Maxwell's Curl Equations Research Articles

Mimetic Finite Difference Methods for Maxwell's Equations and the Equations of Magnetic Diffusion - Abstract

We have constructed mimetic finite difference methods for both the TE and TM modes for 2-D Maxwell's curl equations and equations of magnetic diffusion with discontinuous coefficients on nonorthogonal, nonsmooth grids. The discrete operators were derived using the discrete vector and tensor analysis to satisfy discrete analogs of the main theorems of vector analysis. Because the finite difference methods satisfy these theorems, they do not have spurious solutions and the "divergence-free" conditions for Maxwell's equations are automatically satisfied. The tangential components of the electric field and the normal components of magnetic flux used in the FDM are continuous even across discontinuities. This choice guarantees that problems with strongly discontinuous coefficients are treated properly. Furthermore on rectangular grids the method reduces to the analytically correct averaging for discontinuous coefficients. We verify that the convergence rate was between first and second order on the arbitrary quadrilateral grids and demonstrate robustness of the method in numerical examples.

A wavelet formulation of the finite-difference method: full-vector analysis of optical waveguide junctions

We have developed an efficient, large-stencil finite-difference scheme of the time-dependent Maxwell's curl equations based on the wavelet-collocation formulation in the time-domain. The proposed scheme enables, for the first time within a limited computational resource, full-vector analysis of three-dimensional rib waveguides that are typically used in integrated planar optical devices. The formulation takes advantage of compactly-supported interpolating bases to expand and represent the electric and magnetic fields. Moreover, unlike the well-known beam propagation methods, the numerical scheme is based on the first-principle algorithm with no explicit approximation, and thus rigorous and versatile for various types of boundary conditions. We demonstrate the efficiency of the method by first analyzing a straight rib-waveguide and examining the convergence of the results. Then we investigate a Y-shaped junction structure that is electrically too large to analyze with the conventional finite-difference time-domain scheme.

Analysis of Optical Force and Optical Potential Well for Trapping Particles near a Light-Illuminated Slit Using a Finite-Difference Time-Domain Method

We analyze numerically the optical force and the optical potential well for trapping a circular-cylinder type object in a near zone of a light-illuminated metal slit. Our analysis is based on the finite-difference formulas of the optical force and the optical potential energy to be developed, a finite-difference solution of Maxwell's curl equations, and an absorbing boundary condition to define the problem in a finite region. We show that trapping forces induced by the field gradient produced in the vicinity of a slit can provide a two-dimensional potential well for trapping nanometric particles against a gravitational force. The results may be used to predict the properties of gradient-force traps in a more complicated system.

Electromagnetic plane–wave response characteristics of non–axially excited slabs of dielectric thin–film helicoidal bianisotropic mediums

Thin–film helicoidal bianisotropic mediums (TFHBMs) are rotationally non–homogeneous mediums whose microstructure consists of parallel helical columns. We characterize here the plane–wave response of non–axially excited dielectric TFHBM slabs. The constitutive relations for the dielectric TFHBMs are substituted into the time–harmonic Maxwell curl equations, and a 4 × 4 matrix ordinary differential equation is obtained. The piecewise homogeneity approximation method is shown to be reasonably robust to solve the differential equation. Bragg reflection by non–axially excited dielectric TFHBM slabs is investigated, and transmission characteristics are examined. The effects of several key illumination and constitutive parameters on the response of non–axially excited dielectric TFHBM slabs are studied to elucidate trends in functional relationships. A TFHBM bilayer is presented as an example of a device based on non–axially excited dielectric TFHBMs.

Generalized material models in TLM .I. Materials with frequency-dependent properties

This paper presents the fundamentals of a unified approach for the treatment of general material properties in time-domain simulation based on transmission-line modeling (TLM). Linear frequency-dependent isotropic materials are dealt with in the first instance. The iteration schemes for one-dimensional (1-D) and three-dimensional (3-D) models are developed from Maxwell's curl equations and the constitutive relations. Results are presented showing the accuracy of this approach.

Breakdown of the Area Theorem: Carrier-Wave Rabi Flopping of Femtosecond Optical Pulses

By solving Maxwell's curl equations coupled to a two-level atom, a theoretical study of carrier-wave Rabi flopping of femtosecond optical pulses of only several carrier-cycles time duration is reported. For pulse areas of $2\ensuremath{\pi}$, the usual self-induced transparency regime is essentially recovered. However, for larger pulse areas, carrier-wave Rabi flopping occurs that manifests in local carrier reshaping and subsequently to the production of significantly higher spectral components on the propagating pulse. These new features are not predicted by employing the area theorem or a slowly-varying-envelope approximation for the amplitude and phase terms---which is the usual approach.

Finite-difference time-domain simulation of GPR data

Simulation of digital ground penetrating radar (GPR) wave propagation in two-dimensional (2-D) media is developed, tested, implemented, and applied using a time-domain staggered-grid finite-difference (FD) numerical method. Three types of numerical algorithms for constructing synthetic common-shot, constant-offset radar profiles based on an actual transmitter-to-receiver configuration and based on the exploding reflector concept are demonstrated to mimic different types of radar survey geometries. Frequency-dependent attenuation is also incorporated to account for amplitude decay and time shift in the recorded responses. The algorithms are based on an explicit FD solution to Maxwell's curl equations. In addition, the first-order TE mode responses of wave propagation phenomena are considered due to the operating frequency of current GPR instruments. The staggered-grid technique is used to sample the fields and approximate the spatial derivatives with fourth-order FDs. The temporal derivatives are approximated by an explicit second-order difference time-marching scheme. By combining paraxial approximation of the one-way wave equation ( A 2) and the damping mechanisms (sponge filter), we propose a new composite absorbing boundary conditions (ABC) algorithm that effectively absorb both incoming and outgoing waves. To overcome the angle- and frequency-dependent characteristic of the absorbing behaviors, each ABC has two types of absorption mechanism. The first ABC uses a modified Clayton and Enquist's A 2 condition. Moreover, a fixed and a floating A 2 ABC that operates at one grid point is proposed. The second ABC uses a damping mechanism. By superimposing artificial damping and by alternating the physical attenuation properties and impedance contrast of the media within the absorbing region, those waves impinging on the boundary can be effectively attenuated and can prevent waves from reflecting back into the grid. The frequency-dependent characteristic of the damping mechanism can be used to adjust the width of the absorbing zone around the computational domain. By applying any combination of absorbing mechanism, non-physical reflections from the computation domain boundary can be effectively minimized. The algorithm enables us to use very thin absorbing boundaries. The model can be parameterized through velocity, relative electrical permittivity (dielectric constants), electrical conductivity, magnetic permeability, loss tangent, Q values, and attenuation. According to this scheme, widely varying electrical properties of near-surface earth materials can be modeled. The capability of simulating common-source, constant-offset and zero-offset gathers is also demonstrated through various synthetic examples. The synthetic cases for typical GPR applications include buried objects such as pipes of different materials, AVO analysis for ground water exploration, archaeological site investigation, and stratigraphy studies. The algorithms are also applied to iterative modeling of GPR data acquired over a gymnasium construction site on the NCCU campus.

Mixed finite-difference/Whitney-elements time domain (FD/WE-TD) method

A mixed finite-difference/Whitney-elements time-domain (FD/WE-TD) method is proposed for the analysis of transient electromagnetic field problems. The method consists in discretizing the spatial region in two parts composed respectively by structured and unstructured elements. In the structured mesh Maxwell's curl equations are numerically solved by Yee's algorithm while in the unstructured mesh the wave equation is solved in terms of electric field by the WETD method. The proposed method leads to an explicit-implicit solution scheme which is convenient to model curved boundaries and complex configurations without a significant increase of the computational cost with respect to the basic FDTD method. The FD/WE-TD method is a valid alternative of the subgridding algorithms and nonuniform grid models used in the FDTD calculations. The nonphysical wave reflection introduced by the finite elements is calculated for simple canonical configurations.

Parallel processing for the simulation of problems involving scattering of electromagnetic waves

The three-dimensional simulation of the scattering of electromagnetic waves by obstacles of arbitrary geometrical shape is considered. Maxwell's curl equations are solved in the time domain on an unstructured mesh of linear tetrahedral elements. The discrete equations are formed by a Galerkin procedure, with a Lax-Wendroff consistent numerical flux function and explicit time stepping. Parallelisation of the procedure is achieved by mesh decomposition and the use of a single program multiple data model. The approach is first validated and is then applied to problems involving the scattering of plane electromagnetic waves by a complete aircraft.

Reciprocity for electromagnetic–acoustic fields in compressible magnetoplasmas with anisotropic pressure

Maxwell's curl equations and the linearized balance equations for momentum and pressure tensor for a plasma in an external magnetic field are combined in a system of first-order partial differential equations for the field quantities. With the corresponding adjoint system, the Lagrange identity is established, leading to a symmetry relation between the Green functions for the original and the adjoint fields. With temporal mapping the adjoint field can be transformed into a ‘Lorentz-adjoint’ (physical) field, leading to a reciprocity relation between the Green functions for the original and for the Lorentz-adjoint fields.

Analysis of singly and doubly periodic absorbers by frequency-domain finite-difference method

In this paper, the theory and numerical algorithms of a periodic finite-difference frequency domain (PFDFD) method are presented for the analysis of arbitrarily shaped inhomogeneously filled, singly and doubly periodic absorbers. The PFDFD method is based on a finite-difference solution of the Maxwell's curl equations plus an absorbing boundary condition, metallic backing, and Floquet's periodic condition to define the problem in a finite region. Compared to the integral equation (IE) and moment method techniques, the PFDFD algorithm is simpler to develop, more efficient to model electrically large geometries, and more flexible to analyze absorbers with both electric and magnetic loadings. The theory and algorithms proposed in the paper have been validated via numerous examples and agree very well with those of other techniques and measurements.

Divergence Preserving Discrete Surface Integral Methods for Maxwell's Curl Equations Using Non-orthogonal Unstructured Grids

Several new discrete surface integral methods for solving Maxwell's equations in the time-domain are presented. These methods, which allow the use of general non-orthogonal mixed-polyhedral unstructured grids, are direct generalizations of the canonical staggered-grid finite difference method. These methods are conservative in that they locally preserve "divergence" or charge. Employing mixed polyhedral cells (hexahedral, tetrahedral, etc.), these methods allow more accurate modeling of non-rectangular structures and objects because the traditional "stair stepped" boundary approximations associated with the orthogonal grid based finite difference methods can be avoided. Numerical results demonstrating the accuracy of these new methods are presented.

Hybrid finite element solutions of time dependent Maxwell's curl equations

Hybrid finite element (FE) solutions of transient electromagnetic field are proposed. Time dependent Maxwell's curl equations are solved separately using edge elements for the electric field approximation and nodal-based elements for the magnetic field approximation. Two different formulations are presented: the first adopts a unique mesh, the second two complementary FE meshes. The solution of the resulting equations systems is performed by explicit methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Field-to-wire coupling using the finite element-time domain (FE-TD) method

A new numerical method is proposed to analyze transient induced effects in thin wires excited by external fields. Maxwell's curl equations and transmission line equations are solved by the point-matching finite element-time domain (FE-TD) method. The method is based on a finite element discretization of space and a finite difference discretization of time. The leap-frog scheme is adopted and an explicit solution scheme is obtained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

FD-TD analysis of Vivaldi flared horn antennas and arrays

This paper summarizes a detailed computational study of Vivaldi flared horn antenna designs including single element, double element crossed-pair subarray elements, and linear arrays using the finite-difference time-domain (FD-TD) method. FD-TD, which numerically solves Maxwell's time dependent curl equations, accounts for the complex geometrical and electrical characteristics associated with this antenna design and array implementation. Validations have been made relative to a moment method (MM) model of an electrically large linearly tapered slot antenna. Also, good correlation is shown to exist in the primary features of the antenna patterns between computed and measured data for all cases. This work has achieved a number firsts for the FD-TD methodology. It represents the first exact computational model of a single quad Vivaldi slot antenna; the first exact computational model of a phased array of Vivaldi quad elements; and the first FD-TD model to demonstrate grating lobes for a phased array antenna of any sort. Lastly, this research represents an extensive study of the largest grid-based antenna models conducted to date. >

On Two Numerical Techniques for Light Scattering by Dielectric Agglomerated Structures.

Smoke agglomerates are made of many soot sphcres, and their light scattering response is of interest in fire research. The numerical techniques chiefly used for theoretical scattering studies are the method of moments and the coupled dipole moment. The two methods have been obtained in this tutorial paper directly from the monochromatic Maxwell curl equations and shown to be equivalent. The effects of the finite size of the primary spheres have been numerically delineated.

Maxwell's Equations in the 21st Century

Maxwell's equations in the 21st Century The finite-difference time-domain method is a novel method for solving Maxwell's curl equations, especially when parallel-processing techniques are applied. The next generation of computers will bring a revolution by exploiting the use of parallel processing in computation to the maximum.

FDTD analysis of magnetized ferrites: an approach based on the rotated Richtmyer difference scheme

Electromagnetic wave propagation in magnetized ferrites is modeled by solving Maxwell's time-dependent curl equations coupled with the equation of motion of the magnetization vector. A discretization approach based on the rotated Richtmyer finite-difference scheme is proposed. The new approach has been used to calculate the phase constants of transversally magnetized ferrite-loaded waveguides. The numerical dispersion equation for TE/sub n0/ modes is derived. The results obtained with this approach for a ferrite-filled and a ferrite-slab loaded waveguide are compared with those obtained with Yee's scheme extended for the treatment of ferrites and with the exact results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Combined tangential-normal vector elements for computing electric and magnetic fields

A direct method for computing electric and magnetic fields in two dimensions is developed. This method determines both the fields and fluxes directly from Maxwell's curl and divergence equations without introducing potential functions. This allows both the curl and the divergence of the field to be set independently in all elements. The technique is based on a novel type of vector finite element that simultaneously interpolates to the tangential component of the electric or magnetic flux. Continuity conditions are imposed across element edges. This guarantees the continuity of the field and flux at the mid-point of each edge, and its also guarantees that for all edges the average values of the tangential component of the field and of the normal component of the flux are identical.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A class of symmetrical condensed node TLM methods derived directly from Maxwell's equations

A series of general transmission line matrix (TLM)-type methods, which include the symmetrical condensed node method, are derived directly from Maxwell's curl equations without recourse to transmission line models. Written as a system of conservation laws, Maxwell's equations in 3-D plus time are decomposed along the orthogonal characteristic directions of a rectangular grid. The Riemann invariants in this method correspond to the voltage pulses of the TLM method. A new method of handling inhomogeneous media is proposed based on a new transfer event. The dispersive nature of these schemes is also investigated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>