Three-dimensional Maxwell Equations Research Articles

A preconditioned dual–primal finite element tearing and interconnecting method for solving three-dimensional time-harmonic Maxwell's equations

A new preconditioned dual–primal nonoverlapping domain decomposition method is proposed for the finite element solution of three-dimensional large-scale electromagnetic problems. With the aid of two Lagrange multipliers, the new method converts the original volumetric problem to a surface problem by using a higher-order transmission condition at the subdomain interfaces to significantly improve the convergence of the iterative solution of the global interface equation. Similar to the previous version, a global coarse problem related to the degrees of freedom at the subdomain corner edges is formulated to propagate the residual error to the whole computational domain at each iteration, which further increases the rate of convergence. In addition, a fully algebraic preconditioner based on matrix splitting is constructed to make the proposed domain decomposition method even more robust and scalable. Perfectly matched layers (PMLs) are considered for the boundary truncation when solving open-region problems. The influence of the PML truncation on the convergence performance is investigated by examining the convergence of the transmission condition for an interface inside the PML. Numerical examples including wave propagation and antenna radiation problems truncated with PMLs are presented to demonstrate the validity and the capability of this method.

Electromagnetic momentum and the energy–momentum tensor in a linear medium with magnetic and dielectric properties

In a continuum setting, the energy–momentum tensor embodies the relations between conservation of energy, conservation of linear momentum, and conservation of angular momentum. The well-defined total energy and the well-defined total momentum in a thermodynamically closed system with complete equations of motion are used to construct the total energy–momentum tensor for a stationary simple linear material with both magnetic and dielectric properties illuminated by a quasimonochromatic pulse of light through a gradient-index antireflection coating. The perplexing issues surrounding the Abraham and Minkowski momentums are bypassed by working entirely with conservation principles, the total energy, and the total momentum. We derive electromagnetic continuity equations and equations of motion for the macroscopic fields based on the material four-divergence of the traceless, symmetric total energy–momentum tensor. We identify contradictions between the macroscopic Maxwell equations and the continuum form of the conservation principles. We resolve the contradictions, which are the actual fundamental issues underlying the Abraham–Minkowski controversy, by constructing a unified version of continuum electrodynamics that is based on establishing consistency between the three-dimensional Maxwell equations for macroscopic fields, the electromagnetic continuity equations, the four-divergence of the total energy–momentum tensor, and a four-dimensional tensor formulation of electrodynamics for macroscopic fields in a simple linear medium.

A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise

In this paper we investigate a stochastic multi-symplectic method for stochastic Maxwell equations with additive noise. Based on the stochastic version of variational principle, we find a way to obtain the stochastic multi-symplectic structure of three-dimensional (3-D) stochastic Maxwell equations with additive noise. We propose a stochastic multi-symplectic scheme and show that it preserves the stochastic multi-symplectic conservation law and the local and global stochastic energy dissipative properties, which the equations themselves possess. Numerical experiments are performed to verify the numerical behaviors of the stochastic multi-symplectic scheme.

Inverse diffraction grating of Maxwell’s equations in biperiodic structures

Consider a time-harmonic electromagnetic plane wave incident on a perfectly conducting biperiodic surface (crossed grating). The diffraction is modeled as a boundary value problem for the three-dimensional Maxwell equation. The surface is assumed to be a small and smooth deformation of a planar surface. In this paper, a novel approach is developed to solve the inverse diffraction grating problem in the near-field regime, which is to reconstruct the surface with resolution beyond Rayleigh's criterion. The method requires only a single incident field with one polarization, one frequency, and one incident direction, and is realized by using the fast Fourier transform. Numerical results show that the method is simple, efficient, and stable to reconstruct biperiodic surfaces with subwavelength resolution.

Homogenization of Quasi-static Maxwell's Equations

This paper studies the homogenization of quasi-static and nonlinear Maxwell's equations in grain-oriented (GO) silicon steel laminations. GO silicon steel laminations have multiple scales, and the ratio of the largest scale to the smallest scale can be up to $10^6$. Direct solution of three-dimensional nonlinear Maxwell's equations is very challenging and unrealistic for large electromagnetic devices. Based on the magnetic vector potential and the magnetic field, respectively, we propose two macroscale models for the quasi-static Maxwell's equations. We prove that microscale solutions converge to the solutions of the macroscale models weakly in $\boldsymbol{H}(\mathbf{curl},\Omega)$ and strongly in $\boldsymbol{L}^2(\Omega)$ as the thickness of lamination tends to zero. The well-posedness of the homogenized model is established by using weighted norms. Numerical experiments are carried out for a benchmark problem from the International Compumag Society, TEAM Workshop Problem $21^c$-M1 [Z. Cheng, N. Takaha...

Modelling of thin-film silicon solar cells

Computer modelling has become increasingly important in the design and optimization of surface textures that are used in thin-film solar cells to manipulate scattering of light. Light scattering at rough interfaces together with efficient back reflector is a standard approach to enhance absorption of light in the absorber layers of thin-film silicon solar cells. Substrates with randomly textured surface are commonly used to introduce rough interfaces into solar cells. Scalar scattering theory was used to describe light scattering at random nano-textured interfaces and to optimize random texture in single junction solar cells. Recently, substrates with periodic surface features have been investigated as an alternative to randomly surface-textured substrates. Three-dimensional Maxwell equation solvers were used to analyse and optimize such periodic textures when embedded in single and multi-junction solar cells. In both random and periodic cases, opto-electrical modelling was employed to assess the spectral response, to evaluate optical losses and to simulate current density–voltage characteristic.

Unconditionally Stable Fundamental LOD-FDTD Method With Second-Order Temporal Accuracy and Complying Divergence

An unconditionally stable fundamental locally one-dimensional (LOD) finite-difference time-domain (FDTD) method with second-order temporal accuracy and complying divergence (CD) (denoted as LOD2-CD-FDTD) is presented for three-dimensional (3-D) Maxwell's equations. While the conventional LOD-FDTD method does not have complying divergence, the LOD2-CD-FDTD method has complying divergence in a manner analogous to the conventional explicit FDTD method. The update procedures for a family of LOD-FDTD methods that employ similar splitting matrix operators are presented. By extending the previous concept of achieving second-order temporal accuracy for the LOD2-FDTD method via implicit output processing, we hereby propose novel, explicit output processing that not only retains second-order temporal accuracy, but also complying divergence for the LOD2-CD-FDTD method. The current source implementation for the LOD2-CD-FDTD method involves source-incorporation in only the first procedure. To further enhance efficiency, the LOD2-CD-FDTD method is formulated into the fundamental LOD2-CD-FDTD method with efficient matrix-operator-free right-hand sides. Subsequently, detailed implementation for the fundamental LOD2-CD-FDTD method is presented. Analytical proof is provided to ascertain the second-order temporal accuracy of the LOD2-CD-FDTD method. Numerical results and examples are also presented to validate the divergence-complying property of the LOD2-CD-FDTD method.

Analytic theory of fiber-optic Raman polarizers

The Raman polarizer is a Raman amplifier which not only amplifies but also repolarizes light. We propose a relatively simple and analytically tractable model - the ideal Raman polarizer, for describing the operation of this device. The model efficiently determines key device parameters such as the degree of polarization, the alignment parameter, the gain and the RIN variance.

Bidirectional scatter measurements of a guided-mode resonant filter photonic crystal structure

This work investigates the Bidirectional Scatter Distribution Function (BSDF) at incident angles other than normal and at 544-nm wavelength of a Guided Mode Resonance Filter (GMRF) Photonic Crystal (PC) structure designed for normally incident light at 532 nm. Strong out-coupling of PC diffraction orders into both the transmitted and reflected hemispheres was observed specifically at a 25.7° incidence angle, which we attribute to this incident angle/wavelength pair being a good match to the ( ± 1, 0) PC grating mode. BSDF measurements at incident angles of 15° and 35° also displayed some out-coupled diffraction, though much lower in magnitude, and are also attributed to being a weaker match to the ( ± 1, 0) PC grating mode. Three-dimensional finite-difference time-domain Maxwell's equation simulations demonstrate that since this GMRF was designed for complete destructive interference of the transmitted light upon normal incidence, stronger out-coupling of the diffraction is expected for modal solutions as the angle of incidence increases.

Numerical modeling of the propagation of electromagnetic radiation in 3D photonic crystals

We describe the mathematical model and program BAL 3D that enables one to solve numerically three-dimensional Maxwell equations in inhomogeneous media. We present the results of calculations of the electromagnetic-radiation propagation in a 3D quasiperiodic medium (photonic crystal). We found the positions of stop bands in photonic crystals and calculate the reflection coefficients for composite materials consisting of artificial opals with pores filled with water. We compare our results with the experimental data.

Near-Field Diffraction from a Binary Microaxicon

We study binary axicons of period 4, 6, and 8 μm fabricated by photolithography with a 1 μm resolution, 500 nm depth, and 4 mm diameter. Near-field diffraction focal spots varying in diameter from 3.5λ to 4.5λ (for the axicon of period T=4 μm) and from 5λ to 8λ (for the axicon with T=8 μm) are experimentally found on the optical axis at a distance of up to 40 μm from the axicon for the wavelength λ=0.532 μm. The first focal spot is found at distance 2 μm (T=4 μm), with the period of the focal spots being 2 μm (T=4 μm) and 4 μm (T=8 μm). Diffraction of linearly polarized plane and diverging waves is simulated using FullWAVE (RSoft) and a proprietary program BOR-FDTD, which implement finite-difference schemes to solve three-dimensional Maxwell's equations in the Cartesian and cylindrical coordinates. The numerically simulated values for diameters of the near-field focal spots for the axicon of period T=4 μm are in good agreement with the experimental values.

On the Air–Sea Boundary in Transient Marine CSEM Detection Modeling of Subseafloor Hydrocarbon Reservoirs

The published scientific literature provides extensive results and discussion of approximate finite-difference time-domain (FDTD) computational modeling of marine controlled-source electromagnetics (CSEM) detection of hydrocarbon reservoirs buried under the seafloor. “Approximate” here refers to the neglection of displacement currents in the calculation to reduce the computational burden of the simulation. This leads to the widely used approximate continuation boundary conditions at the ocean-air interface to avoid the free-space region in the simulation where inclusion of displacement currents is required. However, an analysis of when the use of such continuation boundary conditions sufficiently reduces the accuracy of the calculated results is lacking in the published literature. This letter addresses this issue and reports the application of the complete and standard, three-dimensional full-vector Maxwell's equations FDTD method to modeling CSEM hydrocarbon detection. We provide accurate results for shallow and deep-water CSEM problems and determine that the continuation boundary condition is inadequate at large (>; 4.5 km) source-to-receiver distances in deep-water detection problems and at all distances in shallow-water problems.

Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations

We consider a standard Adaptive Edge Finite Element Method (AEFEM) based on arbitrary order Nedelec edge elements, for three-dimensional indefinite time-harmonic Maxwell equations. We prove that the AEFEM gives a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops provided the initial mesh is fine enough. Using the geometric decay, we show that the AEFEM yields the best-possible decay rate of the error plus oscillation in terms of the number of degrees of freedom. The main technical contribution of the paper is in the establishment of a quasi-orthogonality and a localized a posteriori error estimator.

On the possibility of high-level transient coronal mass ejection-induced ionospheric current coupling to electric power grids

[1] Short-time (<10 s) ionospheric current fluctuations are hypothesized to occur due to a coronal mass ejection that can induce high-level transient voltages on long overhead power transmission lines. This hypothesis is first supported by reviewing published literature and recent magnetometer measurements, both of which indicate the existence of such rapid ionospheric current fluctuations. Then results are reported of a new full-vector three-dimensional Maxwell's equations finite-difference time-domain (FDTD) model of the global Earth-ionosphere system during a solar storm. This model naturally treats transient electromagnetic fields and waves unlike previous quasi-DC steady state or sinusoidal steady state analyses. Furthermore, this model accounts for the geometrical and electrical properties of the entire global Earth-ionosphere system, including details of the global topography and bathymetry, rather than focusing upon a particular continental region as for previous analyses. The FDTD modeling results provide new and useful information, such as that the lithosphere conductivity has almost no impact on the induced transient (short timescale) surface geoelectric fields resulting from capacitive or electric field coupling mechanisms. This is counter to the previously analyzed magnetic field coupling mechanisms involving the (long timescale) induction of geomagnetically induced currents, which heavily depend on the lithosphere conductivity structure. The modeling results presented herein are sufficiently significant to warrant measurements of rapid ionospheric current fluctuations at timescales much shorter than considered at present, even less than 1 s.

A domain decomposition method for the parallelization of a three-dimensional Maxwell solver based on a constrained formulation

Abstract The numerical solution of very large three-dimensional electromagnetic field problems are challenging for various applications in the industry. In this paper, we propose a nonoverlapping domain decomposition approach for solving the three-dimensional Maxwell equations on MIMD computers, based on a mixed variational formulation. It is especially well adapted for the solution of the Vlasov–Maxwell equations, widely used to simulate complex devices like particle injectors or accelerators. This approach has the important property that it leads to reuse without modification most of an existing sequential code. The continuity at the interfaces is imposed by duality using Lagrange multipliers. Hence, the resulting parallel algorithm requires only to add an external preconditioned Uzawa solver. We present the results of some numerical experiments on a parallel distributed memory machine.

Retrieval of Electron Return Time from High-order Harmonics Generated in a Mixture of He and Ne Gases

In terms of single-atom induced dipole moment by Lewenstein model, we present the macroscopic high-order harmonic generation from mixed He and Ne gases with different mixture ratios by solving three-dimensional Maxwell's equation of harmonic held. And then we show the validity of mixture formulation by Wagner et al. [Phys. Rev. A 76 (2007) 061403(R)] in macroscopic response level. Finally, using least squares fitting we retrieve the electron return time of short trajectory by formulation in Kanai et al. [Phys. Rev. Lett. 98 (2007) 153904] when the gas jet is put after the laser focus.

Destructive and Constructive Interference of High-Order Harmonic Generation in Mixture of He and Ne Gases

Using the single-atom induced dipole moment under strong field approximation as a source, we suggest a model to simulate the macroscopic high-order harmonic generation (HHG) from the mixed gases (He and Ne) interacting with intense infrared laser by solving the three-dimensional Maxwell's equation of the harmonic field. Regular destructive interference (DI) and constructive interference (CI) are observed in the macroscopic HHG spectra when the gas jet is put at a good phase-matching position. A semiclassical model of short and long electron trajectories is applied to interpret the DI and CI of HHG qualitatively.

Multiscale Computations for 3D Time-Dependent Maxwell's Equations in Composite Materials

We discuss multiscale computations of the initial-boundary value problem for three-dimensional time-dependent Maxwell's equations in composite materials. A multiscale algorithm is developed, and the convergence results are presented. Numerical simulations are carried out to validate the theoretical results reported in the present paper.

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.

Reflectance of surfactant-templated mesoporous silica thin films: Simulations versus experiments

In this study, cubic and hexagonal mesoporous amorphous silica thin films were synthesized using evaporation-induced self-assembly process followed by calcination leaving highly ordered spherical or cylindrical pores in a silica matrix. The films featured pores with diameter between 4 and 11 nm, lattice parameter from 7.8 to 24 nm, and porosity between 22% and 45%. All films were dehydrated prior to reflectance measurements except for one film which was fully hydrated. The present study compares the spectral reflectance measured experimentally between 400 and 900 nm with that computed numerically by solving three-dimensional Maxwell's equations in mesoporous silica thin films with the same morphology as those synthesized. The matrix was assumed to have the same optical properties as bulk fused silica. The pore optical properties were either those of air or liquid water whether the film was dehydrated or hydrated, respectively. Excellent agreement was found between experimental and numerical reflectance for both cubic and hexagonal mesoporous silica films. This study experimentally validates our simulation tool and offers the prospect of ab-initio design of nanocomposite materials with arbitrary optical properties without using effective medium approximation or mixing rules.