based on 2D multipole magnetization printing of micro-magnets is proposed and analyzed to overcome current limitations. The fixtures are designed to be printed on a standard 35 μm PCB. The dependence of the magnetizing field with respect to the geometrical parameter of the fixture is analyzed. Maps of the required current for the magnetizing fields are also given. Some design recommendations to optimize the magnetizing field and to minimize current, thus the heat, are given.Universidad de Alcal

Variational Principles of Fluid Mechanics and Electromagnetism: Imposition and Neglect of the Lin Constraint

Variational principles in classical fluid mechanics and electromagnetism have sprinkled the literature since the eighteenth century. Even so, no adequate variational principle in the Eulerian description of matter was had until 1968 when an Eulerian variational principle was introduced which reproduces Euler\u27s equation of fluid dynamics. Although it successfully produces the appropriate equation of motion for a perfect fluid, the variational principle requires imposition of a constraint which was not fully understood at the time the variational principle was introduced. That constraint is the Lin constraint. The Lin constraint has subsequently been utilized by a number of authors who have sought to develop Eulerian variational principles in both fluid mechanics and electromagnetics (or plasmadynamics). How-ever, few have sought to fully understand the constraint. This dissertation first reviews the work of earlier authors concerning the development of variational principles in both the Eulerian and Lagrangian nomenclatures. In the process, it is shown rigorously whether or not the Euler-Lagrange equations which result from the variational principles are equivalent to the generally accepted equations of motion. In particular, it is shown in the case of several Eulerian variational principles that imposition of the Lin constraint results in Euler-Lagrange equations which are equivalent to the generally accepted equations of motion. On the other hand, it is shown that neglect of the Lin constraint results in Euler-Lagrange equations restrictive of the generally accepted equations of motion. In an effort to improve the physical motivation behind introduction of the Lin constraint a new variational constraint is developed based on the concept of surface forces within a fluid. The new constraint has the advantage of producing Euler-Lagrange equations which are globally correct whereas the Lin constraint itself allows only local equivalence to the standard classical equations of fluid motion. Several additional items of interest regarding variational principles are presented. It is shown that a quantity often referred to as the canonical momentum of a charged fluid is not always a constant of the motion of the fluid. This corrects an error which has previously appeared in the literature. In addition, it is demonstrated that there does not exist an unconstrained Eulerian variational principle giving rise to the generally accepted equations of motion for both a perfect fluid and a cold, electromagnetic fluid

Fast methods for inverse wave scattering problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 125-137).Inverse wave scattering problems arise in many applications including computerized/diffraction tomography, seismology, diffraction/holographic grating design, object identification from radar singals, and semiconductor quality control. Efficient algorithms exist for some inverse wave scattering problems in the low- and high-frequency regime or with weak scatterers. However, inverse wave scattering problems in the resonance regime with strong scatterers still pose many challenges. This thesis proposes algorithms for inverse wave scattering problems in the resonance regime with strong scatterers. These problems are part of, for instance, grating design, object identification, and semiconductor quality control. The proposed methods are (a) a spectrally convergent Nyström method for periodic structures in 2-D; (b) a fast Jacobian approximation method accompanying a Nyström method; (c) a fast and accurate method for evaluating the potential integrals in the 3-D mixed-potential integral operator with the Rao-Wilton-Glisson basis function; and (d) optimization with parameterized reduced-order models. The Nyström method and the method to evaluate the potential integrals accelerate scattered field evaluations by solving integral equations efficiently. The Jacobian approximation method and optimization with parameterized reduced-order models efficiently couple algorithms to evaluate scattered fields due to a guess of the scatterer and optimization methods to improve the guess. The Nyström and the Jacobian approximation methods are used to identify the parameters characterizing a periodic dielectric grating in 2-D. The method to evaluate the potential integrals and optimization with parameterized reduced-order models are applied to the problem of identifying simple discrete geometries in 3-D.by Jung Hoon Lee.Ph.D

On the well-posedness of the Maxwell system for linear bianisotropic media

The time dependent Maxwell system is supplemented with the constitutive relations of linear bianisotropc media and is treated as a neutral integro-differential equation in a Hilbert space. By using the theory of abstract Volterra equations and strongly continuous semigroups we obtain general well-posedness results for the corresponding mathematical problem

Nonlinear optics

Nonlinear light-matter interactions have been drawing attention of physicists since the 1960's. Quantum mechanics played a significant role in their description and helped to derive important formulas showing the dependence on the intensity of the electromagnetic field. High intensity light is able to generate second and third harmonics which translates to generation of electromagnetic field with multiples of the original frequency. In comparison with the linear behaviour of light, the nonlinear interactions are smaller in scale. This makes perturbation methods well suited for obtaining solutions to equations in nonlinear optics. In particular, the method of multiple scales is deployed in paper 3, where it is used to solve nonlinear dispersive wave equations. The key difference in our multiple scale solution is the linearity of the amplitude equation and a complex valued frequency of the mode. Despite the potential ill-posedness of the amplitude equation, the multiple scale solution remained a valid approximation of the solution to the original model. The results showed great potential of this method and its promising wider applications. Other methods use pseudo-spectral methods which require an orthogonal set of eigenfunctions (modes) used to create a substitute for the usual Fourier transform. This mode transform is only useful if it succeeds to represent target functions well. Papers 1 and 2 deal with investigating such modes called resonant and leaky modes and their ability to construct a mode transform. The modes in the first paper are the eigenvalues for a quantum mechanical system where an external radiation field is used to excite an electron trapped in an electrical potential. The findings show that the resonant mode expansion converges inside the potential independently of its depth. Equivalently, leaky modes are obtained in paper 2 which are in close relation to resonant modes. Here, the modes emerge from a system where a channel is introduced with transparent boundaries for simulation of one-directional optical beam propagation. Artificial index material is introduced outside the channel which gives rise to leaky modes associated with such artificial structure. The study is showing that leaky modes are well suited for function representation and thus solving the nonlinear version of this problem. In addition, the transparent boundary method turns out to be useful for spectral propagators such as the unidirectional pulse propagation equation in contrast to a perfectly matched layer

Guided Wave Resonant Optical Structures and LED Micro Resonators for Biosensing Applications

Integrated opto-electronic and nanophotonic devices for sensing application in the fields of medicine, microbiology, environmental, safety and defense have attracted considerable attention due to their potential for achieving greater compactness, shorter response times and higher sensitivities as compared to non-optical sensing systems. Optical cavity resonant devices such as Fabry--Perot interferometers have been extensively used in lasing applications and optical sensing has been accomplished through many similar technologies.;Fiber optic and planar waveguide based resonant devices which use evanescent waves for detection of refractive index changes are one of the most widely used approaches for photonic sensors. In this work we investigate the simulations, fabrication and characterization of resonant optical cavity devices for sensing applications. Morphology Dependent Resonances (MDRs) of planar, micro-spherical and micro-cylindrical cavities were reviewed for resonance line widths, spacing between modes, and density of resonances and experimental observations of internal and external field distributions. We focus on planar thin film stacked resonant waveguide geometries, microsphere-waveguide coupled resonances, cylindrical Gallium Nitride (GaN) microdisks for passive detection of Whispering Gallery Modes (WGMs) and electrically pumped active Resonant Cavity (RC) LED disk geometries for Vertical Cavity Modes (VCMs) as structures of interest.;Advances in stacked thin film coupled waveguide sensors enhance the selectivity and sensitivity by measuring the changes of the resonant optical modes and provide an integrated platform for label-free molecular detection. The effective surface loading detection sensitivity of the planar coupled alumina waveguide transducer was determined to be 20 pg/mm2 with a bulk index sensitivity of 5.6x10-4 Refractive Index Units (RIU) for aqueous sucrose solutions. For circular geometry based resonators, as the physical device size approaches the wavelength of light the MDRs are enhanced by retaining longer photon path length times and enhancing detection due to its high Q factors. Circular micro-cavities not only modify the optical resonances but also distribute the resonant frequencies as compared to a planar macro-cavity. The waveguide-coupled microspheres were experimentally detected to have a minimum surface coverage limit of 0.192%. Passive WGMs in GaN micro-disks showed a variation in mode spacing of 3nm to 7nm (lambda2/2piRn) as disk radius was reduced from 4.5microm to 2microm. Micro-cylindrical Distributed Bragg Reflector (DBR) RCLEDs were designed for layer thicknesses and Multi Quantum Well (MQW) placement to enhance VCMs and LED emission output. Experimental and simulated LED spectral minima matched at 432 nm and 451 nm confirming VCMs related to (lambda/2) cavity resonances