Maxwell's Curl Equations Research Articles

Wave modes in a collision-free multi-species magnetoplasma with anisotropic background and perturbation pressures

The momentum balance equations and the balance equations for the pressure tensors of a collision-free multi-species magnetoplasma are linearized and Fourier-transformed, together with Maxwell's curl equations. They are combined in an algebraic dispersion system for the electric wave vector. The vanishing determinant of this system is the dispersion equation. It is of degree 2 + 3s in the square of the refractive index if s is the number of different particle species.

Smoothed Particle ElectroMagnetics: A mesh-free solver for transients

In this paper an advanced mesh-free particle method for electromagnetic transient analysis, is presented. The aim is to obtain efficient simulations by avoiding the use of a mesh such as in the most popular grid-based numerical methods. The basic idea is to obtain numerical solutions for partial differential equations describing the electromagnetic problem by using a set of particles arbitrarily placed in the problem domain. The mesh-free smoothed particle hydrodynamics method has been adopted to obtain numerical solution of time domain Maxwell's curl equations. An explicit finite difference scheme has been employed for time integration. Details about the numerical treatment of electromagnetic vector fields components are discussed. Two case studies in one and in two dimensions are reported. In order to validate the new proposed methodology, named as Smoothed Particle ElectroMagnetics, a comparison with the standard finite difference time domain method results is performed. The intrinsic adaptive capability of the proposed method, has been exploited by introducing irregular particles distribution.

Electrically controlled optical bandgap in a structurally chiral material

The propagation of light along the axis of non-homogeneity of a structurally chiral medium (SCM) under the influence of a low-frequency (dc) electric field aligned along the same axis is investigated. The Pockels effect is assumed to occur, and the SCM is taken to possess locally a 4¯2m point group symmetry. The frequency-domain Maxwell curl equations are cast in a 4×4 matrix representation, and the Oseen transformation is invoked. The band structure is analyzed, as are also the eigenvectors as well as the transfer matrix. Finally, the reflection and transmission of a plane wave due to a SCM of finite thickness is considered. The low-frequency electric field is shown to control the bandgap.

Modeling of magnetic loss in the finite-element time-domain method

The modeling of magnetically lossy media in the finite-element time-domain (FETD) method is substantially different from the simple treatment in the frequency-domain finite-element method. Two FETD formulations are presented and their numerical implementations are discussed. One is applicable to the case where each element is homogeneous, and the other can be applied to inhomogeneous elements. Verification is provided through comparison with results obtained from the traditional finite-difference time-domain method that solves the two Maxwell curl equations. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 46: 165–168, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20932

Model-Order Reduction of Finite-Element Approximations of Passive Electromagnetic Devices Including Lumped Electrical-Circuit Models

A methodology is presented for the development of reduced-order macromodels for multiport passive electromagnetic devices that include embedded lumped elements. The proposed methodology utilizes a discrete state-space model for the electromagnetic device, generated through the application of the finite-element method for the spatial discretization of Maxwell's curl equations. The incorporation of lumped resistors, inductors, and capacitors is effected through the direct stamping of the state-space voltage-current relationship for these elements in the matrices of the generated state-space form of the discrete model. The conditions necessary for the discrete model to be passive are discussed. The subsequent reduction of the discrete state-space model is effected through the application of a Krylov-subspace-based model-order reduction scheme that guarantees the passivity of the generated multiport macromodel, provided that the original state-space model is passive. The proposed methodology is demonstrated and validated through its application for the generation of reduced-order macromodels for a coaxial cable circuit and a microstrip directional coupler circuit.

Rigorous Electromagnetic Modeling of Radiative Interactions with Microstructures Using the Finite Volume Time-Domain Method

A rigorous electromagnetic model is developed to predict the radiative properties of patterned silicon wafers. For nonplanar structures with a characteristic length close to the wavelength of incident radiation, Maxwell's equations must be used to describe the associated radiative interaction and they are solved by the unstructured finite volume time-domain (FVTD) method. The basic idea of the FVTD method is to cast the two Maxwell curl equations in a conservative form, and then treat the six scalar components of the electromagnetic fields as conserved quantities via a finite volume approach. In the die area, only one period of the structure is modeled due to its periodicity in geometry. To truncate a computational domain in an open space, the Mur boundary condition is applied to absorb outgoing waves. With the steady state time-harmonic electromagnetic fields known, the Poynting vector is used to calculate the radiative properties. To validate the present model, a wave scattering problem from a cylinder is first considered and the predicted results are found to be essentially identical to the analytical solution. After that, radiative interactions with a nonplanar structure and a patterned wafer consisting of the periphery and die area are investigated, and predicted reflectivities and absorptivities are found to match other available solutions very well, indicating that the present finite volume approach in the time domain is accurate to predict radiative interaction with microstructures.

Uniaxial approach to time‐domain computations using electromagnetic potentials

AbstractA new approach to transient electromagnetic (EM) analysis in terms of wave potentials is presented. It is based on the EM field and source scalarization in a non‐uniform medium. The algorithm computes two scalar wave functions of space‐time, which constitute a complete 3‐D solution. They represent the time‐dependent values of two collinear (electric and magnetic) vector potentials of known fixed direction throughout the computational volume. Auxiliary source functions are also computed but only at the location of material non‐uniformities. A finite‐difference implementation in the case of metallic boundaries and inclusions is considered. The algorithm offers approximately 30% reduction of the computation time and memory requirements in comparison with the FDTD method based on the Maxwell curl equations. The technique is illustrated with a number of waveguide examples. Copyright © 2004 John Wiley & Sons, Ltd.

A low-order unstructured-mesh approach for computational electromagnetics in the time domain.

Maxwell's curl equations in the time domain are solved using an explicit linear finite-element approach implemented on unstructured tetrahedral meshes. For the simulation of scattering problems, a perfectly matched layer is added at the artificial far-field boundary, created by the truncation of the physical domain prior to the numerical solution. The complete solution procedure is parallelized. The computational challenges that are encountered when attempting simulations at higher frequencies suggest that the implementation of a hybrid algorithm could have certain advantages. The hybrid approach adopted uses a combination of the finite-element procedure and the well-known low operation count/low storage finite-difference time-domain method. Examples are included to demonstrate the numerical performance of the techniques that are described.

Accurate implementation of current sources in the ADI-FDTD scheme

This letter presents a new method of implementing current sources in the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method. The current source condition is derived systematically, starting with the fully implicit Crank-Nicolson (CN) FDTD scheme for Maxwell's curl equations and building the ADI-FDTD scheme as a second-order-in-time perturbation of the CN-FDTD scheme. We demonstrate that the accuracy of this new source condition is superior to previously proposed techniques.

Microwave imaging in the time domain of buried multiple scatterers by using an FDTD-based optimization technique

In this paper, a microwave imaging technique for reconstructing underground multiple scatterers is presented. The electromagnetic properties of buried objects are estimated by postprocessing total-field measurements, which are obtained when the domain of investigation is illuminated by wide-band electromagnetic waves. The solution of this limited-angle inverse scattering problem is based on the differential formulation of the direct problem. The scatterers are reconstructed by applying an iterative technique, which combines the finite-difference time-domain (FDTD) method and the Polak-Ribiere optimization algorithm. An augmented cost functional is defined taking into account the fulfillment of the Maxwell's curl equations by means of Lagrange multipliers. The Frechet derivatives of the functional with respect to the scatterer properties are derived from the stationary condition. Moreover, it is proven that the Lagrange multipliers fulfill the Maxwell's curl equations. In numerical results, the presented technique is applied to the reconstruction of scatterers buried in earth. In general, these scatterers can be dielectric, lossy, or magnetic.

A new Ka-band microstrip Y-junction circulator with a ferrite sphere

A novel microstrip circulator with a magnetized ferrite sphere is proposed for millimetre wave communications. A three-dimensional finite difference time-domain (FDTD) approach for the analysis of this ferrite sphere-based microstrip circulator is presented. The electromagnetic fields inside the ferrite junction are calculated by using special updating equations derived from the equation of motion of the magnetization vector and Maxwell's curl equations in consistency. Frequency-dependent insertion loss, isolation and reflection loss of circulator are calculated over a wide band of frequencies with a single FDTD run. Experimental results at the Ka-band are presented and compared with theoretical simulation. Good agreement is found between them.

TIME-DOMAIN INVERSE SCATTERING USING LAGRANGE MULTIPLIERS: AN ITERATIVE FDTD-BASED OPTIMIZATION TECHNIQUE

In this paper, we present an electromagnetic inverse scattering technique in the time domain, which is based on the minimization of an augmented cost functional. This functional is composed of an error term representing the discrepancy between the measured and the estimated values of the electromagnetic field, a regularization term related to the first-order Tikhonov's regularization scheme and, finally, an equality-constraints term associated with the fulfillment of the Maxwell's curl equations. These equality constraints are introduced by applying Lagrange multipliers. The minimization of the functional is carried out by applying the Polak-Ribière nonlinear conjugate-gradient algorithm. The required Fr´echet derivatives of the augmented cost functional with respect to the functions that describe the scatterer properties are derived by an analysis based on the calculus of variations. Actually, it is proven that the Lagrange multipliers are waves satisfying the Maxwell's curl equations. Consequently, during each iteration of the minimization procedure, we apply the finite-difference time-domain method and the perfectly-matched-layer absorber to calculate both the electromagnetic fields and the Lagrange multipliers. The presented technique is successfully applied to the reconstruction of multiple two-dimensional scatterers, which can be dielectric, lossy, and magnetic.

Wave modes in a magnetoplasma with anisotropic perturbation pressure

The momentum balance equation and the balance equation for the pressure tensor of a collision-free one-species magnetoplasma are linearized and Fourier-transformed, together with Maxwell's curl equations. They are combined in an algebraic dispersion system for the average perturbation velocity of the plasma particles. To do this a set of five fourth-order projectors is introduced, constructed with the aid of a commutative ‘ring product’ of two second-order projectors. It is shown that the trace-free part of the anisotropic perturbation pressure tensor, usually ignored for the sake of simplicity in such analyses, is of the same order of magnitude as the trace of this tensor. The vanishing determinant of the dispersion system is the dispersion equation of fifth degree in the square of the refractive index.

Development of the Microstrip Circulator with a Magnetized Ferrite Sphere in Millimeter Wave Band

In this paper, a novel microstrip circulator with a magnetized ferrite sphere is proposed for various millimeter wave communications. A three-dimensional Finite-Difference Time-Domain (FDTD) approach for the analysis of this ferrite sphere based microstrip circulator is first presented. The electromagnetic fields inside the ferrite junction are calculated using special updating equations derived from the equation of motion of the magnetization vector and Maxwell's curl equations in consistency. Frequency dependent insertion loss, isolation and reflection loss of circulator are calculated over a wide band of frequencies with a single FDTD run. Experimental results at Ka band are presented and compared with theoretical simulations. As a result, a good agreement is found between them.

Coupled Maxwell-pseudospin equations for investigation of self-induced transparency effects in a degenerate three-level quantum system in two dimensions: Finite-difference time-domain study

We extend to more than one spatial dimension the semiclassical full-wave vector Maxwell-Bloch equations for the purpose of achieving an adequate and rigorous description of ultrashort pulse propagation in optical waveguides containing resonant nonlinearities. Our considerations are based on the generalized pseudospin formalism introduced by Hioe and Eberly [Phys. Rev. Lett. 47, 838 (1981)] for treatment of the resonant coherent interactions of ultrashort light pulses with discrete-multilevel systems. A self-consistent set of coupled curl Maxwell-pseudospin equations in two spatial dimensions and time for the special case of a degenerate three-level system of quantum absorbers is originally derived. Maxwell's curl equations are considered to be coupled via macroscopic medium polarization to the three-level atom model for the resonant medium. Two distinct sets of pseudospin equations are obtained corresponding to the TE- and TM-polarized optical waves. For the case of TM polarization, the electromagnetic wave is polarized in a general direction in the plane of incidence inducing two dipole transitions in a degenerate three-level system by each E-field component along the propagation axis and in transverse direction. We introduce a dipole-coupling interaction Hamiltonian allowing Rabi flopping of the population difference along and perpendicular to the propagation axis with frequencies depending on the corresponding field components. The relationship between the induced polarization and the state vector components that describe the evolution of the discrete-level system is derived in order to couple the quantum system equations to the Maxwell's curl equations. The pseudospin equations are phenomenologically extended to include relaxation effects by introducing nonuniform decay times corresponding to the various dipole transitions occurring in a three-level system. The system has been discretized using finite differences on a Yee grid and solved numerically by an iterative predictor-corrector finite-difference time-domain method. Self-induced transparency soliton propagation through a degenerate three-level quantum system of absorbers in two spatial dimensions and time is demonstrated in planar parallel-mirror waveguide geometries.

Finite element‐based model order reduction of electromagnetic devices

AbstractIn this paper an efficient algorithm is presented for the development of compact and passive macro‐models of electromagnetic devices through the systematic reduction of the order of discrete models for these devices obtained through the use of finite elements. The proposed methodology is founded on a new finite element formulation that casts Maxwell's curl equations in a state‐space form. Such state‐space representations are very compatible with a class of robust model order reduction techniques based on Krylov subspaces. However, the advantage of this compatibility appears to be hindered by the fact that the state matrix of the discretized Maxwellian system is of dimension almost twice that obtained from the finite element approximation of the electromagnetic vector wave equation. It is shown in this paper that the apparent penalty in both memory and computation efficiency can be avoided by a proper selection of the finite element expansion functions used for the discretization of the electromagnetic fields. More specifically, it is shown that the proper selection of expansion functions renders the state‐space form of the Maxwellian system equivalent to the discrete problem obtained from the approximation of the vector wave equation using tangentially continuous vector finite elements. This equivalence is then used to effect Krylov‐based model order reduction directly from the finite element approximation of the vector wave equation. In particular, a passive model order reduction algorithm is used for this purpose. The proposed reduced order macro‐modelling algorithm is demonstrated through its application to a variety of microwave passive components. Copyright © 2002 John Wiley & Sons, Ltd.

Time-Domain Analysis of Millimeter Wave Circulation Around a Magnetised Ferrite Sphere

An efficient numerical method has been devised for the study of wave circulated by a magnetised ferrite sphere. It is a finite-difference time-domain formulation that incorporates Mur's absorbing boundary conditions and a perfectly matched layer. The electromagnetic fields inside the ferrite body are calculated using special updating equations derived from the equation of motion of the magnetization vector and Maxwell's curl equations in consistency. The electromagnetic fields inside ferrite and the power-density distribution on the ferrite's surface normal to the bias external magnetic field are obtained in a wide frequency band with a single time domain run. It is observed that an incident plane wave would circulate around the magnetised ferrite body in an open space as if the ferrite were placed inside a waveguide / microstrip junction circulators.

A new finite element model for reduced order electromagnetic modeling

This paper introduces a new formulation suitable for direct model order reduction of finite element approximations of electromagnetic systems using Krylov subspace methods. The proposed formulation utilizes a finite element model of Maxwell's curl equations to generate a state-space representation of the electromagnetic system most suitable for the implementation of model order reduction techniques based on Krylov subspaces. It is shown that, with a proper selection of the finite element interpolation functions for the fields, the proposed formulation is equivalent to the commonly used approximation of the vector wave equation with tangentially continuous vector finite elements. This equivalence is exploited to improve the computational efficiency of the model order reduction process.

Three-dimensional analysis of light propagation through uncoated near-field fibre probes.

The near-field emission from uncoated tapered fibre probes is investigated for different probe geometries. The three-dimensional model calculations are based on Maxwell's curl equations and describe the propagation of a 10 fs optical pulse (lambda = 805 nm) through tapers of different lengths and different diameters of the taper exit. The numerical evaluation is done with a finite difference time domain code. Two tapers with cone angles of 50 degrees, with taper lengths of 1.5 microm and 1.0 microm and exit diameters of 100 nm and 520 nm, respectively, are considered. We find that without sample the short taper with large exit diameter optimizes both light transmission and spatial resolution. In the presence of a sample with a high dielectric constant, however, the spatial near-field distribution changes drastically for both taper geometries. We find a pronounced increase in spatial resolution, down to about 250 nm inside the medium. This collimation of the near-field distribution arises from interferences between emitted and reflected light from the sample surface and from a collimation effect that the field experiences in the high-index semiconductor material. The combination of high spatial resolution and transmission and collection efficiencies makes such probes interesting for spectroscopic investigations, as demonstrated by recent experiments.

Effective permittivities for second-order accurate FDTD equations at dielectric interfaces

In Yee's finite-difference time-domain (FDTD) scheme, effective permittivities are often used to account for offsets of dielectric interfaces from grid nodes. The specific values of these effective permittivities must be chosen in such a way that the second-order accuracy of the scheme is preserved. It is shown in this work that, contrary to more elaborate techniques proposed recently for the development of these effective permittivities, a rigorous application of the integral forms of Maxwell's curl equations on the Yee's lattice leads to the desired values in a straightforward fashion. Numerical experiments in a two-dimensional (2-D) cavity are used to verify that the calculated effective permittivities preserve the second-order accuracy of the FDTD scheme.