Abstracts of Presentations

Talks

Alain Bossavit LGEP

Frequency-dependent homogenization in electromagnetism: Application to Metamaterials

Metamaterials are crystal-like composites, where high-frequency electromagnetic fields propagate in surprising ways. Contrary to what happens in standard homogenization theory, effective constants for the equivalent homogeneous medium, in the Full Maxwell case, depend on the exciting frequency, with unexpected results: for instance, effective permeability and permittivity may become negative (that is, their real parts may), over a specific range of frequencies, hence a negative refractive index, with all the exotic properties that entails.

Metamaterials, in this respect, challenge standard homogenization theory. The latter does provide effective homogenized coefficients mu_eff and eps_eff. But these are averages (a kind of mix of arithmetic and harmonic mean) of the physical, spatially periodic eps and mu. These being essentially positive, one fails to comprehend how negative homogenized coefficients could result from such a procedure.

Learning from the physicists gives the essential cue in this respect: the "negative index" behavior is restricted to a relatively narrow band of frequency, and is caused by "internal resonance" phenomena, due to deliberate arrangement of materials inside the cell in order to create the equivalent of RLC circuits. Hence the necessity for a "frequency-dependent" homogenization procedure. In the version I propose, the cell-problem's differential operator is akin to omega**2 + curlcurl, instead of the -divgrad of standard theory, so that computed effective coefficients *do* depend on frequency. Energetic considerations, based on the variational formulation of the problem, then easily show that, indeed, effective coefficients, complex-valued, can cross the imaginary axis at definite frequencies, thus acquiring negative real parts.

In the spirit of the workshop, this is presented as an exercise in "discrete exterior calculus" in presence of crystal-like symmetry. The cell problem can be understood as building two equivalent networks, one electric, one magnetic, in interaction, both obeying Kirchhoff's laws. This interpretation makes easier the detection (and enhancement, via shape optimization) of the desired "internal resonances", and might help establish a common language with the specialists.

Luca Dal Negro Boston University

Light propagation in deterministically aperiodic structures: theoretical opportunities from an experimental viewpoint.

Deterministic aperiodic dielectric structures (DADS) have a non-periodic refractive index modulation and share distinctive physical properties with both periodic dielectric materials, i.e., the formation of large energy band-gaps, and disordered random media, i.e., the presence of localized states with strong light-matter coupling. However, unlike the disordered random materials, DADS are generated according to simple mathematical prescriptions and possess perfect long-range order without translational invariance. Interestingly, DADS exhibit new fascinating physical properties that originate from the distinctive interplay between the global lack of translational invariance and the presence of well defined internal symmetries associated with the long-range order. In particular, it has been theoretically demonstrated that aperiodic dielectrics show rich (self-similar) fractal transmission spectra, critical light localization with multi-fractal wave functions, large field enhancement effects and anomalous light transport properties. However, despite the fascinating behavior of this novel class of photonics materials, their theoretical as well as experimental study is still in its infancy. The purpose of my talk is to survey, mainly within an experimentalist's perspective, the current status of activities in the field of aperiodic deterministic structures hoping to stimulate novel theoretical opportunities.

Jozef Dodziuk City University of New York

Whitney forms and their applications in global analysis and topology

We review the definition of basic properties of Whitney forms with emphasis on a) their approximation properties; and b) parallels between the discrete and continuous setup. We also describe their applications to proving equivalence of analytic and topological invariants (de Rham theory, analytic and Reidemeister torsion, analytic and combinatorial $L^2$ Betti numbers, analytic and combinatorial Novikov-Shubin invariants).

Christophe Geuzaine Case Western Reserve University

Convergent numerical solution of wave scattering problems at high frequencies

The calculation of wave scattering from acoustically or electrically large objects is one of the most challenging problems in computational science. Nevertheless, current state-of-the-art algorithms are limited by the competing demands of accuracy (which requires an increasing number of degrees of freedom to resolve the fields on the scale of the wavelength) and efficiency (which favors coarse discretizations). In this talk we will present a new strategy for the numerical solution of the Helmholtz equation that does not require to discretize the fields on the scale of the wavelength, while retaining error-controllability and high-order convergence. Our approach is based on the use of an ansatz for the unknown density in a boundary integral formulation of the problem and on a generalization of the method of stationary phase. It leads to a rigorously convergent algorithm with a bounded computational complexity as the frequency tends to infinity, that can be viewed as a natural link between high-frequency engineering approximations such as Kirchhoff's formula, and mathematically rigorous integral equation methods.

Steven G. Johnson Massachusetts Institute of Technology

Photonic-crystals, from order to disorder: perturbative methods in nanophotonics

Photonic crystals are periodic dielectric structures in which light can behave much differently than a homogeneous medium. We give an overview of some of the interesting properties and applications of these media, from switching in subwavelength microcavities, to slow-light devices, to guiding light in air. Some of the most interesting and challenging problems occur, however, when the periodicity is broken, either by design or by the inevitable fabrication imperfections. We focus especially on small perturbations that have important effects, from slow-light tapers to surface-roughness disorder, and show that many classic perturbative approaches must be rethought for high-contrast nanophotonics. The combination of strong periodicity with large field discontinuities at interfaces causes standard methods to fail but succumbs to new generalizations, while some problems remain open.

Tomasz Kaczynski Universite de Sherbrooke

Topological feature extraction in cubical grids and homology algorithms

Cubical sets, that is, finite unions of n-dimensional pixels interpreted as cubes, are a convenient structure for representing any large data of a geometric nature, among others, information contained in digital images. Moreover, the graph of a map can also be represented by a cubical set in the product space. Homology is an algebraic tool for extracting information about specific features of an image or a multidimensional structure. Where the compared objects have the same global topology but we want to distinguish their local features, relative homology becomes useful. In this talk, we first present two stages of that feature extraction: Generating a cubical chain complex directly from the data, and computing its homology. We next discuss recent progress in efficiency of reduction algorithms.

Lauri Kettunen TUT Finland, Institute of Electromagnetics

On the geometry of Whitney elements

Whitney elements, and especially so called edge elements are commonly employed in computational electromagnetism. However, very little attention is paid on Whitney's own definition of so called 'elementary flat cochains in a complex', and in the literature there are only very few papers on what Whitney elements are all about. In this presentation we examine Whitney elements relying on Whitney's own framework and examine some implications in practical computing.

Stefan Kurz Robert Bosch GmbH

A framework for Maxwell’s equations in non-inertial frames based on differential forms

Joint work with Bernd Flemisch and Barbara Wohlmuth Universität Stuttgart

In many engineering applications the interaction between the electromagnetic field and moving bodies is of great interest. It is natural to use a Lagrangian description, where the unknowns are defined on a mesh which moves and deforms together with the considered objects. What is the correct form of Maxwell’s equations and the material laws under such circumstances? The aim of the present paper is to tackle this question by using the language of differential forms. We first provide a review of the formulations of electrodynamics in terms of vector fields, as well as differential forms in the (1+3)- and four-dimensional setting. In order to keep both Maxwell’s and the constitutive equations as simple as possible, we set up two reference frames in addition to the laboratory frame. In the natural material frame, the (1+3)-Maxwell’s equations have their simple form, whereas in the co-moving inertial frame, the material laws are canonical. In contrast to existing literature [1,2] these frames are both retained to benefit from their individual advantages.

It remains to construct transformation laws connecting the considered frames. To achieve this, we use a (1+3)-decomposition in terms of general projection operators which do primarily not depend on an underlying metric or on the choice of a spatial coordinate system [3-5]. The desired transformation laws are established by comparing the different decompositions of an arbitrary p-form with respect to the considered frames. We provide an interpretation of these laws in terms of vector fields, and consider the low frequency limit, which is the most relevant case for an implementation into numerical codes.

For the description of low frequency electromagnetism, all rigid frames are equivalent. This goes beyond the standard principle of Galilean relativity, where only inertial frames are regarded as equivalent. We emphasize that this property is restricted to the low frequency limit of Maxwell’s equations, it justifies the usual analysis of induction machines from the rotor’s point of view. The proper treatment in the general case is demonstrated by means of an example in rotating coordinates, where the classical paradox by Schiff [6] is resolved.

References

[1] T. Mo, “Theory of electrodynamics in media in non-inertial frames and applications,” J. Math. Phys., vol. 11, no. 8, pp. 2589–2610, Aug. 1970.
[2] J. V. Bladel, Relativity and Engineering, ser. Springer Series in Electrophysics. Berlin: Springer-Verlag, 1984.
[3] F. Hehl and Y. Obukhov, Foundations of Classical Electrodynamics. Boston: Birkhäuser, 2003.
[4] M. Fecko, “On 3+1 decompositions with respect to an observer field via differential forms,” J. Math. Phys., vol. 38, no. 9, pp. 4542–4560, 1997.
[5] J. Kocik, “Relativistic observer and Maxwell’s equations: an example of a non-principal Ehresmann connection,“ Preprint P-98-10-029, Department of Physics, UIUC, Urbana, IL61801, http://www.math.siu.edu/Kocik/pracki/obs-gph.pdf.
[6] L. Schiff, “A question in general relativity,” Proc. Nat. Acad. Sci. USA, vol. 25, pp. 391–395, 1939.

Andre Nicolet Institut Fresnel UMR CNRS 6133

Twisted waveguides and geometrical transformations

The study of microstructured optical fibres also known as photonic crystal fibres is an appealing application for waveguide modeling in electromagnetism (determination of propagating modes). Beside very specific models such as, for instance, the multipole method that gives extremely efficient algorithms when all the elements of the geometry are supposed to be of circular cross section, there is room for more general models based on finite element modelling. For instance, during the manufacturing process, appears a presently uncontrollable twist along the fibre. Where the classical axisymmetrical fibres are naturally impervious to such a phenomenon, it seems to heavily affect the performances of microstructured fibres. At first sight, this phenomenon also destroys the invariance of the problem along an axis and jeopardizes the 2D approach but we propose a model that restores the 2D character of the problem via a (non-orthogonal) system of helicoidal coordinates where the twisted structure is constant along one co-ordinate (the one along the axis of twist), let say the w-co-ordinate. Using differential geometry and the weak formulation of the problem, it is a classical result to show that any change of co-ordinates can be taken into account with equivalent material properties. For general transformations, these material properties may be inhomogeneous and anisotropic. They in fact involve the metric tensor expressed in the new coordinates. This is the case for the helicoidal co-ordinates, but the good surprise is that though the Jacobian matrix depends on the w-coordinate, the equivalent material properties do not! This restores the 2D nature of the problem where the very concept of propagation mode is preserved. The main complication is the totally anisotropic and inhomogeneous nature of the media that calls for a finite element solution. The numerical model is then a full wave model with complete anisotropy tensors (9 non zero entries) that prevent any reduction in the products crossing transverse and longitudinal components. Edge elements are used for the transverse component and nodal elements for the longitudinal one. As we are working in a non-orthogonal system of co-ordinates, the use of Whitney forms for discretisation is fundamental to guarantee a correct covariant discrete formulation. Given the pulsation $\omega$, a generalized quadratic eigenvalue problem is obtained for the propagation constant $\gamma$. There is no trick available to reduce this problem to a linear one except the direct linearization. The obtained numerical eigenproblems are the solved using Arpack. As the practical twist is extremely weak (one single twist on several centimeters when the diameter of the fibre is of the order of the micrometer), we are currently designing asymptotic models where the small parameter is related to the twist parameter and that can be tested with respect to the direct approach explained here above.

Lassi Paivarinta Helsinki

Link to abstract

Francesca Rapetti Universite de Nice Sophia-Antipolis

Construction of two transfer operators between nested grids in the case of Whitney finite elements (node-, edge-, face- or volume-based).

These transfer operators, instances of what is called ``chain map" in Homology, have duals acting on cochains, that is to say, arrays of degrees of freedom in the context of the finite-element discretization of variational problems. We show how these duals can act as restriction/prolongation operators in a multigrid approach to such problems, especially those involving vector fields (e.g., electromagnetism). The duality between the operation of mesh refinement of a simplicial complex and that of restriction/prolongation of degrees of freedom from one mesh to a nested one is thus analysed and explained. We use the language of $p$-forms, with frequent explanatory references to the more traditional vector-fields formalism.

Claudio Rebbi Boston University

Lattice Gauge Theory

All models of particle interactions are based on quantized gauge field theories, where the gauge field is either the electromagnetic field or one of its non-Abelian generalizations. Since the early eighties particle theorists have resorted to the lattice discretization of a gauge field theory, coupled with a stochastic simulation of quantum fluctuations, to obtain otherwise unattainable numerical predictions for the theory's observables. In this talk I will review the basic aspects of the lattice discretization of a gauge field theory and give a quick overview of current state-of-the-art lattice gauge theory simulations.

Kenneth Shepard Columbia University

On-chip inductance: friend or foe?

Modelling the inductance of on-chip wires presents unique challenges to successfully modelling clock, power supply, and signalling interconnect. By the same token, on-chip inductance either in the form of explicit spirals or transmission lines presents unique opportunities for circuit innovation including resonant clock distributions and low latency interconnections.

Samuli Siltanen TUT

The d-bar reconstruction method for electrical impedance tomography

In electrical impedance tomography (EIT) one applies a set of electric voltage distributions at the boundary of an unknown physical body, measures the resulting currents through the boundary and reconstructs electric conductivity inside the body.A regularized reconstruction method for two-dimensional EIT is presented and applied to measured data. The method is based on the d-bar method first introduced for 2D EIT by A. Nachman. Also, the application of the d-bar method to 3D EIT is discussed.

John Stachel Boston University

The Generally Covariant Form of Maxwell's Equations.

Written in terms of two bivector fields (E&B, D&H)and/or their duals, Maxwell's equations are generally covariant. It is the choice of constitutive relations between the two fields that reduces the symmetry group. Various choices of constitutive relations will be discussed, as well as the generalization of this approach to Yang-Mills fields.

Saku Suuriniemi TUT, Institute of Electromagnetics

Homology groups in electrical engineering: use, computation, and experiences

Homology groups can be and are used in a variety of ways to pose and analyze electromagnetic boundary value problems (BVP's). This is both a great possibility and a challenge: Homology groups enable automated checking of consistency of BVP's, saving time by accurate diagnoses at problem definition time. Techniques in some formulations, such as cuts for scalar potentials, can be automatically performed with no possibility of error. A challenge is the somewhat complicated computation of the groups and some open questions about its complexity bounds. Moreover, homology groups are used by both machines and humans, and humans require certain additional qualitative requirements for the homology group generators: The generators should be "easy to grasp" and "intuitive", and there are no precise statements for these qualities, which makes it hard to attain them. In this presentation, a general purpose homology solver is taken as a starting point, its functioning and the open questions are briefly exposed. Then a survey to its applications follows, and the qualitative requirements by humans is addressed in the end.

Sebastien Tordeux ETH Zurich

Gunther Uhlmann University of Washington

Electrical Impedance Tomography and Travel Time Tomography

In inverse boundary problems one attempts to determine the properties of a medium by making measurements at the boundary of the medium. In the lecture we will concentrate on two inverse boundary problems, Electrical Impedance Tomography and Travel Tomography in anisotropic media. These problems arise in medical imaging, geophysics and other fields. We will also discuss a surprising connection between these two inverse problems. Travel Time Tomography, consists in determining the index of refraction or sound speed of a medium by measuring the travel times of waves going through the medium. In differential geometry this is known as the boundary rigidity problem. In this case the information is encoded in the boundary distance function which measures the lengths of geodesics joining points of the boundary of a compact Riemannian manifold with boundary. The inverse boundary problem consists in determining the Riemannian metric from the boundary distance function. Calderon's inverse boundary problem consists in determining the electrical conductivity inside a body by making voltage and current measurements at the boundary. This inverse problem is also called Electrical Impedance Tomography (EIT). The boundary information is encoded in the Dirichlet-to-Neumann (DN) map and the inverse problem is to determine the coefficients of the conductivity equation (an elliptic partial differential equation) knowing the DN map. A connection between these two inverse problems has led to a solution of the boundary rigidity problem in two dimensions for simple Riemannian metrics. We will also discuss a reconstruction method in two dimensions for the sound speed from first arrival times of waves.

Ursula van Rienen Rostock University

Some aspects of computational bioelectromagnetics

Computational bioelectromagnetism covers a wide range of scientific problems. Within these, there are a number of challenges which still have to be tackled. First, scales have to be defined - starting on the microscopic level with cells and their membranes and ending at macroscopic level with human body models. The nature of the models used on the different levels also needs to be defined. Other questions include how to get the appropriate material properties. The list goes on ....
Some of these questions will be touched on in the talk.

Scott Wilson University of Minnesota

Cochain Algebra and Approximation

We describe some of the algebraic structure on cochains (linear Whitney forms) including the cochain product and Hodge star operator. We give several results on their approximation to the wedge product and smooth Hodge star operator on smooth forms.

Afra Zomorodian Stanford University

Persistent Homology: Theory and Applications

In this talk, I motivate and describe persistent homology, an algebraic tool that reveals the underlying structure of a multi-scale view of a sampled space. After describing the theory, I present a few applications briefly, including shape description, bilipid membrane fusion, and localization. Time permitting, I will also talk about recent results on multidimensional persistence.

Posters

Peter Traneus Anderson
GE Healthcare, Lawrence MA

An Electromagnetic Tracking System Using Printed-Circuit Coils

Six-degree-of-freedom electromagnetic tracking systems traditionally use arrays of tiny coils so that the coils' mutual inductances may be modeled using dipole magnetic field models. Here, we describe a tracking system which uses coils which are too large to be modeled as dipoles. We present a model for these large coils, and compare modeled to measured mutual inductances

Markus Clemens
Helmut-Schmidt-University, Elec. Eng. Dept, Hamburg, Germany

Timo Euler
Technische Universität Darmstadt,
Institut für Theorie Elektromagnetischer Felder

POLYGONAL FINITE ELEMENTS (with Rolf Schuhmann und Thomas Weiland)

In mesh-based electromagnetic simulations, triangular or quadrilateral elements are widely used for 2-dimensional domains. The Finite Integration Technique (FIT) and the Finite Element Method (FEM) employing Whitney elements transfer the geometrical properties of the gradient, curl, and divergence operator (div curl = 0 and curl grad = 0) onto the discrete operators. Now the question arises whether these properties can also be fulfilled using more general mesh elements allowing for a completely new meshing flexibility. The full benefit of such general elements comes into play in 3-dimensional simulations, as the use of special prism and pyramid elements e.g. in the electromagnetic community shows.A general framework for the derivation of nodal, edge, and facial basis functions for arbitrarily shaped 2-dimensional polygonal elements will be presented. The commuting diagram properties responsible for the preservation of the geometrical properties will bedescribed. These basis functions can be directly employed in the standard FEM or used to derive the material matrices within the framework of the FIT.Numerical examples show the wide application range for these elements in different electromagnetic formulations.

Ardavan Farjadpour Massachusetts Institute of Technology

FDTD, quadratic convergence, and interfaces

Standard finite-difference time-domain (FDTD) methods in electromagnetism, which have nominally quadratic convergence with resolution for homogeneous media, achieve only linear or erratic convergence when high-contrast material interfaces are included. We demonstrate, however, how smooth quadratic convergence can be restored by a simple sub-pixel material averaging technique, based on perturbation and effective-medium theories, that computes an effective dielectric tensor for grid voxels that intersect sharp interfaces. We compare with previous schemes for sub-pixel averaging and show that the new method is the only one that attains quadratic convergence for arbitrary interface orientations and polarizations. Finally, we discuss additional difficulties that arise for dielectric corners.

Francois Henrotte IEM, RWTH Aachen

Electromagnetism from the point of view of energy

A system of electromagnetic (EM) energy reservoirs and flows is defined, of which the fundamental variables are the EM fields B and D, and the EM potentials A and V. The structure of this energy diagram determines a macroscopic theory of electromagnetism, in the sense that the associated conservation laws, in the material manifold, are equivalent to the combination of the classical Maxwell system with a full set of general macroscopic constitutive relations. It is shown that the classical metric free Maxwell system of equation can then be isolated by defining a second pair of EM fields, namely E and H, as additional fundamental quantities. This operation however cuts off unnecessarily all energy related aspects from the theory, and conceals the fact that the conservation equations were essentially valid in the material manifold. The classical theory then proceeds by spelling out the symmetry properties of the metric free equations and invoking a relativity principle. This leads to building artificial 4-tensors; the Faraday, form for instance, mixes up into one single mathematic quantity two physical fields of different nature. Moreover, the invariance group found this way, i.e. the Lorentz group, is too restrictive. The mapping by any diffeomorphism of the conservation equations obtained by the energy approach delivers indeed a complete set of governing equations for electromagnetic phenomena in arbitrary reference frames, even accelerated ones. The covariant equations obtained this way involve a Lie derivative, slightly modified to account for Einstein's simultaneity argument, and invariance is indeed found back when the general diffeomorphism is restricted to the subgroup of Lorentz transformations.

P Robert Kotiuga Boston University

EIT and MREIT in 3-D: SVD of the Dirichlet to Neumann map, Conditioning, and Asymptotics

In this presentation, Weyl asymptotics are used to make several general statements about the distribution of the eigenvalues of the Dirichlet to Neumann (DN) map. In particular, they demonstrate that the conditioning of electrical impedance tomography improves with dimension. The analysis shows that the practical difficulties of collecting data in 3-d should not deter one from expecting good reconstructions. In addition, the explicit computation of the SVD of the DN map is outlined. This, in turn, gives a very concrete understanding of how to obtain optimal reconstructions based on low rank approximations.

Weitzenbock Identities and Variational Formulations in Nanophotonics and Micromagnetics

Pairs of non-identical variational formulations, having identical Euler-Lagrange equations, but non-identical natural boundary conditions, are examined with a view to nano-scale applications. In particular, difference in the variational formulations is due to a boundary term, which has a concrete topological interpretation, and which may be non-zero on an arbitrarily small surface surrounding a defect. As a result, the significance of the difference between these variational formulations becomes more significant as the computational domain shrinks.

Melvin Leok University of Michigan, Ann Arbor.

Discrete Exterior Calculus and its Applications to Computational Electromagnetism

We will describe a cochain based formulation of discrete exterior calculus on simplicial complexes, and show how it can be used to construct a discretization of the variational formulation of the Maxwell equations. Interestingly, the corresponding discrete Euler-Lagrange equations are equivalent to the formulation of the Maxwell equations in terms of the Laplace-deRham operator on Lorentzian metric spaces. In particular, that means that a direct discretization of the Maxwell equations using the operators of discrete exterior calculus exhibits the desirable geometric conservation properties typical of variational integrators.

Krishna Naishadham MIT LIncoln Laboratory

State-Space Spectral Estimation of Characteristic Electromagnetic Responses,

Estimation of model parameters from scattered electromagnetic fields, either computed or measured, is an important problem in many areas, such as antenna design, radar target identification, bioelectromagnetics, VLSI interconnect analysis, etc. This paper presents a robust spectral estimation method, based on state-space control theory, to coherently process wideband frequency domain scattered fields of a given object and extract specific modal (or characteristic) responses associated with wave propagation along the object. The estimation problem is formulated in terms of well-known range processing used in radar imaging. Thus, the data is modeled in terms of complex sinusoids, whose amplitude is scaled by the decay constants of the modes and whose phase yields the range associated with scattering centers pertinent to modal propagation. The decay constants and the range are determined from eigen-decomposition of an open-loop state matrix, computed from singular value decomposition of the state equations that represent the complex sinusoid model. Unlike other approaches, such as Prony's and pencil-of-functions, that require the system poles to be inside the unit circle, the complex poles yielding the decay constants in the proposed approach can be located anywhere in the z-plane, and can vary with frequency, in order to capture the dynamic wideband behavior of the scattering mechanism. The method is illustrated by application to mode extraction for perfectly conducting and dielectric-coated cylindrically stratified scatterers.

Diego Rivera Northeastern University

Improving locality over sparse algorithms by using pattern recognition techniques

Sparse computations are an essential part of many scientific disciplines, including fluid dynamics, structural engineering, and econometric models, among others. A barrier encountered in these kinds of applications, the computational time required. Parallel computer have been shown to especially effective for this class of applications, though the sparsity of the data reduces the effectiveness of these platforms. If we can reduce inter node communications, properly partition the data sets, and modifying the data layout, we can significantly improve the performance of solving sparse matrices on parallel computers. When dealing with sparse matrices, selecting the most appropriate preconditioner can produce a much more efficient solution. However, to pick the proper optimization heuristic and best preconditioner heavily depends on the structure of nonzero elements in the sparse matrix. We are working on the development and implementation of an efficient framework for solving sparse linear systems. We have defined a set of features that lets us delineate several relationships between: 1) structural and numerical characteristics of the user data set and 2) architectural characteristics of the computation platform used. The framework will allow us to make suitable optimizations for any sparse matrix structure by using pattern recognition techniques.

Alejandro Rodriguez MIT

Single-photon All-Optical switching using waveguide-cavity QED

We demonstrate the possibility of single-photon all-optical switching in a waveguide-cavity QED framework using electromagnetically induced transparency (EIT). An anaytical model of a system consisting of a photonic crystal (PhC) waveguide, a microcavity and a four-level EIT atom is solved exactly. We extend the previous Finite Difference Time Domain (FDTD) semiclassical calculations of this phenomenon by using a quantum model and analyzing (using experimentally accessible parameters) the switching properties of the system. It is observed that switching can take place even with small Rabi splitting, though a larger Rabi-splitting is preferred for switching applications.

Ruben Specogna University of Udine

Symmetric positive-definite Ohm's constitutive matrices for discrete eddy-current problems (Joint work with L. Codecasa and F. Trevisan)

We examine the construction of a symmetric positive definite Ohm's matrix for eddy currents problems, when a discrete approach is used. To this aim we will construct a set of bases vector functions both on the primal and on the dual complex. These vector functions will be defined both for tetrahedra and prisms.

Ursula van Rienen Rostock University

P1. "Progress in the 3D Space-Charge Calculations in the GPT Code"

Abstract:Precise and fast 3D space-charge calculations for bunches of charged particles are of growing importance in recent accelerator designs. One of the possible approaches is the particle-mesh method computing the potential of the bunch in the rest frame by means of Poisson's equation. In that, the charge of the particles are distributed on a mesh. Fast methods for solving Poisson's equation are the direct solution applying Fast Fourier Methods (FFT) and a finite difference discretization combined with a multigrid method for solving the resulting linear system of equations. Both approaches have been implemented in the tracking code ASTRA. In this paper the properties of these two algorithms are discussed. Numerical examples will demonstrate the advantages and disadvantages of each method, respectively.Supported by DESY, Hamburg.

P2. "Characterization of Electrodes for Deep Brain Stimulation"

Several neurological diseases, such as Parkinson´s disease, are characterized by a pathologic synchronization of ensembles of oscillatory neurons in particular brain areas, where the normal neuronal activity is supposed to be uncorrelated. The electric stimulation of these brain areas with pulsed signals, the so-called Deep Brain Stimulation (DBS), has a positive therapeutic influence on the symptoms of the diseases. Different electrode geometries allow for different tissue volumes to be stimulated. For every geometry a factor (cell constant) can numerically be determined. It relates the electrode impedances to the specific medium conductivity when comparing electrodes of different shapes and geometries. The cell constant also allows for determining the limiting stimulation currents in order to avoid damage resulting from excessive electric field strengths in the tissue.

P3. "Computation of Land Mine Signatures using Domain Decomposition with Lagrange Multipliers"

A good knowledge of the electromagnetic fields of metal detectors which are used for mine detection is mandantory for reducing the false alarm rate, originating in the disqualification of the metal detector to decide whether there is a mine or just another metal object. Therefore the comparison of the measured signal from the detector with the pre-calculated signatures of standard mines was proposed. In order to set up a database of mine signatures it is mandatory to use different resolutions in disretising the typically very small metal parts in land mines, the soil and the detector coil. The latter is about two orders magnitudes larger and apart from the mine. Here, a domain decomposition method with Lagrange mulitpliers is used. Supported by BMBF, Bonn.