Time-harmonic Maxwell Equations Research Articles

Computation of Surface-Enhanced Infrared Absorption Spectra of Particles at a Surface through the Finite Element Method

The finite element method (FEM) was used to solve the time-harmonic Maxwell equations to calculate full infrared (IR) absorption spectra of organic substances near particles adsorbed to a silicon/air interface mimicking a surface-enhanced infrared absorption spectroscopy (SEIRAS) experiment in internal reflection geometry. An enhancement factor attributed to the presence of the metal has been defined, and its dependence on the particle size and arrangement has been studied. The enhancement factor was found to increase with increasing particle size. Contributions of parts of the surface to the overall absorbance has been analyzed through the electric field patterns and by placing a probe material in the region to be analyzed. For isolated particles, the gap between particle and surface has a large contribution to the overall spectrum. Further, large contributions were found from the regions between touching particles. An experiment probing gold particles on a silicon surface using attenuated total internal reflection infrared (ATR-IR) spectroscopy with a silicon crystal was performed and was compared to computations. The experimentally derived enhancement factor in the given geometry was of the same magnitude as the computed enhancement factor.

First-Principles Full-Vectorial Eigenfrequency Computations for Axially Symmetric Resonators

Starting from the time-harmonic Maxwell's equations in cylindrical coordinates, we derive and solve the finite-difference (FD) eigenvalue equations for determining vector modes of axially symmetric resonator structures such as disks, rings, spheres and toroids. Contrary to the most existing implementations, our FD scheme is readily adapted for both eigenmode and eigenfrequency calculations. An excellent match of the FD solutions with the analytically calculated mode indices of a microsphere resonator provides a numerical confirmation of the mode-solver accuracy. The comparison of the presented FD technique with the finite-element method highlights the relative strengths of both techniques and advances the FD mode-solver as an important tool for cylindrical resonator design.

Optimal Control of Three-Dimensional State-Constrained Induction Heating Problems with Nonlocal Radiation Effects

The paper is concerned with a class of optimal heating problems in semiconductor single crystal growth processes. To model the heating process, time-harmonic Maxwell equations are considered in the system of the state. Due to the high temperatures characterizing crystal growth, it is necessary to include nonlocal radiation boundary conditions and a temperature-dependent heat conductivity in the description of the heat transfer process. The first goal of this paper is to prove existence and uniqueness of the state. The regularity analysis associated with the time-harmonic Maxwell equations is also studied. In the second part of the paper, existence and uniqueness of the solution of the corresponding linearized equation are shown. With this result at hand, the differentiability of the control-to-state operator is derived. Finally, based on the theoretical results, first order necessary optimality conditions for an associated optimal control problem are established.

On an inverse problem in electromagnetism with local data: stability and uniqueness

In this paper we prove a stable determination of the coefficients of the time-harmonic Maxwell equations from local boundary data. The argument --due to Isakov-- requires some restrictions on the domain.

Compact Imbeddings in Electromagnetism with Interfaces between Classical Materials and Metamaterials

In a metamaterial, the electric permittivity and/or the magnetic permeability can be negative in given frequency ranges. We investigate the solution of the time-harmonic Maxwell equations in a composite material, made up of classical materials, and metamaterials with negative electric permittivity, in a two-dimensional bounded domain $\Omega$. We study the imbedding of the space of electric fields into $L^2(\Omega)^2$. In particular, we extend the famous result of Weber, proving that it is compact. This result is obtained by studying the regularity of the fields. We first isolate their most singular part, using a decomposition à la Birman and Solomyak. With the help of the Mellin transform, we prove that this singular part belongs to $H^s(\Omega)^2$ for some $s>0$. Finally, we show that the compact imbedding result holds as soon as no ratio of permittivities between two adjacent materials is equal to $-1$.

Homogenization of time harmonic Maxwell equations and the frequency dispersion effect

We perform homogenization of the time-harmonic Maxwell equations in order to determine the effective dielectric permittivity ε h and effective electric conductivity σ h . We prove that ε h and σ h depend on the pulsation ω; this phenomenon is known as the frequency dispersion effect. Moreover, the macroscopic Maxwell equations also depend on ω; they are different for small and large values of ω.

Stable determination of the electromagnetic coefficients by boundary measurements

The goal of this paper is to prove a stable determination of the coefficients for the time-harmonic Maxwell equations, in a Lipschitz domain, by boundary measurements.

Effect of Medium Chirality on the Rate of Resonance Energy Transfer

Resonance energy transfer (RET) between two chromophores in an absorptive and dispersive chiral medium is investigated using a quantum electrodynamical formulation. To accurately describe such an environment involves the introduction of electric displacement and auxiliary magnetic field operators that are solutions of the Drude-Born-Fedorov equations and the time-harmonic Maxwell equations. Perturbation theory within the electric and magnetic dipole approximation is used in the derivation of the probability amplitude for energy transfer. Expressions for the contributions to the RET rate arising from the pure electric dipole term, replacing one of the chiral chromophores by its corresponding enantiomer in the mixed electric-magnetic dipole term, and the pure magnetic dipole contribution are obtained. In the near-zone limit in a nonabsorptive medium, the medium chirality amplifies the pure electric dipole contribution to the rate relative to that in a racemic mixture and also increases the discriminatory contribution, but to a lesser extent relative to the pure electric dipole term. On the other hand, under the same conditions, the medium chirality does not affect the pure magnetic dipole contribution to the rate. Measurements of the rate could be used to obtain information on the magnitude of the chirality admittance or concentration of chiral species, by comparing with the rate in an environment comprised of a racemic mixture of the enantiomers. This method could allow for the analysis of macroscopically heterogeneous systems that are comprised of enantiomers and where the chromophores experiencing RET are located in regions of interest.

Influence of medium chirality on electric dipole–dipole resonance energy transfer

Electric dipole–dipole resonance energy transfer taking place between two chromophores in an absorptive and dispersive chiral medium is studied. Quantized electromagnetic field operators in this environment are first obtained from the time-harmonic Maxwell equations and the Drude–Born–Fedorov equations. Second-order time-dependent perturbation theory and the Fermi Golden rule are used to calculate the transfer rate. A complicated dependence on the permittivity, permeability and chirality admittance of the medium is found. In the near-zone, the rate is amplified in a medium with negligible absorption comprised of one enantiomer relative to that in a racemic mixture.

Optimal Penalty Parameters for Symmetric Discontinuous Galerkin Discretisations of the Time-Harmonic Maxwell Equations

We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are (i) valid in the pre-asymptotic regime; (ii) solely depend on the geometry and the polynomial order; and (iii) are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order (p = 1, 2, 3, 4) hierarchic H(curl)-conforming polynomial basis functions.

Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics

Non-overlapping domain decomposition (DD) methods provide efficient algorithms for solving time-harmonic Maxwell equations. It has been shown that the convergence of DD algorithms can be improved significantly by using high order transmission conditions. In this paper, we extend a newly developed second-order transmission condition (SOTC), which involves two second-order transverse derivatives, to facilitate fast convergence in the non-conformal DD algorithms. However, the non-conformal nature of the DD methods introduces an additional technical difficulty, which results in poor convergence in many real-life applications. To mitigate the difficulty, a corner-edge penalty method is proposed and implemented in conjunction with the SOTC to obtain truly robust solver performance. Numerical results verify the analysis and demonstrate the effectiveness of the proposed methods on a few model problems. Finally, drastically improved convergence, compared to the conventional Robin transmission condition, was observed for an electrically large problem of practical interest.

Nonoverlapping Domain Decomposition with Second Order Transmission Condition for the Time-Harmonic Maxwell's Equations

Nonoverlapping domain decomposition methods have been shown to provide efficient iterative algorithms for the solution of the time-harmonic Maxwell's equations. Convergence of the algorithms depends strongly upon the nature of the transmission conditions that communicate information between adjacent subdomains. In this work, we introduce a new second order transmission condition that improves convergence. In contrast to previous high order interface conditions, the new condition uses two second order transverse derivatives to improve the convergence of evanescent modes without deteriorating the convergence of propagating modes. An analysis using a splitting of the field into traverse electric and magnetic components demonstrates the improved convergence provided by the transmission condition. Numerical experiments demonstrate the effectiveness of the algorithm.

Homogenization of Maxwell's Equations in a Split Ring Geometry

We analyze the time harmonic Maxwell equations in a complex three-dimensional geometry. The scatterer $\Omega\subset\mathbb{R}^3$ contains a periodic pattern of small wire structures of high conductivity, and the single element has the shape of a split ring. We rigorously derive effective equations for the scatterer and provide formulas for the effective permittivity and permeability. The latter turns out to be frequency dependent and has a negative real part for appropriate parameter values. This magnetic activity is the key feature of a left-handed metamaterial.

Debye sources and the numerical solution of the time harmonic Maxwell equations

AbstractIn this paper, we develop a new representation for outgoing solutions to the time‐harmonic Maxwell equations in unbounded domains in ℝ3. This representation leads to a Fredholm integral equation of the second kind for solving the problem of scattering from a perfect conductor, which does not suffer from spurious resonances or low‐frequency breakdown, although it requires the inversion of the scalar surface Laplacian on the domain boundary. In the course of our analysis, we give a new proof of the existence of nontrivial families of time‐harmonic solutions with vanishing normal components that arise when the boundary of the domain is not simply connected. We refer to these as k‐Neumann fields, since they generalize, to nonzero wave numbers, the classical harmonic Neumann fields. The existence of k‐Neumann fields was established earlier by Kress. © 2009 Wiley Periodicals, Inc.

A decoupling-based imaging method for inverse medium scattering for Maxwell's equations

We present an iterative reconstruction method for an inverse medium scattering problem (IMP) for the three-dimensional time-harmonic Maxwell equations. The goal here is to determine the electromagnetic properties of a nonmagnetic unknown inhomogeneous object. The data are near-field measurements of scattered electric fields for multiple illuminations at a fixed frequency. We use the concept of the generalized induced source (GIS) to recast the intertwined vector equations of Maxwell into decoupled scalar scattering problems. To treat the nonlinearity of the IMP, we apply the localized nonlinear approximation due to Habashy and co-workers. In this framework, we derive a fast reconstruction method based on the Kaczmarz algorithm. Besides, for the underlying approximation we present a uniqueness result for determining the contrast function. Numerical experiments in 3D with synthetic and real data show the scope and limitations of the method.

Weighted regularization for composite materials in electromagnetism

In this paper, a weighted regularization method for the time-harmonic Maxwell equations with perfect conducting or impedance boundary condition in composite materials is presented. The computational domain Ω is the union of polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularities near geometric singularities, which are the interior and exterior edges and corners. The variational formulation of the weighted regularized problem is given on the subspace of H(curl;Ω) whose fields u satisfy wα div(eu) ∈ L 2(Ω) and have vanishing tangential trace or tangential trace in L2(δΩ). The weight function w(x) is equivalent to the distance of x to the geometric singularities and the minimal weight parameter α is given in terms of the singular exponents of a scalar transmission problem. A density result is proven that guarantees the approximability of the solution field by piecewise regular fields. Numerical results for the discretization of the source problem by means of Lagrange Finite Elements of type P1 and P2 are given on uniform and appropriately refined two-dimensional meshes. The performance of the method in the case of eigenvalue problems is addressed. © EDP Sciences, SMAI 2009.

Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements

A few years ago, Costabel and Dauge proposed a variational setting, which allows one to solve numerically the time-harmonic Maxwell equations in 3D polyhedral geometries, with the help of a continuous approximation of the electromagnetic field. In order to remove spurious eigenmodes, their method required a parameterization of the variational formulation. In order to avoid this difficulty, we use a mixed variational setting instead of the parameterization, which allows us to handle the divergence-free constraint on the field in a straightforward manner. The numerical analysis of the method is carried out, and numerical examples are provided to show the efficiency of our approach.

New block triangular preconditioner for linear systems arising from the discretized time-harmonic Maxwell equations

In this paper, based on the preconditioners presented by Rees and Greif [T. Rees, C. Greif, A preconditioner for linear systems arising from interior point optimization methods, SIAM J. Sci. Comput. 29 (2007) 1992–2007], we present a new block triangular preconditioner applied to the problem of solving linear systems arising from finite element discretization of the mixed formulation of the time-harmonic Maxwell equations ( k = 0 ) in electromagnetic problems, since linear systems arising from the corresponding equations and methods have the same matrix block structure. Similar to spectral distribution of the preconditioners presented by Rees and Greif, this paper analyzes the corresponding spectral distribution of the new preconditioners considered in this paper. From the views of theories and applications, the presented preconditioners are as efficient as the preconditioners presented by Rees and Greif to apply. Moreover, numerical experiments are also reported to illustrate the efficiency of the presented preconditioners.

Reliable Fast Frequency Sweep for Microwave Devices via the Reduced-Basis Method

In this paper, a reduced-basis-approximation-based model-order reduction for fast and reliable frequency sweep in the time-harmonic Maxwell's equations is detailed. Contrary to what one may expect by observing the frequency response of different microwave circuits, the electromagnetic field within these devices does not drastically vary as frequency changes in a band of interest. Thus, instead of using computationally inefficient, large dimension, numerical approximations such as finite- or boundary-element methods for each frequency in the band, the point in here is to approximate the dynamics of the electromagnetic field itself as frequency changes. A much lower dimension, reduced-basis approximation sorts this problem out. Not only rapid frequency evaluation of the reduced-order model is carried out within this approach, but also special emphasis is placed on a fast determination of the error measure for each frequency in the band of interest. This certifies the accurate response of the reduced-order model. The same scheme allows us, in an offline stage, to adaptively select the basis functions in the reduced-basis approximation and automatically select the model-order reduction process whenever a preestablished accuracy is required throughout the band of interest. Finally, real-life applications will illustrate the capabilities of this approach.

Optimal Error Estimates for Nédélec Edge Elements for Time-Harmonic Maxwell's Equations

In this paper, we obtain optimal error estimates in both L 2 -norm and H(curl)-norm for the Nedelec edge flnite element approximation of the time-harmonic Maxwell's equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L 2 error estimates into the L 2 estimate of a discrete divergence-free function which belongs to the edge flnite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nedelec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.