Two-dimensional Maxwell's Equations Research Articles

Electromagnetic wave propagation in polarizable wet media: application to a TDR probe

The time domain reflectometry (TDR) probe is a technique applied to soil moisture measurement. This probe is a waveguide composed of two or three parallel metallic rods inserted in the ground, between which one applies a step function of potential. The main objective of this paper is to contribute to the interpretation of the signals obtained at the entrance of the probe. The theoretical approach is formulated using two-dimensional Maxwell equations and we propose to take into account the transient polarization of the medium. We have simulated the waveguide in this medium, using the finite element method.

Multigrid ADI Method for Two-Dimensional Electromagnetic Simulations

We propose a multigrid alternating-direction implicit (ADI) method for solving two-dimensional Maxwell's equations. This method is based on interpreting the ADI method as an iterative solver for the Crank-Nicolson (CN) scheme. By introducing a special procedure to solve the residual equation within the ADI framework, multigrid methods are incorporated into the iterative ADI method. The accuracy and efficiency of the proposed method are further demonstrated by numerical examples.

Effective optical properties of non-absorbing nanoporous thin films

Numerous effective medium models have been proposed for the effective optical properties of nanoporous media. However, validations of these models against experimental data are often contradictory and inconclusive. This issue was numerically investigated by solving the two-dimensional Maxwell's equations in non-absorbing nanoporous thin films with various morphologies. It was found that below a certain critical film thickness, the effective index of refraction depends on the porosity and on the pore size, shape and spatial distribution. For thick enough films the effective index of refraction depends solely on porosity and on the indices of refraction of the two constitutive phases. The numerical results agree very well with a recent model obtained by applying the Volume Averaging Theory to the Maxwell's equations. However, commonly used models systematically and sometimes significantly underpredict the numerical results.

Perfectly matched layers for Maxwell’s equations in second order formulation

We consider the two-dimensional Maxwell's equations in domains external to perfectly conducting objects of complex shape. The equations are discretized using a node-centered finite-difference scheme on a Cartesian grid and the boundary condition are discretized to second order accuracy employing an embedded technique which does not suffer from a ''small-cell'' time-step restriction in the explicit time-integration method. The computational domain is truncated by a perfectly matched layer (PML). We derive estimates for both the error due to reflections at the outer boundary of the PML, and due to discretizing the continuous PML equations. Using these estimates, we show how the parameters of the PML can be chosen to make the discrete solution of the PML equations converge to the solution of Maxwell's equations on the unbounded domain, as the grid size goes to zero. Several numerical examples are given.

Carleman estimate for Maxwell's equations in anisotropic media and the observability inequality

We establish a Carleman estimate for two-dimensional Maxwell's equations in isomagnetic anisotropic media and prove an energy estimate by boundary data.

Electrophysical and Thermal Parameters of Thermal Plasma in a High-Frequency Induction Discharge

A semiempirical model is proposed for the calculation of electrical, magnetic and thermal parameters of the plasma of a high-frequency induction discharge. The model is based on solving a system of two-dimensional Maxwell equations written for a cylindrical coordinate system.

Exact solution of Maxwell’s equations for optical interactions with a macroscopic random medium

We report what we believe to be the first rigorous numerical solution of the two-dimensional Maxwell equations for optical propagation within, and scattering by, a random medium of macroscopic dimensions. Our solution is based on the pseudospectral time-domain technique, which provides essentially exact results for electromagnetic field spatial modes sampled at the Nyquist rate or better. The results point toward the emerging feasibility of direct, exact Maxwell equations modeling of light propagation through many millimeters of biological tissues. More generally, our results have a wider implication: Namely, the study of electromagnetic wave propagation within random media is moving toward exact rather than approximate solutions of Maxwell's equations.

A Staggered Upwind Embedded Boundary (SUEB) Method to Eliminate the FDTD Staircasing Error

In spite of its flexibility and second-order accuracy in a homogeneous medium, Yee's finite-difference time-domain (FDTD) method suffers from serious degradation when treating material interfaces, greatly reducing its accuracy in the presence of inhomogeneous media and perfect conductors. Indeed, such so-called staircasing approximation may lead to local zeroth-order and global first-order errors. In this work, an embedded FDTD scheme, the staggered upwind embedded boundary (SUEB) method, is developed for the solution of oneand two-dimensional Maxwell's equations. This simple embedded technique uses upwind conditions in the FDTD method to correctly represent the location and physical conditions of material and metallic boundaries, hence eliminating problems caused by the staircasing approximation. Accuracy analysis has been made to show that the SUEB method maintains a second-order accuracy globally. Since the entire problem has been embedded into the simple staggered grid similar to that employed by the Yee's scheme, extra effort is only needed when treating the grid points close to the interfaces. Therefore, little additional computational cost is needed over Yee's scheme. The SUEB method has been validated by analytical solutions for plane wave normally incident to a planar boundary and for the TM wave propagation in the presence of a dielectric cylinder and a perfectly electrically conducting cylinder.

Numerical evaluation of AC losses in HTS wires with 2D FEM formulated by self magnetic field

The ac losses of high critical-temperature superconducting (HTS) wires are numerically calculated by means of a finite element method (FEM), which is formulated with a self magnetic field due to an induced current as unknown. The numerical model is straight HTS wires carrying an alternating transport current in an external ac magnetic field perpendicular to the wire axis. In this situation, the electromagnetic field around the wires is given by two-dimensional (2D) Maxwell's equations. It is also assumed that the transport property is represented by either the critical state model or the power-law model, in which the electric field is proportional to the power of the current density. The obtained losses are compared with conventional theoretical curves in several simple geometries.

Unconditionally stable Crank-Nicolson scheme for solving two-dimensional Maxwell's equations

The Crank-Nicolson method is an unconditionally stable, implicit numerical scheme with second-order accuracy in both time and space. When applied to solve Maxwell's equations in two-dimensions, the resulting matrix is block tri-diagonal, which is very expensive to solve. The Douglas-Gunn algorithm is used to subdivide the update procedure into two sub-steps. At each sub-step only a tri-diagonal matrix needs to be solved for one field component. The other two field components are updated explicitly in one step. The numerical dispersion relations are given for the original Crank-Nicolson scheme and for the Douglas-Gunn modification. The predicted numerical dispersion is shown to agree with numerical experiments, and its numerical anisotropy is shown to be much smaller than that of the ADI-FDTD.

Investigation of numerical errors of the two-dimensional ADI-FDTD method [for Maxwell's equations solution

We previously proposed the ADI-FDTD method as a means of solving two-dimensional Maxwell's equations. The algorithm used in this method is unconditionally stable, which means the time-step size ran be set arbitrarily when this method is used. The limitation of the time-step size is not dependent on the Courant-Friedrich-Levy (CFL) condition, but on numerical errors such as numerical dispersion. In this paper, we investigate the numerical errors of the method quantitatively and discuss the calculation accuracy and efficiency of the method. We found that a large time-step size results in high numerical dispersion. However, the limit of the time-step size due to numerical dispersion is greater than the CPL limit if the size of the local minimum cell in the computational domain is much smaller than the other cells and the wavelength. In that case, because the large time-step size reduces the number of time-loop iterations, the ADI-FDTD method is more efficient than the conventional FDTD method in terms of computer resources such as central processing unit time.

Discharge impedance of solenoidal inductively coupled plasma discharge.

The discharge impedance is calculated for a solenoidal inductively coupled plasma (ICP) discharge, which is one of the important sources for plasma processing. To calculate this impedance, the electromagnetic field quantities are obtained by solving the two-dimensional Maxwell equations in a realistic geometry. Also considered in the calculation is the anomalous skin effect which is regarded as a collisionless heating mechanism of ICP discharge. The results show that the discharge impedance is a function of various discharge parameters, such as plasma density, electron temperature, antenna position, collision frequency, excitation frequency, and chamber geometry.

Extremely low-precision integer cellular array algorithm for computational electromagnetics

We present an algorithm for computational electromagnetics using extremely low-precision integer variables (only a single or a few bits) operating on a cellular array topology. We show, through the enforcement of field and energy conservation rules, that lossless linear wave behavior can be simulated enabling solution of the two-dimensional Maxwell's equations. Since the algorithm operates using only a few bits per cell and does not require floating-point operations, it is ideally suited for implementation on a fine grain computing engine.

Numerical Stability Analysis of the Thompson-FDTD Method

The stability criterion for the numerical solutions of two-dimensional Maxwell's equations obtained by the method of difference-Thompson transformation (DTTR) combined with the finite difference time domain (FDTD) method is obtained for the first time.

Theory of nonequilibrium inductive high-frequency discharge in a gas flow

The global nonequilibrium flow in the discharge chamber of an induction plasma generator is modeled. The problem for an equilibrium discharge was considered in [2, 3]. Here, on the basis of a numerical solution of the combined system of Navier-Stokes, Maxwell, energy, ionization kinetics and electron-gas energy balance equations, the structure of the nonequilibrium discharge is analyzed and the results obtained within the framework of the local one-dimensional approach [1] and on the basis of global numerical modeling of the flow are compared. As distinct from [2, 3], in finding the electromagnetic field distribution in the discharge chamber the boundary-value problem for the two-dimensional Maxwell equations is solved.