Linear Maxwell Equations Research Articles

Bifurcation and asymptotics of cubically nonlinear transverse magnetic surface plasmon polaritons

Linear Maxwell equations for transverse magnetic (TM) polarized fields support single frequency surface plasmon polaritons (SPPs) localized at the interface of a metal and a dielectric. Metals are typically dispersive, i.e. the dielectric function depends on the frequency. We prove the bifurcation of localized SPPs in dispersive media in the presence of a cubic nonlinearity and provide an asymptotic expansion of the solution and the frequency. The problem is reduced to a system of nonlinear differential equations in one spatial dimension by assuming a plane wave dependence in the direction tangential to the (flat) interfaces. The number of interfaces is arbitrary and the nonlinear system is solved in a subspace of functions with the H1-Sobolev regularity in each material layer. The corresponding linear system is an operator pencil in the frequency parameter due to the material dispersion. The studied bifurcation occurs at a simple isolated eigenvalue of the pencil. For geometries consisting of two or three homogeneous layers we provide explicit conditions on the existence of eigenvalues and on their simpleness and isolatedness. Real frequencies are shown to exist in the nonlinear setting in the case of PT-symmetric materials. We also apply a finite difference numerical method to the nonlinear system and compute bifurcating curves.

Wellposedness and regularity for linear Maxwell equations with surface current

We study linear time-dependent Maxwell equations on a cuboid consisting of two homogeneous subcuboids. At the interface, we allow for nonzero surface charge density and surface current. This model is a first step towards a detailed mathematical analysis of the interaction of single-layer materials with electromagnetic fields. The main results of this paper provide several wellposedness and regularity statements for the solutions of the Maxwell system. To prove the statements, we employ extension arguments using interpolation theory, as well as semigroup theory and regularity theory for elliptic transmission problems.

Analysis of a Peaceman-Rachford ADI scheme for Maxwell equations in heterogeneous media

The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than H2-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equations in detail in an appropriate functional analytical framework. The theoretical findings are complemented by a numerical experiment indicating that the proven convergence rate is indeed observable and optimal.

A space–time Trefftz DG scheme for the time-dependent Maxwell equations in anisotropic media

In this paper we are concerned with Trefftz discretizations of the time-dependent Maxwell equations in anisotropic media in three-dimensional domains. We propose a class of space–time Trefftz DG methods, which include the Trefftz variational formulation from Egger et al. (2015). We prove the error estimates of the approximate solutions with respect to the meshwidth and the condition number of the coefficient matrices. Furthermore, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent linear nonhomogeneous Maxwell equations in anisotropic media. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy.

Development of capacitance sensor for void fraction measurement in a packed bed of spheres

A capacitance void fraction sensor (CVS) is applied to measure the volumetric averaged void fraction in a packed bed of spheres. The void fraction in the packed bed is one of the most important parameters to evaluate cooling characteristics in a porous debris bed during a severe accident of nuclear reactors, and the quantitative void fraction measuring technique for such porous flow channels should be developed. The CVS is a very simple method, and the void fraction is estimated from the electrical capacitance measured between the electrodes installed on the pipe. Generally, the linear relationship or Maxwell equation could be applied to estimate the void fraction from the capacitance measured by the CVS. However, the electrical field in the packed bed becomes complex due to the existence of spheres. Therefore, they may not be applied to the void fraction estimation in the packed bed. In this study, the CVS with a ring-type electrode configuration is used for the sphere-packed beds, and the applicability of the CVS is investigated. At first, the particle size and the pipe diameter are varied in the packed test section, and X-ray transmission imaging is used to clarify the relation between the void fraction and the capacitance in the packed bed. Then, it is found that the void fraction can be obtained by the coefficient in Maxwell's equation, depending on the packed bed properties. Finally, the measurement accuracy of the CVS for the sphere-packed bed is estimated by comparing it with a volumetric method, and the availability of the proposed method is shown.

Modal analysis of electromagnetic resonators: User guide for the MAN program

All electromagnetic systems, in particular resonators or antennas, have resonances with finite lifetimes. The associated eigenstates, also called quasinormal modes, are essentially non-Hermitian and determine the optical responses of the system. We introduce MAN (Modal Analysis of Nanoresonators), a software with many open scripts, which computes and normalizes the quasinormal modes of virtually any electromagnetic resonator, be it composed of dispersive, anisotropic, or non-reciprocal materials. MAN reconstructs the scattered field in the basis formed by the quasinormal modes of the resonator and provides a transparent interpretation of the physics.The software is implemented in MATLAB and has been developed over the past ten years. MAN features many toolboxes that illustrate how to use the software for various emblematic computations in low and high frequency regimes. A specific effort has been devoted to interface the solver with the finite-element software COMSOL Multiphysics. However, MAN can also be used with other frequency-domain numerical solvers. This article introduces the program and summarizes the relevant theoretical background. MAN includes a comprehensive set of classical models and toolboxes that can be downloaded from the web. Program summaryProgram title: MAN (Model Analysis of Nanoresonators)CPC Library link to program files:https://doi.org/10.17632/wdz5pd4sb4.1Licensing provisions: GNU General Public License 3Developer's repository link:https://doi.org/10.5281/zenodo.7400937Programming language: MatlabNature of problem: Compute and normalize the quasinormal modes of resonators and antennas of any type, be they composed of dispersive, anisotropic, or non-reciprocal materials, or be they operated at high or low frequencies. Use the quasinormal modes to reconstruct the resonator responses in the near field and far field.Solution method: Two methods are gathered. The quasinormal modes are computed and normalized either by solving the linearized Maxwell equations or searching poles in the complex frequency plane. The reconstruction is then performed analytically with closed form expressions of the excitation coefficients. A presentation of the software capabilities is found in the video (https://www.youtube.com/watch?v=SOdQsXdBGv0). To see how to use the software, please also watch the demo (https://www.youtube.com/watch?v=BSf9A1qyUBE).

Analysis of a dimension splitting scheme for Maxwell equations with low regularity in heterogeneous media

We analyze a dimension splitting scheme for the time integration of linear Maxwell equations in a heterogeneous cuboid. The domain contains several homogeneous subcuboids and serves as a model for a rectangular embedded waveguide. Due to discontinuities of the material parameters and irregular initial data, the solution of the Maxwell system has regularity below H^1. The splitting scheme is adapted to the arising singularities and is shown to converge with order one in L^2. The error result only imposes assumptions on the model parameters and the initial data, but not on the unknown solution. To achieve this result, the regularity of the Maxwell system is analyzed in detail, giving rise to sharp explicit regularity statements. In particular, the regularity parameters are given in explicit terms of the largest jump of the material parameters. The analysis is based on semigroup theory, interpolation theory, and regularity analysis for elliptic transmission problems.

Regularity theory for nonautonomous Maxwell equations with perfectly conducting boundary conditions

In this work we study linear Maxwell equations with time- and space-dependent matrix-valued permittivity and permeability on domains with a perfectly conducting boundary. This leads to an initial boundary value problem for a first-order hyperbolic system with characteristic boundary. We prove a priori estimates for solutions in Hm. Moreover, we show the existence of a unique Hm-solution if the coefficients and the data are sufficiently regular and satisfy certain compatibility conditions. Since the boundary is characteristic for the Maxwell system, we have to exploit the divergence conditions in the Maxwell equations in order to derive the energy-type Hm-estimates. A combination of these estimates with several regularization techniques then yields the existence of solutions in Hm.

A uniformly exponentially stable ADI scheme for Maxwell equations

A modified alternating direction implicit scheme for the time integration of linear isotropic Maxwell equations with strictly positive conductivity on cuboids is constructed. A key feature of the proposed scheme is its uniform exponential stability, being achieved by coupling the Maxwell system with an additional damped PDE and adding artificial damping to the scheme. The implicit steps in the resulting time integrator further decouple into essentially one-dimensional elliptic problems, requiring only linear complexity. The convergence of the scheme to the solution of the original Maxwell system is analyzed in the abstract time-discrete setting, providing an error bound in a space related to H−1.

The optical analyses of the multilayer transparent electrode and the formation of ITO/Mesh-Ag/ITO multilayers for enhancing an optical transmittance

As a highly conductive and transparent electrode, ITO/Ag/ITO multilayers are fabricated using an in-line sputtering method. Optimal thickness conditions have been investigated in terms of optical transmittance and electrical conductance. Considering the optical properties, the experimental characteristics are analyzed based on theoretical phenomena and compared with the simulated results. The simulations are based on the finite-difference-time-domain (FDTD) method in solving linear Maxwell equations. The results showed that ITO/Ag/ITO structures with respective thicknesses of 39.2 nm/10.7 nm/39.2 nm are most suitable with an average transmittance of 87% calculated for wavelengths ranging from 400 to 800 nm and a sheet resistance of about 7.1 Ω∕□. However, even with the optimum thickness conditions of the ‘ITO/Ag/ITO’ multilayers, the transmittances on some ranges of the visible wavelength were much lower than those of a single ITO layer. In order to improve the transmittance, Ag layer was formed with mesh structure. The transmittance and the sheet resistance in the ITO/Mesh-Ag/ITO multilayer structure were analyzed depending on the open ratio. As a result, a trade off in the open ratio was necessary in order to obtain the transmittance as high as possible and the sheet resistance as possible low. By the experiments, we found out that the transmittance was nearly same as the single ITO layer and the sheet resistance was about 30 Ω∕□ when the open ratio was about 85%.

Diagnostics of electromagnetic waves in the vacuum

Under interaction of plasma flows there are electromagnetic waves are generated in the plasma. They can be distributed in different directions, interact with the plasma and can be changed by amplitude and motion direction. The study of the generating electromagnetic waves in the plasma is impossible due to the nonlinear character of such interaction when modeling. In many problems to solve this problem one places a vacuum domain next to the plasma region, where the generated by the plasma electromagnetic waves are studied. Waves in vacuum follow to the linear Maxwell equations and, therefore, it is easy to determine their frequencies and amplitudes using the Fourier analysis. To study this matter, it is sufficient to consider the problem in the 2D case. The aim of the current work is the development of the technique to determine directions and amplitudes of all electromagnetic waves in the vacuum which are present in a computational domain at definite time.

Upwind discontinuous Galerkin space discretization and locally implicit time integration for linear Maxwell’s equations

This paper is dedicated to the full discretization of linear Maxwell's equations, where the space discretization is carried out with a discontinuous Galerkin (dG) method on a locally refined spatial grid. For such problems explicit time integrators are inffcient due to their strict CFL condition stemming from the fine mesh elements in the spatial grid. In the last years this issue of so-called grid-induced stiffness was successfully tackled with locally implicit time integrators. So far, these methods were limited to unstabilized (central fluxes) dG methods. However, stabilized (upwind fluxes) dG schemes provide many benefits and thus are a popular choice in applications. In this paper we construct a new variant of a locally implicit time integrator based on an upwind fluxes dG discretization on the coarse part of the grid. In contrast to our earlier analysis of a central fluxes locally implicit method, we now use an energy technique to rigorously prove its stability and provide error bounds with optimal rates in space and time.

Mechanical, Electrical and Magnetic Properties of Ferrogels with Embedded Iron Oxide Nanoparticles Obtained by Laser Target Evaporation: Focus on Multifunctional Biosensor Applications.

Hydrogels are biomimetic materials widely used in the area of biomedical engineering and biosensing. Ferrogels (FG) are magnetic composites capable of functioning as magnetic field sensitive transformers and field assisted drug deliverers. FG can be prepared by incorporating magnetic nanoparticles (MNPs) into chemically crosslinked hydrogels. The properties of biomimetic ferrogels for multifunctional biosensor applications can be set up by synthesis. The properties of these biomimetic ferrogels can be thoroughly controlled in a physical experiment environment which is much less demanding than biotests. Two series of ferrogels (soft and dense) based on polyacrylamide (PAAm) with different chemical network densities were synthesized by free-radical polymerization in aqueous solution with N,N’-methylene-diacrylamide as a cross-linker and maghemite Fe2O3 MNPs fabricated by laser target evaporation as a filler. Their mechanical, electrical and magnetic properties were comparatively analyzed. We developed a giant magnetoimpedance (MI) sensor prototype with multilayered FeNi-based sensitive elements deposited onto glass or polymer substrates adapted for FG studies. The MI measurements in the initial state and in the presence of FG with different concentrations of MNPs at a frequency range of 1–300 MHz allowed a precise characterization of the stray fields of the MNPs present in the FG. We proposed an electrodynamic model to describe the MI in multilayered film with a FG layer based on the solution of linearized Maxwell equations for the electromagnetic fields coupled with the Landau-Lifshitz equation for the magnetization dynamics.

Anisotropic metamaterial waveguide driven by a cold and relativistic electron beam

We study the interaction of a cold and relativistic electron beam with a cylindrical waveguide loaded by an anisotropic and dispersive metamaterial layer. The general dispersion relation for the transverse magnetic (TM) mode, through the linear fluid model and Maxwell equations decomposition method, is derived. The effects of some metamaterial parameters on dispersion relation are presented. A qualitative discussion shows the possibility of monomodal propagation band widening and obtaining more control on dispersion relation behavior. Especially for epsilon negative near zero metamaterials, these effects are considerable. Finally, the anisotropy and metamaterial layer thickness impacts on wave growth rate for different metamaterials are considered. The results demonstrate that we can control both wave growth rate and voltage of saturation peak by metamaterial parameters.

Error analysis of an ADI splitting scheme for the inhomogeneous Maxwell equations

In this paper we investigate an alternating direction implicit (ADI) time integration scheme for the linear Maxwell equations with currents, charges and conductivity. We show its stability and efficiency. The main results establish that the scheme converges in a space similar to \begin{document}$H^{-1}$\end{document} with order two to the solution of the Maxwell system. Moreover, the divergence conditions in the system are preserved in \begin{document}$H^{-1}$\end{document} with order one.

Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

This paper is focused on following time‐harmonic Maxwell equation: urn:x-wiley:mma:media:mma4615:mma4615-math-0001 where is a bounded Lipschitz domain, is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as , we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor and permittivity tensor , ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.

A model for asymmetric magnetoimpedance effect in multilayered bimagnetic films

The magnetoimpedance in three-layered bimagnetic film structure is studied theoretically. The structure consists of the soft and hard magnetic films separated by highly conductive non-magnetic layer. A model to describe the magnetoimpedance effect in the film structure based on a simultaneous solution of linearized Maxwell equations and Landau–Lifshitz equation is proposed. It is shown that magnetostatic coupling between the magnetic layers results in the asymmetry in the field dependence of the film impedance. The magnetostatic coupling is described in terms of an effective bias field appearing in the soft magnetic layer. The calculated field and frequency dependences of the film impedance are shown to be in a qualitative agreement with previous results of experimental studies of the asymmetric magnetoimpedance in NiFe/Cu/Co film structures. The results obtained may be useful for development of weak magnetic-field sensors.

Error Analysis of a Second-Order Locally Implicit Method for Linear Maxwell's Equations

In this paper we consider the full discretization of linear Maxwell's equations on spatial grids which are locally refined. For such problems, explicit time integration schemes become inefficient because the smallest mesh width results in a strict CFL condition. Recently locally implicit time integration methods have become popular in overcoming the problem of so-called grid-induced stiffness. Various such schemes have been proposed in the literature and have been shown to be very efficient. However, a rigorous analysis of such methods is still lacking. In fact, the available literature focuses on error bounds which are valid on a fixed spatial mesh only but deteriorate in the limit where the smallest spatial mesh size tends to zero. Moreover, some important questions cannot be answered without such an analysis. For example, there has been no study of which elements of the spatial mesh enter the CFL condition. In this paper we provide such a rigorous analysis for a locally implicit scheme proposed by Verw...

Non-perturbative aspects of particle acceleration in non-linear electrodynamics

We undertake an investigation of particle acceleration in the context of non-linear electrodynamics. We deduce the maximum energy that an electron can gain in a non-linear density wave in a magnetised plasma, and we show that an electron can “surf” a sufficiently intense Born-Infeld electromagnetic plane wave and be strongly accelerated by the wave. The first result is valid for a large class of physically reasonable modifications of the linear Maxwell equations, whilst the second result exploits the special mathematical structure of Born-Infeld theory.

Time-dependent current source identification for numerical simulations of Maxwell's equations

This paper discusses an inverse source problem for time-dependent linear Maxwell's equations. We propose a numerical procedure to compute a current density source that gives a specified electromagnetic field. The efficiency of our method is shown by numerical tests, first in 1D and then we move on to a 3D example with antennas. In that case, we show potential applications for fault detection.